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Introduction to EconometricsLecture 1: Review of Probability Theory & Introduction to
Causal Inference
Zhaopeng Qu
Business School,Nanjing University
Sep. 18, 2020
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 1 / 100
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Outlines
1 Review the Last Lecture
2 A Short Review of Probability Theory
3 Causal Inference in Social Science
4 Experimental Design as a Benchmark
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 2 / 100
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Review the Last Lecture
Review the Last Lecture
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 3 / 100
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Review the Last Lecture
The Last Lecture
A Scientific Framework of Making Rational ChoiceEconometrical Analysis plays a key role
What is EconometricsEconometrics and Big Data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 4 / 100
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Review the Last Lecture
The Last Lecture
A Scientific Framework of Making Rational ChoiceEconometrical Analysis plays a key role
What is EconometricsEconometrics and Big Data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 4 / 100
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Review the Last Lecture
The Last Lecture
A Scientific Framework of Making Rational ChoiceEconometrical Analysis plays a key role
What is EconometricsEconometrics and Big Data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 4 / 100
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Review the Last Lecture
The Last Lecture
A Scientific Framework of Making Rational ChoiceEconometrical Analysis plays a key role
What is EconometricsEconometrics and Big Data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 4 / 100
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Review the Last Lecture
The Last Lecture
Logistics to the CourseEvaluation(you care about most)
Class Participation (10%)Homework(30%)Two-student team Project Report(20%)Final Exam: (40%)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 5 / 100
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Review the Last Lecture
The Last Lecture
Logistics to the CourseEvaluation(you care about most)
Class Participation (10%)Homework(30%)Two-student team Project Report(20%)Final Exam: (40%)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 5 / 100
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Review the Last Lecture
The Last Lecture
Logistics to the CourseEvaluation(you care about most)
Class Participation (10%)Homework(30%)Two-student team Project Report(20%)Final Exam: (40%)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 5 / 100
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Review the Last Lecture
The Last Lecture
Logistics to the CourseEvaluation(you care about most)
Class Participation (10%)Homework(30%)Two-student team Project Report(20%)Final Exam: (40%)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 5 / 100
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Review the Last Lecture
The Last Lecture
Logistics to the CourseEvaluation(you care about most)
Class Participation (10%)Homework(30%)Two-student team Project Report(20%)Final Exam: (40%)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 5 / 100
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Review the Last Lecture
The Last Lecture
Logistics to the CourseEvaluation(you care about most)
Class Participation (10%)Homework(30%)Two-student team Project Report(20%)Final Exam: (40%)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 5 / 100
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Review the Last Lecture
The Last Lecture
The Structure of Economic DataData Structure
Cross-sectional dataTime series dataPooled cross-sectional dataPanel data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 6 / 100
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Review the Last Lecture
The Last Lecture
The Structure of Economic DataData Structure
Cross-sectional dataTime series dataPooled cross-sectional dataPanel data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 6 / 100
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Review the Last Lecture
The Last Lecture
The Structure of Economic DataData Structure
Cross-sectional dataTime series dataPooled cross-sectional dataPanel data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 6 / 100
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Review the Last Lecture
The Last Lecture
The Structure of Economic DataData Structure
Cross-sectional dataTime series dataPooled cross-sectional dataPanel data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 6 / 100
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Review the Last Lecture
The Last Lecture
The Structure of Economic DataData Structure
Cross-sectional dataTime series dataPooled cross-sectional dataPanel data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 6 / 100
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Review the Last Lecture
The Last Lecture
The Structure of Economic DataData Structure
Cross-sectional dataTime series dataPooled cross-sectional dataPanel data
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 6 / 100
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A Short Review of Probability Theory
A Short Review of Probability Theory
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 7 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
A Fundamental Axiom of Econometrics
1 Any economy can be viewed as a stochastic process governedby some probability law.
2 Economic phenomenon, as often summarized in form of data,can be reviewed as a realization of this stochastic datagenerating process.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 8 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
A Fundamental Axiom of Econometrics
1 Any economy can be viewed as a stochastic process governedby some probability law.
2 Economic phenomenon, as often summarized in form of data,can be reviewed as a realization of this stochastic datagenerating process.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 8 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probabilities and the Sample Space
Random Phenomena, Outcomes and ProbabilitiesThe mutually exclusive potential results of a random process are calledthe outcomes(结果）.The probability of an outcome is the proportion of the time that theoutcome occurs in the long run.
The Sample Space and Random Event（样本空间与随机事件）The set of all possible outcomes is called the sample space.An event is a subset of the sample space, that is, an event is a set ofone or more outcomes.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 9 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probabilities and the Sample Space
Random Phenomena, Outcomes and ProbabilitiesThe mutually exclusive potential results of a random process are calledthe outcomes(结果）.The probability of an outcome is the proportion of the time that theoutcome occurs in the long run.
The Sample Space and Random Event（样本空间与随机事件）The set of all possible outcomes is called the sample space.An event is a subset of the sample space, that is, an event is a set ofone or more outcomes.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 9 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probabilities and the Sample Space
Random Phenomena, Outcomes and ProbabilitiesThe mutually exclusive potential results of a random process are calledthe outcomes(结果）.The probability of an outcome is the proportion of the time that theoutcome occurs in the long run.
The Sample Space and Random Event（样本空间与随机事件）The set of all possible outcomes is called the sample space.An event is a subset of the sample space, that is, an event is a set ofone or more outcomes.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 9 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probabilities and the Sample Space
Random Phenomena, Outcomes and ProbabilitiesThe mutually exclusive potential results of a random process are calledthe outcomes(结果）.The probability of an outcome is the proportion of the time that theoutcome occurs in the long run.
The Sample Space and Random Event（样本空间与随机事件）The set of all possible outcomes is called the sample space.An event is a subset of the sample space, that is, an event is a set ofone or more outcomes.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 9 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probabilities and the Sample Space
Random Phenomena, Outcomes and ProbabilitiesThe mutually exclusive potential results of a random process are calledthe outcomes(结果）.The probability of an outcome is the proportion of the time that theoutcome occurs in the long run.
The Sample Space and Random Event（样本空间与随机事件）The set of all possible outcomes is called the sample space.An event is a subset of the sample space, that is, an event is a set ofone or more outcomes.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 9 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probabilities and the Sample Space
Random Phenomena, Outcomes and ProbabilitiesThe mutually exclusive potential results of a random process are calledthe outcomes(结果）.The probability of an outcome is the proportion of the time that theoutcome occurs in the long run.
The Sample Space and Random Event（样本空间与随机事件）The set of all possible outcomes is called the sample space.An event is a subset of the sample space, that is, an event is a set ofone or more outcomes.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 9 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Random VariablesRandom Variables(R.V.)A random variable (r.v.) is a function that maps from the sample space ofan experiment to the real line or X : Ω R
A random variable is a numerical summary of a random outcome.They are numeric representation of uncertain events.(thus we can usemath!)Notation: R.V.s are usually denoted by upper case letters (e.g. X),particular realizations are denoted by the corresponding lowercaseletters (e.g. x = 3)
ExampleTossing a coin 5 times
but not a random variable because it‘s not numeric.X(ω) = number of heads in the five tosses. X(HTHTT) = 2
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 10 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Random VariablesRandom Variables(R.V.)A random variable (r.v.) is a function that maps from the sample space ofan experiment to the real line or X : Ω R
A random variable is a numerical summary of a random outcome.They are numeric representation of uncertain events.(thus we can usemath!)Notation: R.V.s are usually denoted by upper case letters (e.g. X),particular realizations are denoted by the corresponding lowercaseletters (e.g. x = 3)
ExampleTossing a coin 5 times
but not a random variable because it‘s not numeric.X(ω) = number of heads in the five tosses. X(HTHTT) = 2
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 10 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Random VariablesRandom Variables(R.V.)A random variable (r.v.) is a function that maps from the sample space ofan experiment to the real line or X : Ω R
A random variable is a numerical summary of a random outcome.They are numeric representation of uncertain events.(thus we can usemath!)Notation: R.V.s are usually denoted by upper case letters (e.g. X),particular realizations are denoted by the corresponding lowercaseletters (e.g. x = 3)
ExampleTossing a coin 5 times
but not a random variable because it‘s not numeric.X(ω) = number of heads in the five tosses. X(HTHTT) = 2
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 10 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Random VariablesRandom Variables(R.V.)A random variable (r.v.) is a function that maps from the sample space ofan experiment to the real line or X : Ω R
A random variable is a numerical summary of a random outcome.They are numeric representation of uncertain events.(thus we can usemath!)Notation: R.V.s are usually denoted by upper case letters (e.g. X),particular realizations are denoted by the corresponding lowercaseletters (e.g. x = 3)
ExampleTossing a coin 5 times
but not a random variable because it‘s not numeric.X(ω) = number of heads in the five tosses. X(HTHTT) = 2
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 10 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Random VariablesRandom Variables(R.V.)A random variable (r.v.) is a function that maps from the sample space ofan experiment to the real line or X : Ω R
A random variable is a numerical summary of a random outcome.They are numeric representation of uncertain events.(thus we can usemath!)Notation: R.V.s are usually denoted by upper case letters (e.g. X),particular realizations are denoted by the corresponding lowercaseletters (e.g. x = 3)
ExampleTossing a coin 5 times
but not a random variable because it‘s not numeric.X(ω) = number of heads in the five tosses. X(HTHTT) = 2
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 10 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Random VariablesRandom Variables(R.V.)A random variable (r.v.) is a function that maps from the sample space ofan experiment to the real line or X : Ω R
A random variable is a numerical summary of a random outcome.They are numeric representation of uncertain events.(thus we can usemath!)Notation: R.V.s are usually denoted by upper case letters (e.g. X),particular realizations are denoted by the corresponding lowercaseletters (e.g. x = 3)
ExampleTossing a coin 5 times
but not a random variable because it‘s not numeric.X(ω) = number of heads in the five tosses. X(HTHTT) = 2
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 10 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probability DistributionsUncertainty over the value of ω. We’ll use probability to formalizethis uncertainty.The probability distribution of a r.v. gives the probability of all of thepossible values of the r.v.
PX(X = x) = P (ω ∈ Ω : X(ω) = x)
ExampleTossing two coins: let X be the number of heads.ω P(ω) X(ω)
TT 1/4 0HT 1/4 1TH 1/4 1HH 1/4 2
x P(X = x)0 1/41 1/22 1/4
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 11 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probability DistributionsUncertainty over the value of ω. We’ll use probability to formalizethis uncertainty.The probability distribution of a r.v. gives the probability of all of thepossible values of the r.v.
PX(X = x) = P (ω ∈ Ω : X(ω) = x)
ExampleTossing two coins: let X be the number of heads.ω P(ω) X(ω)
TT 1/4 0HT 1/4 1TH 1/4 1HH 1/4 2
x P(X = x)0 1/41 1/22 1/4
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 11 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probability DistributionsUncertainty over the value of ω. We’ll use probability to formalizethis uncertainty.The probability distribution of a r.v. gives the probability of all of thepossible values of the r.v.
PX(X = x) = P (ω ∈ Ω : X(ω) = x)
ExampleTossing two coins: let X be the number of heads.ω P(ω) X(ω)
TT 1/4 0HT 1/4 1TH 1/4 1HH 1/4 2
x P(X = x)0 1/41 1/22 1/4
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 11 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Distributional Functions of R.V.
It is cumbersome to derive the probabilities of X each time we needthem, so it is helpful to have a function that can give us theprobability of values or sets of values of X.
DefinitionThe cumulative distribution function or c.d.f of a r.v. X, denotedFX(x), is defined by
FX(x) ≡ PX(X ≤ x)
The c.d.f tells us the probability of a r.v. being less than some givenvalue.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 12 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Distributional Functions of R.V.
It is cumbersome to derive the probabilities of X each time we needthem, so it is helpful to have a function that can give us theprobability of values or sets of values of X.
DefinitionThe cumulative distribution function or c.d.f of a r.v. X, denotedFX(x), is defined by
FX(x) ≡ PX(X ≤ x)
The c.d.f tells us the probability of a r.v. being less than some givenvalue.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 12 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Distributional Functions of R.V.
It is cumbersome to derive the probabilities of X each time we needthem, so it is helpful to have a function that can give us theprobability of values or sets of values of X.
DefinitionThe cumulative distribution function or c.d.f of a r.v. X, denotedFX(x), is defined by
FX(x) ≡ PX(X ≤ x)
The c.d.f tells us the probability of a r.v. being less than some givenvalue.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 12 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Distribution Functions of R.V.
We have two kinds of r.v.s
DefinitionA r.v. X, is discrete if its range(the set of values it can take) is finite(X ∈ x1, x2, ...xk) or countably infinite(X ∈ x1, x2, ...)
eg: the number of computer crashes before deadline
DefinitionA r.v. X, is continuous if it can contain all real numbers in a interval.There are an uncountably infinite number of possible realizations.
eg: commuting times from home to school
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 13 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Distribution Functions of R.V.
We have two kinds of r.v.s
DefinitionA r.v. X, is discrete if its range(the set of values it can take) is finite(X ∈ x1, x2, ...xk) or countably infinite(X ∈ x1, x2, ...)
eg: the number of computer crashes before deadline
DefinitionA r.v. X, is continuous if it can contain all real numbers in a interval.There are an uncountably infinite number of possible realizations.
eg: commuting times from home to school
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 13 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Distribution Functions of R.V.
We have two kinds of r.v.s
DefinitionA r.v. X, is discrete if its range(the set of values it can take) is finite(X ∈ x1, x2, ...xk) or countably infinite(X ∈ x1, x2, ...)
eg: the number of computer crashes before deadline
DefinitionA r.v. X, is continuous if it can contain all real numbers in a interval.There are an uncountably infinite number of possible realizations.
eg: commuting times from home to school
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 13 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Distribution Functions of R.V.
We have two kinds of r.v.s
DefinitionA r.v. X, is discrete if its range(the set of values it can take) is finite(X ∈ x1, x2, ...xk) or countably infinite(X ∈ x1, x2, ...)
eg: the number of computer crashes before deadline
DefinitionA r.v. X, is continuous if it can contain all real numbers in a interval.There are an uncountably infinite number of possible realizations.
eg: commuting times from home to school
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 13 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Distribution Functions of R.V.
We have two kinds of r.v.s
DefinitionA r.v. X, is discrete if its range(the set of values it can take) is finite(X ∈ x1, x2, ...xk) or countably infinite(X ∈ x1, x2, ...)
eg: the number of computer crashes before deadline
DefinitionA r.v. X, is continuous if it can contain all real numbers in a interval.There are an uncountably infinite number of possible realizations.
eg: commuting times from home to school
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 13 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probability Distribution of a Discrete R.V.
Probability mass functionProbability mass function (p.m.f.) describes the distribution of r.v. whenit is discrete:
fX(xk) = P (X = xk) = px, k = 1, 2, ..., n
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 14 / 100
. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probability Distribution of a Discrete R.V.
c.d.f of a discrete r.vthe c.d.f of a discrete r.v. is denoted as
FX(x) = P(X ≤ x) =∑Xk≤x
fX(xk)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 16 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probability Distribution of a Continuous R.V.
Probability density functionThe probability density function or p.d.f., for a continuous random variableX is the function that satisfies for any interval, B
P(X ∈ B) =∫
BfX(x)dx
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 17 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probability Distribution of a Continuous R.V.Specifically, for a subset of the real line(a, b):P(a < X < b) =
∫ ba fX(x)dx, thus the probability of a region is the
area under the p.d.f. for that region.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 18 / 100
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A Short Review of Probability Theory Probabilities, the Sample Space and Random Variables
Probability Distribution of a Continuous R.V.Cumulative probability distributionjust as it is for a discrete random variable, except using p.d.f to calculatethe probability of x,
F(X) = P(X ≤ x) =∫ x
−∞fX(x)dx
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 19 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Distributions
Probability distributions describe the uncertainty about r.v.s. Thecdf/pmf/pdf give us all the information about the distribution ofsome r.v., but we are quite often interested in some feature of thedistribution rather than the entire distribution.
What is the difference between these two density curves? How mightwe summarize this difference?
There are two simple indictors:1 Central tendency: where the center of the distribution is.
Mean/expectation (均值或期望）2 Spread: how spread out the distribution is around the center.
Variance/standard deviation（方差或标准差）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 20 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Distributions
Probability distributions describe the uncertainty about r.v.s. Thecdf/pmf/pdf give us all the information about the distribution ofsome r.v., but we are quite often interested in some feature of thedistribution rather than the entire distribution.
What is the difference between these two density curves? How mightwe summarize this difference?
There are two simple indictors:1 Central tendency: where the center of the distribution is.
Mean/expectation (均值或期望）2 Spread: how spread out the distribution is around the center.
Variance/standard deviation（方差或标准差）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 20 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Distributions
Probability distributions describe the uncertainty about r.v.s. Thecdf/pmf/pdf give us all the information about the distribution ofsome r.v., but we are quite often interested in some feature of thedistribution rather than the entire distribution.
What is the difference between these two density curves? How mightwe summarize this difference?
There are two simple indictors:1 Central tendency: where the center of the distribution is.
Mean/expectation (均值或期望）2 Spread: how spread out the distribution is around the center.
Variance/standard deviation（方差或标准差）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 20 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Distributions
Probability distributions describe the uncertainty about r.v.s. Thecdf/pmf/pdf give us all the information about the distribution ofsome r.v., but we are quite often interested in some feature of thedistribution rather than the entire distribution.
What is the difference between these two density curves? How mightwe summarize this difference?
There are two simple indictors:1 Central tendency: where the center of the distribution is.
Mean/expectation (均值或期望）2 Spread: how spread out the distribution is around the center.
Variance/standard deviation（方差或标准差）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 20 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Distributions
Probability distributions describe the uncertainty about r.v.s. Thecdf/pmf/pdf give us all the information about the distribution ofsome r.v., but we are quite often interested in some feature of thedistribution rather than the entire distribution.
What is the difference between these two density curves? How mightwe summarize this difference?
There are two simple indictors:1 Central tendency: where the center of the distribution is.
Mean/expectation (均值或期望）2 Spread: how spread out the distribution is around the center.
Variance/standard deviation（方差或标准差）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 20 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Distributions
Probability distributions describe the uncertainty about r.v.s. Thecdf/pmf/pdf give us all the information about the distribution ofsome r.v., but we are quite often interested in some feature of thedistribution rather than the entire distribution.
What is the difference between these two density curves? How mightwe summarize this difference?
There are two simple indictors:1 Central tendency: where the center of the distribution is.
Mean/expectation (均值或期望）2 Spread: how spread out the distribution is around the center.
Variance/standard deviation（方差或标准差）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 20 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Distributions
Probability distributions describe the uncertainty about r.v.s. Thecdf/pmf/pdf give us all the information about the distribution ofsome r.v., but we are quite often interested in some feature of thedistribution rather than the entire distribution.
What is the difference between these two density curves? How mightwe summarize this difference?
There are two simple indictors:1 Central tendency: where the center of the distribution is.
Mean/expectation (均值或期望）2 Spread: how spread out the distribution is around the center.
Variance/standard deviation（方差或标准差）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 20 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
The Expected Value of a Random Variable
The expected value of a random variable X, denoted E(X) or µx, isthe long-run average value of the random variable over many repeatedtrials or occurrences. it is a natural measure of central tendency.For a discrete r.v., X ∈ x1, x2, ..., xk
µX = E[X] =k∑
j=1
xjpj
it is computed as a weighted average of the value of r.v., where theweights are the probability of each value occurring.For a continuous r.v., X, use the integral
µX = E[X] =∫ +∞
−∞xfX(x)dx
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 21 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
The Expected Value of a Random Variable
The expected value of a random variable X, denoted E(X) or µx, isthe long-run average value of the random variable over many repeatedtrials or occurrences. it is a natural measure of central tendency.For a discrete r.v., X ∈ x1, x2, ..., xk
µX = E[X] =k∑
j=1
xjpj
it is computed as a weighted average of the value of r.v., where theweights are the probability of each value occurring.For a continuous r.v., X, use the integral
µX = E[X] =∫ +∞
−∞xfX(x)dx
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 21 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
The Expected Value of a Random Variable
The expected value of a random variable X, denoted E(X) or µx, isthe long-run average value of the random variable over many repeatedtrials or occurrences. it is a natural measure of central tendency.For a discrete r.v., X ∈ x1, x2, ..., xk
µX = E[X] =k∑
j=1
xjpj
it is computed as a weighted average of the value of r.v., where theweights are the probability of each value occurring.For a continuous r.v., X, use the integral
µX = E[X] =∫ +∞
−∞xfX(x)dx
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 21 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Expectation
1 Additivity: expectation of sums are sums of expectations
E[X + Y] = E[X] + E[Y]
2 Homogeneity: Suppose that a and b are constants. Then
E[aX + b] = aE[X] + b
3 Law of the Unconscious Statistician, or LOTUS, if g(x) is afunction of a discrete random variable, then
E [g(X)] =∑
x g(x)fX(x) when x is discrete∫g(x)fX(x)dx when x is continuous
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 22 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Expectation
1 Additivity: expectation of sums are sums of expectations
E[X + Y] = E[X] + E[Y]
2 Homogeneity: Suppose that a and b are constants. Then
E[aX + b] = aE[X] + b
3 Law of the Unconscious Statistician, or LOTUS, if g(x) is afunction of a discrete random variable, then
E [g(X)] =∑
x g(x)fX(x) when x is discrete∫g(x)fX(x)dx when x is continuous
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 22 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Expectation
1 Additivity: expectation of sums are sums of expectations
E[X + Y] = E[X] + E[Y]
2 Homogeneity: Suppose that a and b are constants. Then
E[aX + b] = aE[X] + b
3 Law of the Unconscious Statistician, or LOTUS, if g(x) is afunction of a discrete random variable, then
E [g(X)] =∑
x g(x)fX(x) when x is discrete∫g(x)fX(x)dx when x is continuous
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 22 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
The Variance of a Random Variable
Besides some sense of where the middle of the distribution is, we alsowant to know how spread out the distribution is around that middle.
DefinitionTheVariance of a random variable X, denoted var(X)or σ2
X
σ2X = Var(X) = E[(X − µX)
2]
The Standard Deviation of X, denoted σX, is just the square root of thevariance.
σX =√
Var(X)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 23 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
The Variance of a Random Variable
Besides some sense of where the middle of the distribution is, we alsowant to know how spread out the distribution is around that middle.
DefinitionTheVariance of a random variable X, denoted var(X)or σ2
X
σ2X = Var(X) = E[(X − µX)
2]
The Standard Deviation of X, denoted σX, is just the square root of thevariance.
σX =√
Var(X)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 23 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Variance
If a and b are constants, then we have the following properties:1 V(b) = 02 V(aX + b) = a2V(X)3 V(X) = E[X2]− (E[X])2
ExampleBernoulli Distribution:
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 24 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Variance
If a and b are constants, then we have the following properties:1 V(b) = 02 V(aX + b) = a2V(X)3 V(X) = E[X2]− (E[X])2
ExampleBernoulli Distribution:
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 24 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Variance
If a and b are constants, then we have the following properties:1 V(b) = 02 V(aX + b) = a2V(X)3 V(X) = E[X2]− (E[X])2
ExampleBernoulli Distribution:
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 24 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Variance
If a and b are constants, then we have the following properties:1 V(b) = 02 V(aX + b) = a2V(X)3 V(X) = E[X2]− (E[X])2
ExampleBernoulli Distribution:
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 24 / 100
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A Short Review of Probability Theory Expected Values, Mean, and Variance
Properties of Variance
If a and b are constants, then we have the following properties:1 V(b) = 02 V(aX + b) = a2V(X)3 V(X) = E[X2]− (E[X])2
ExampleBernoulli Distribution:
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 24 / 100
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A Short Review of Probability Theory Multiple Random Variables
Why multiple random variables?
We are going to want to know what the relationships are betweenvariables.“The objective of science is the discovery of the relations”—Lord KelvinIn most cases,we often want to explore the relationship between twovariables in one study.
eg. Mortality and GDP growth
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 25 / 100
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A Short Review of Probability Theory Multiple Random Variables
Why multiple random variables?
We are going to want to know what the relationships are betweenvariables.“The objective of science is the discovery of the relations”—Lord KelvinIn most cases,we often want to explore the relationship between twovariables in one study.
eg. Mortality and GDP growth
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 25 / 100
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A Short Review of Probability Theory Multiple Random Variables
Why multiple random variables?
We are going to want to know what the relationships are betweenvariables.“The objective of science is the discovery of the relations”—Lord KelvinIn most cases,we often want to explore the relationship between twovariables in one study.
eg. Mortality and GDP growth
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 25 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Probability Distribution
Consider two discrete random variables X and Y with a jointprobability distribution,Then the joint probability mass function of (X,Y) describes theprobability of any pair of values:
fX,Y(x, y) = P(X = x,Y = y) = pxy
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 26 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Probability Distribution
Consider two discrete random variables X and Y with a jointprobability distribution,Then the joint probability mass function of (X,Y) describes theprobability of any pair of values:
fX,Y(x, y) = P(X = x,Y = y) = pxy
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 26 / 100
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A Short Review of Probability Theory Multiple Random Variables
Marginal Probability Distribution
The marginal distribution: often need to figure out the distribution ofjust one of the r.v.s.
fY(y) = P(Y = y) =∑
xfX,Y(x, y)
Intuition: sum over the probability that Y = y for all possible valuesof x.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 27 / 100
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A Short Review of Probability Theory Multiple Random Variables
Marginal Probability Distribution
The marginal distribution: often need to figure out the distribution ofjust one of the r.v.s.
fY(y) = P(Y = y) =∑
xfX,Y(x, y)
Intuition: sum over the probability that Y = y for all possible valuesof x.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 27 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Probability Density Function
Consider two continuous random variables X and Y with a jointprobability distribution, then the joint probability density functionof (X,Y) is a function, denoted as fX,Y(x, y) such that:
1 fX,Y(x, y) ≥ 02
∫ +∞−∞
∫ +∞−∞ fX,Y(x, y) dxdy = 1
3 P(a < X < b, c < Y < d) =∫ d
c∫ b
a fX,Y(x, y) dxdy, thus the probabilityin the a, b, c, darea.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 28 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Probability Density Function
Consider two continuous random variables X and Y with a jointprobability distribution, then the joint probability density functionof (X,Y) is a function, denoted as fX,Y(x, y) such that:
1 fX,Y(x, y) ≥ 02
∫ +∞−∞
∫ +∞−∞ fX,Y(x, y) dxdy = 1
3 P(a < X < b, c < Y < d) =∫ d
c∫ b
a fX,Y(x, y) dxdy, thus the probabilityin the a, b, c, darea.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 28 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Probability Density Function
Consider two continuous random variables X and Y with a jointprobability distribution, then the joint probability density functionof (X,Y) is a function, denoted as fX,Y(x, y) such that:
1 fX,Y(x, y) ≥ 02
∫ +∞−∞
∫ +∞−∞ fX,Y(x, y) dxdy = 1
3 P(a < X < b, c < Y < d) =∫ d
c∫ b
a fX,Y(x, y) dxdy, thus the probabilityin the a, b, c, darea.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 28 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Probability Density Function
Consider two continuous random variables X and Y with a jointprobability distribution, then the joint probability density functionof (X,Y) is a function, denoted as fX,Y(x, y) such that:
1 fX,Y(x, y) ≥ 02
∫ +∞−∞
∫ +∞−∞ fX,Y(x, y) dxdy = 1
3 P(a < X < b, c < Y < d) =∫ d
c∫ b
a fX,Y(x, y) dxdy, thus the probabilityin the a, b, c, darea.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 28 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Probability Density Function
Y and X axes denote on the “floor”, height is the value offXY(x, y)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 29 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Probability Density Function
The probability equals to volume above a specific region
P(X,Y) ∈ A) =∫(x,y)∈A
fX,Y(x, y)dxdy
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 30 / 100
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A Short Review of Probability Theory Multiple Random Variables
Continuous Marginal Distribution
the marginal p.d.f of Y by integrating over the distribution of X:
fY(y) =∫ +∞
−∞fX,Y(x, y)dx
the marginal p.d.f of X by integrating over the distribution of Y:
fX(x) =∫ +∞
−∞fX,Y(x, y)dy
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 31 / 100
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A Short Review of Probability Theory Multiple Random Variables
Continuous Marginal Distribution
the marginal p.d.f of Y by integrating over the distribution of X:
fY(y) =∫ +∞
−∞fX,Y(x, y)dx
the marginal p.d.f of X by integrating over the distribution of Y:
fX(x) =∫ +∞
−∞fX,Y(x, y)dy
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 31 / 100
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A Short Review of Probability Theory Multiple Random Variables
Continuous Marginal Distribution
Pile up all of the joint density onto a single dimensionZhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 32 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Cumulative Distribution Function
The joint cumulative distribution function of (X,Y) is
FX,Y(x, y) = P(X ≤ x,Y ≤ y) =∫ y
−∞
∫ x
−∞fX,Y(u, v) dudv
Transform joint c.d.f and joint p.d.f
fX,Y(x, y) =∂2FX,Y(x, y)
∂y
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 33 / 100
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A Short Review of Probability Theory Multiple Random Variables
Joint Cumulative Distribution Function
The joint cumulative distribution function of (X,Y) is
FX,Y(x, y) = P(X ≤ x,Y ≤ y) =∫ y
−∞
∫ x
−∞fX,Y(u, v) dudv
Transform joint c.d.f and joint p.d.f
fX,Y(x, y) =∂2FX,Y(x, y)
∂y
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 33 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Expectations over multiple r.v.s
Expectations over multiple r.v.s
E[g(X,Y)] =∑
x∑
y g(x, y)fX,Y(x, y) if∫x∫
y g(x, y)fX,Y(x, y)dxdy if
Marginal expectation
E[Y] =∑
x∑
y yfX,Y(x, y) if∫x∫
y yf(x, y)dxdy if
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 34 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Expectations over multiple r.v.s
Expectations over multiple r.v.s
E[g(X,Y)] =∑
x∑
y g(x, y)fX,Y(x, y) if∫x∫
y g(x, y)fX,Y(x, y)dxdy if
Marginal expectation
E[Y] =∑
x∑
y yfX,Y(x, y) if∫x∫
y yf(x, y)dxdy if
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 34 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Independence
Independence
Two r.v.s X and Y are independent, which we denote it as X ⊥ Y, if for allsets A and B
P(X ∈ A,Y ∈ B) = P(X ∈ A)P(Y ∈ B)
Intuition: knowing the value of X gives us no information about thevalue of Y.IfX and Y are independent, then
Joint p.d.f is the product of marginal p.d.f, thus fX,Y(x, y) = fX(x)fY(y)Joint c.d.f is the product of marginal c.d.f, thus fX,Y(x, y) = fX(x)fY(y)functions of independent r.v.s are independent, thus h(X) ⊥ g(Y) forany functions h(·) and g(·).
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 35 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Independence
Independence
Two r.v.s X and Y are independent, which we denote it as X ⊥ Y, if for allsets A and B
P(X ∈ A,Y ∈ B) = P(X ∈ A)P(Y ∈ B)
Intuition: knowing the value of X gives us no information about thevalue of Y.IfX and Y are independent, then
Joint p.d.f is the product of marginal p.d.f, thus fX,Y(x, y) = fX(x)fY(y)Joint c.d.f is the product of marginal c.d.f, thus fX,Y(x, y) = fX(x)fY(y)functions of independent r.v.s are independent, thus h(X) ⊥ g(Y) forany functions h(·) and g(·).
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 35 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Independence
Independence
Two r.v.s X and Y are independent, which we denote it as X ⊥ Y, if for allsets A and B
P(X ∈ A,Y ∈ B) = P(X ∈ A)P(Y ∈ B)
Intuition: knowing the value of X gives us no information about thevalue of Y.IfX and Y are independent, then
Joint p.d.f is the product of marginal p.d.f, thus fX,Y(x, y) = fX(x)fY(y)Joint c.d.f is the product of marginal c.d.f, thus fX,Y(x, y) = fX(x)fY(y)functions of independent r.v.s are independent, thus h(X) ⊥ g(Y) forany functions h(·) and g(·).
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 35 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Independence
Independence
Two r.v.s X and Y are independent, which we denote it as X ⊥ Y, if for allsets A and B
P(X ∈ A,Y ∈ B) = P(X ∈ A)P(Y ∈ B)
Intuition: knowing the value of X gives us no information about thevalue of Y.IfX and Y are independent, then
Joint p.d.f is the product of marginal p.d.f, thus fX,Y(x, y) = fX(x)fY(y)Joint c.d.f is the product of marginal c.d.f, thus fX,Y(x, y) = fX(x)fY(y)functions of independent r.v.s are independent, thus h(X) ⊥ g(Y) forany functions h(·) and g(·).
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 35 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Independence
Independence
Two r.v.s X and Y are independent, which we denote it as X ⊥ Y, if for allsets A and B
P(X ∈ A,Y ∈ B) = P(X ∈ A)P(Y ∈ B)
Intuition: knowing the value of X gives us no information about thevalue of Y.IfX and Y are independent, then
Joint p.d.f is the product of marginal p.d.f, thus fX,Y(x, y) = fX(x)fY(y)Joint c.d.f is the product of marginal c.d.f, thus fX,Y(x, y) = fX(x)fY(y)functions of independent r.v.s are independent, thus h(X) ⊥ g(Y) forany functions h(·) and g(·).
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 35 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Independence
Independence
Two r.v.s X and Y are independent, which we denote it as X ⊥ Y, if for allsets A and B
P(X ∈ A,Y ∈ B) = P(X ∈ A)P(Y ∈ B)
Intuition: knowing the value of X gives us no information about thevalue of Y.IfX and Y are independent, then
Joint p.d.f is the product of marginal p.d.f, thus fX,Y(x, y) = fX(x)fY(y)Joint c.d.f is the product of marginal c.d.f, thus fX,Y(x, y) = fX(x)fY(y)functions of independent r.v.s are independent, thus h(X) ⊥ g(Y) forany functions h(·) and g(·).
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 35 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Independence
Theorem (Independence)if X and Y are independent r.v.s, then
E[XY] = E[X]E[Y]
Proof.Skip. you could finish it by yourself.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 36 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Independence
Theorem (Independence)if X and Y are independent r.v.s, then
E[XY] = E[X]E[Y]
Proof.Skip. you could finish it by yourself.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 36 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance
If two variables are not independent, we could still measure thestrength of their dependence by the definition of covariance.
Covariancethe covariance between X and Y is defined as
Cov[X,Y] = E [(X − E[X]) (Y − E[Y])]
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]If X ⊥ Y, Cov[X,Y] = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 37 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance
If two variables are not independent, we could still measure thestrength of their dependence by the definition of covariance.
Covariancethe covariance between X and Y is defined as
Cov[X,Y] = E [(X − E[X]) (Y − E[Y])]
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]If X ⊥ Y, Cov[X,Y] = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 37 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance
If two variables are not independent, we could still measure thestrength of their dependence by the definition of covariance.
Covariancethe covariance between X and Y is defined as
Cov[X,Y] = E [(X − E[X]) (Y − E[Y])]
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]If X ⊥ Y, Cov[X,Y] = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 37 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance
If two variables are not independent, we could still measure thestrength of their dependence by the definition of covariance.
Covariancethe covariance between X and Y is defined as
Cov[X,Y] = E [(X − E[X]) (Y − E[Y])]
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]If X ⊥ Y, Cov[X,Y] = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 37 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance
If two variables are not independent, we could still measure thestrength of their dependence by the definition of covariance.
Covariancethe covariance between X and Y is defined as
Cov[X,Y] = E [(X − E[X]) (Y − E[Y])]
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]If X ⊥ Y, Cov[X,Y] = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 37 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Intuition of Covariance
The conditional probability mass function(conditional p.m.f) of Yconditional of X is
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 38 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Intuition of Covariance
The conditional probability mass function(conditional p.m.f) of Yconditional of X is
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 38 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Independence
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]Cov[aX + b, cY + d] = acCoV[XY]Cov[X,X] = Var[X]
Covariance and IndependenceIf X ⊥ Y, then Cov[X,Y] = 0. thus independence⇒Cov[X,Y] = 0.If Cov[X,Y] = 0, then X ⊥ Y ? NO! Cov[X,Y] = 0 ⇏independence.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 39 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Independence
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]Cov[aX + b, cY + d] = acCoV[XY]Cov[X,X] = Var[X]
Covariance and IndependenceIf X ⊥ Y, then Cov[X,Y] = 0. thus independence⇒Cov[X,Y] = 0.If Cov[X,Y] = 0, then X ⊥ Y ? NO! Cov[X,Y] = 0 ⇏independence.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 39 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Independence
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]Cov[aX + b, cY + d] = acCoV[XY]Cov[X,X] = Var[X]
Covariance and IndependenceIf X ⊥ Y, then Cov[X,Y] = 0. thus independence⇒Cov[X,Y] = 0.If Cov[X,Y] = 0, then X ⊥ Y ? NO! Cov[X,Y] = 0 ⇏independence.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 39 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Independence
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]Cov[aX + b, cY + d] = acCoV[XY]Cov[X,X] = Var[X]
Covariance and IndependenceIf X ⊥ Y, then Cov[X,Y] = 0. thus independence⇒Cov[X,Y] = 0.If Cov[X,Y] = 0, then X ⊥ Y ? NO! Cov[X,Y] = 0 ⇏independence.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 39 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Independence
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]Cov[aX + b, cY + d] = acCoV[XY]Cov[X,X] = Var[X]
Covariance and IndependenceIf X ⊥ Y, then Cov[X,Y] = 0. thus independence⇒Cov[X,Y] = 0.If Cov[X,Y] = 0, then X ⊥ Y ? NO! Cov[X,Y] = 0 ⇏independence.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 39 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Independence
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]Cov[aX + b, cY + d] = acCoV[XY]Cov[X,X] = Var[X]
Covariance and IndependenceIf X ⊥ Y, then Cov[X,Y] = 0. thus independence⇒Cov[X,Y] = 0.If Cov[X,Y] = 0, then X ⊥ Y ? NO! Cov[X,Y] = 0 ⇏independence.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 39 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Independence
Properties of covariances:Cov[X,Y] = E[XY]− E[X]E[Y]Cov[aX + b, cY + d] = acCoV[XY]Cov[X,X] = Var[X]
Covariance and IndependenceIf X ⊥ Y, then Cov[X,Y] = 0. thus independence⇒Cov[X,Y] = 0.If Cov[X,Y] = 0, then X ⊥ Y ? NO! Cov[X,Y] = 0 ⇏independence.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 39 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Correlation
Covariance is not scale-free. Correlation is a special form ofcovariance after dividing out the scales of the respective variables.
CorrelationThe correlation between X and Y is defined as
ρXY =Cov[X,Y]√Var[X]Var[Y]
Correlation properties:−1 ≤ ρ ≤ 1If | ρXY |= 1, then X and Y are perfectly correlated with a linearrelationship: Y = a + bX
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 40 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Correlation
Covariance is not scale-free. Correlation is a special form ofcovariance after dividing out the scales of the respective variables.
CorrelationThe correlation between X and Y is defined as
ρXY =Cov[X,Y]√Var[X]Var[Y]
Correlation properties:−1 ≤ ρ ≤ 1If | ρXY |= 1, then X and Y are perfectly correlated with a linearrelationship: Y = a + bX
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 40 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Correlation
Covariance is not scale-free. Correlation is a special form ofcovariance after dividing out the scales of the respective variables.
CorrelationThe correlation between X and Y is defined as
ρXY =Cov[X,Y]√Var[X]Var[Y]
Correlation properties:−1 ≤ ρ ≤ 1If | ρXY |= 1, then X and Y are perfectly correlated with a linearrelationship: Y = a + bX
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 40 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Correlation
Covariance is not scale-free. Correlation is a special form ofcovariance after dividing out the scales of the respective variables.
CorrelationThe correlation between X and Y is defined as
ρXY =Cov[X,Y]√Var[X]Var[Y]
Correlation properties:−1 ≤ ρ ≤ 1If | ρXY |= 1, then X and Y are perfectly correlated with a linearrelationship: Y = a + bX
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 40 / 100
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A Short Review of Probability Theory Properties of Joint Distributions
Covariance and Correlation
Covariance is not scale-free. Correlation is a special form ofcovariance after dividing out the scales of the respective variables.
CorrelationThe correlation between X and Y is defined as
ρXY =Cov[X,Y]√Var[X]Var[Y]
Correlation properties:−1 ≤ ρ ≤ 1If | ρXY |= 1, then X and Y are perfectly correlated with a linearrelationship: Y = a + bX
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 40 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Probability function
The conditional probability mass functional(conditional p.m.f) ofY conditional of X is
fY|X (y|x) = P(Y = y | X = x) = P(X = x,Y = y)P(X = x) =
fX,Y(x, y)fX(x)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 41 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Density Function
Conditional probability density function:c.d.f. of Y conditional on X is
fY|X (y|x) = fX,Y(x, y)fX(x)
Based on the definition of the conditional p.m.f./p.d.f., we have thefollowing equation
fX,Y(x, y) = fY|X (y|x) fX(x)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 42 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Density Function
Conditional probability density function:c.d.f. of Y conditional on X is
fY|X (y|x) = fX,Y(x, y)fX(x)
Based on the definition of the conditional p.m.f./p.d.f., we have thefollowing equation
fX,Y(x, y) = fY|X (y|x) fX(x)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 42 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Density Function
c.d.f is proportional to joint p.d.f along x0 like a slice of total volume.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 43 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Independence
Conditional IndependenceX and Y are conditional Independent given Z, denoted as X ⊥ Y | Z, if
fX,Y|Z (x, y|z) = fX|Z (x|z) fY|Z (y|z)
X and Y are independent within levels of Z.Example:
X = swimming accidents, Y = ice cream sold.In general, two variable is highly correlated.If conditional on Z = temperature, then they are independent.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 44 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Independence
Conditional IndependenceX and Y are conditional Independent given Z, denoted as X ⊥ Y | Z, if
fX,Y|Z (x, y|z) = fX|Z (x|z) fY|Z (y|z)
X and Y are independent within levels of Z.Example:
X = swimming accidents, Y = ice cream sold.In general, two variable is highly correlated.If conditional on Z = temperature, then they are independent.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 44 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Independence
Conditional IndependenceX and Y are conditional Independent given Z, denoted as X ⊥ Y | Z, if
fX,Y|Z (x, y|z) = fX|Z (x|z) fY|Z (y|z)
X and Y are independent within levels of Z.Example:
X = swimming accidents, Y = ice cream sold.In general, two variable is highly correlated.If conditional on Z = temperature, then they are independent.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 44 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Independence
Conditional IndependenceX and Y are conditional Independent given Z, denoted as X ⊥ Y | Z, if
fX,Y|Z (x, y|z) = fX|Z (x|z) fY|Z (y|z)
X and Y are independent within levels of Z.Example:
X = swimming accidents, Y = ice cream sold.In general, two variable is highly correlated.If conditional on Z = temperature, then they are independent.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 44 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Independence
Conditional IndependenceX and Y are conditional Independent given Z, denoted as X ⊥ Y | Z, if
fX,Y|Z (x, y|z) = fX|Z (x|z) fY|Z (y|z)
X and Y are independent within levels of Z.Example:
X = swimming accidents, Y = ice cream sold.In general, two variable is highly correlated.If conditional on Z = temperature, then they are independent.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 44 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Independence
Conditional IndependenceX and Y are conditional Independent given Z, denoted as X ⊥ Y | Z, if
fX,Y|Z (x, y|z) = fX|Z (x|z) fY|Z (y|z)
X and Y are independent within levels of Z.Example:
X = swimming accidents, Y = ice cream sold.In general, two variable is highly correlated.If conditional on Z = temperature, then they are independent.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 44 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Expectation Function
Conditional ExpectationConditional on X, Y’s Conditional Expectation is
E(Y|X) =∑
yfY|X(y|x) discrete Y∫yfY|X(y|x)dy continuous Y
Conditional Expectation Function(CEF) is a function of x, since X is arandom variable, so CEF is also a random variable.Intuition：期望就是求平均值，而条件期望就是“分组取平均”或“在... 条件下的均值”。
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 45 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Expectation Function
Conditional ExpectationConditional on X, Y’s Conditional Expectation is
E(Y|X) =∑
yfY|X(y|x) discrete Y∫yfY|X(y|x)dy continuous Y
Conditional Expectation Function(CEF) is a function of x, since X is arandom variable, so CEF is also a random variable.Intuition：期望就是求平均值，而条件期望就是“分组取平均”或“在... 条件下的均值”。
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 45 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Expectation Function
Conditional ExpectationConditional on X, Y’s Conditional Expectation is
E(Y|X) =∑
yfY|X(y|x) discrete Y∫yfY|X(y|x)dy continuous Y
Conditional Expectation Function(CEF) is a function of x, since X is arandom variable, so CEF is also a random variable.Intuition：期望就是求平均值，而条件期望就是“分组取平均”或“在... 条件下的均值”。
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 45 / 100
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A Short Review of Probability Theory Conditional Distributions
Properties of Conditional Expectation
1 E[c(X) | X] = c(X) for any function c(X).Thus if we know X, thenwe also know c(X).
eg. E[(X2 + 2X3) | X] =X2 + 2X3
if X and Y are independent r.v.s, then
E[Y | X = x] = E[Y]
if X and Y independent conditional on Z, thus X ⊥ Y | Z ,
E[Y | X = x,Z = z] = E[Y | Z = z]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 46 / 100
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A Short Review of Probability Theory Conditional Distributions
Properties of Conditional Expectation
1 E[c(X) | X] = c(X) for any function c(X).Thus if we know X, thenwe also know c(X).
eg. E[(X2 + 2X3) | X] =X2 + 2X3
if X and Y are independent r.v.s, then
E[Y | X = x] = E[Y]
if X and Y independent conditional on Z, thus X ⊥ Y | Z ,
E[Y | X = x,Z = z] = E[Y | Z = z]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 46 / 100
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A Short Review of Probability Theory Conditional Distributions
Properties of Conditional Expectation
1 E[c(X) | X] = c(X) for any function c(X).Thus if we know X, thenwe also know c(X).
eg. E[(X2 + 2X3) | X] =X2 + 2X3
if X and Y are independent r.v.s, then
E[Y | X = x] = E[Y]
if X and Y independent conditional on Z, thus X ⊥ Y | Z ,
E[Y | X = x,Z = z] = E[Y | Z = z]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 46 / 100
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A Short Review of Probability Theory Conditional Distributions
Properties of Conditional Expectation
1 E[c(X) | X] = c(X) for any function c(X).Thus if we know X, thenwe also know c(X).
eg. E[(X2 + 2X3) | X] =X2 + 2X3
if X and Y are independent r.v.s, then
E[Y | X = x] = E[Y]
if X and Y independent conditional on Z, thus X ⊥ Y | Z ,
E[Y | X = x,Z = z] = E[Y | Z = z]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 46 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Variance
Conditional VarianceConditional on X, Y’s Conditional Expectation is defined asVar(Y|X) = E [(Y − E[Y|X])2 | X]
Usual variance formula applied to conditional distribution.Discrete
V[Y | X] =∑
y(y − E[Y | X])2fY|X (y|x)
ContinuousV[Y | X] =
∫y(y − E[Y | X])2fY|X (y|x)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 47 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Variance
Conditional VarianceConditional on X, Y’s Conditional Expectation is defined asVar(Y|X) = E [(Y − E[Y|X])2 | X]
Usual variance formula applied to conditional distribution.Discrete
V[Y | X] =∑
y(y − E[Y | X])2fY|X (y|x)
ContinuousV[Y | X] =
∫y(y − E[Y | X])2fY|X (y|x)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 47 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Variance
Conditional VarianceConditional on X, Y’s Conditional Expectation is defined asVar(Y|X) = E [(Y − E[Y|X])2 | X]
Usual variance formula applied to conditional distribution.Discrete
V[Y | X] =∑
y(y − E[Y | X])2fY|X (y|x)
ContinuousV[Y | X] =
∫y(y − E[Y | X])2fY|X (y|x)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 47 / 100
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A Short Review of Probability Theory Conditional Distributions
Conditional Variance
Conditional VarianceConditional on X, Y’s Conditional Expectation is defined asVar(Y|X) = E [(Y − E[Y|X])2 | X]
Usual variance formula applied to conditional distribution.Discrete
V[Y | X] =∑
y(y − E[Y | X])2fY|X (y|x)
ContinuousV[Y | X] =
∫y(y − E[Y | X])2fY|X (y|x)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 47 / 100
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A Short Review of Probability Theory Famous Distributions
Families of distributions
There are several important families of distributions:The p.m.f./p.d.f. within the family has the same form, with parametersthat might vary across the family.The parameters determine the shape of the distribution
Statistical modeling in a nutshell: to study probability distributionfunction.
Assume the data, X1,X2, ...,Xn, are independent draws from a commondistribution fθ(x) within a family of distributions (normal, poisson, etc)Use a function of the observed data to estimate the value of theθ : θ(X1,X2, ...,Xn)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 48 / 100
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A Short Review of Probability Theory Famous Distributions
Families of distributions
There are several important families of distributions:The p.m.f./p.d.f. within the family has the same form, with parametersthat might vary across the family.The parameters determine the shape of the distribution
Statistical modeling in a nutshell: to study probability distributionfunction.
Assume the data, X1,X2, ...,Xn, are independent draws from a commondistribution fθ(x) within a family of distributions (normal, poisson, etc)Use a function of the observed data to estimate the value of theθ : θ(X1,X2, ...,Xn)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 48 / 100
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A Short Review of Probability Theory Famous Distributions
Families of distributions
There are several important families of distributions:The p.m.f./p.d.f. within the family has the same form, with parametersthat might vary across the family.The parameters determine the shape of the distribution
Statistical modeling in a nutshell: to study probability distributionfunction.
Assume the data, X1,X2, ...,Xn, are independent draws from a commondistribution fθ(x) within a family of distributions (normal, poisson, etc)Use a function of the observed data to estimate the value of theθ : θ(X1,X2, ...,Xn)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 48 / 100
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A Short Review of Probability Theory Famous Distributions
Families of distributions
There are several important families of distributions:The p.m.f./p.d.f. within the family has the same form, with parametersthat might vary across the family.The parameters determine the shape of the distribution
Statistical modeling in a nutshell: to study probability distributionfunction.
Assume the data, X1,X2, ...,Xn, are independent draws from a commondistribution fθ(x) within a family of distributions (normal, poisson, etc)Use a function of the observed data to estimate the value of theθ : θ(X1,X2, ...,Xn)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 48 / 100
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A Short Review of Probability Theory Famous Distributions
Families of distributions
There are several important families of distributions:The p.m.f./p.d.f. within the family has the same form, with parametersthat might vary across the family.The parameters determine the shape of the distribution
Statistical modeling in a nutshell: to study probability distributionfunction.
Assume the data, X1,X2, ...,Xn, are independent draws from a commondistribution fθ(x) within a family of distributions (normal, poisson, etc)Use a function of the observed data to estimate the value of theθ : θ(X1,X2, ...,Xn)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 48 / 100
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A Short Review of Probability Theory Famous Distributions
Families of distributions
There are several important families of distributions:The p.m.f./p.d.f. within the family has the same form, with parametersthat might vary across the family.The parameters determine the shape of the distribution
Statistical modeling in a nutshell: to study probability distributionfunction.
Assume the data, X1,X2, ...,Xn, are independent draws from a commondistribution fθ(x) within a family of distributions (normal, poisson, etc)Use a function of the observed data to estimate the value of theθ : θ(X1,X2, ...,Xn)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 48 / 100
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A Short Review of Probability Theory Famous Distributions
The Bernoulli DistributionDefinitionX has a Bernoulli distribution if it have a binary values X ∈ 0, 1 andits probability mass function is
fX(x) = P(X = x) =
p if x = 1
1− p if x = 0
Question:What is the Expectation and Variance of X?
E(X) =
k∑j=1
xjpj = 0× (1− p) + 1× p = p
Var(X) = E[X − E(X)]2 = E[X2]− (E[X])2 = p − p2 = p(1− p)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 49 / 100
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A Short Review of Probability Theory Famous Distributions
The Bernoulli DistributionDefinitionX has a Bernoulli distribution if it have a binary values X ∈ 0, 1 andits probability mass function is
fX(x) = P(X = x) =
p if x = 1
1− p if x = 0
Question:What is the Expectation and Variance of X?
E(X) =
k∑j=1
xjpj = 0× (1− p) + 1× p = p
Var(X) = E[X − E(X)]2 = E[X2]− (E[X])2 = p − p2 = p(1− p)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 49 / 100
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A Short Review of Probability Theory Famous Distributions
The Normal Distribution
The p.d.f of a normal random variable X is
fX(x) =1
σ√2π
exp[− 1
2σ2(x − µ)2
], −∞ < X < +∞
if X is normally distributed with expected value µ and variance σ2,denoted as X ∼ N(µ, σ2)
if we know these two parameters, we know everything about thedistribution.
Examples: Human heights, weights, test scores,If X represents wage, income or consumption etc, it will has alog-normal distribution, thus
log(X) ∼ N(µ, σ2)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 50 / 100
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A Short Review of Probability Theory Famous Distributions
The Normal Distribution
The p.d.f of a normal random variable X is
fX(x) =1
σ√2π
exp[− 1
2σ2(x − µ)2
], −∞ < X < +∞
if X is normally distributed with expected value µ and variance σ2,denoted as X ∼ N(µ, σ2)
if we know these two parameters, we know everything about thedistribution.
Examples: Human heights, weights, test scores,If X represents wage, income or consumption etc, it will has alog-normal distribution, thus
log(X) ∼ N(µ, σ2)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 50 / 100
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A Short Review of Probability Theory Famous Distributions
The Normal Distribution
The p.d.f of a normal random variable X is
fX(x) =1
σ√2π
exp[− 1
2σ2(x − µ)2
], −∞ < X < +∞
if X is normally distributed with expected value µ and variance σ2,denoted as X ∼ N(µ, σ2)
if we know these two parameters, we know everything about thedistribution.
Examples: Human heights, weights, test scores,If X represents wage, income or consumption etc, it will has alog-normal distribution, thus
log(X) ∼ N(µ, σ2)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 50 / 100
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A Short Review of Probability Theory Famous Distributions
The Normal Distribution
The p.d.f of a normal random variable X is
fX(x) =1
σ√2π
exp[− 1
2σ2(x − µ)2
], −∞ < X < +∞
if X is normally distributed with expected value µ and variance σ2,denoted as X ∼ N(µ, σ2)
if we know these two parameters, we know everything about thedistribution.
Examples: Human heights, weights, test scores,If X represents wage, income or consumption etc, it will has alog-normal distribution, thus
log(X) ∼ N(µ, σ2)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 50 / 100
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A Short Review of Probability Theory Famous Distributions
The Normal Distribution
The p.d.f of a normal random variable X is
fX(x) =1
σ√2π
exp[− 1
2σ2(x − µ)2
], −∞ < X < +∞
if X is normally distributed with expected value µ and variance σ2,denoted as X ∼ N(µ, σ2)
if we know these two parameters, we know everything about thedistribution.
Examples: Human heights, weights, test scores,If X represents wage, income or consumption etc, it will has alog-normal distribution, thus
log(X) ∼ N(µ, σ2)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 50 / 100
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A Short Review of Probability Theory Famous Distributions
The Normal Distribution
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 51 / 100
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A Short Review of Probability Theory Famous Distributions
The Standard Normal Distribution
A special case of the normal distribution where the mean is zero(µ = 0) and the variance is one (σ2 = σ = 1), then its p.d.f is
fX(x) = ϕ(x)= 1√2π
e− 12
x2 , −∞ < X < +∞
if X is standard normally distributed, then denoted as X ∼ N(0, 1)
The standard normal cumulative distribution function is denoted
Φ(z) = P(Z ≤ z)
where z is a standardize r.v. thus z = x−µXσX
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 52 / 100
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A Short Review of Probability Theory Famous Distributions
The Standard Normal Distribution
A special case of the normal distribution where the mean is zero(µ = 0) and the variance is one (σ2 = σ = 1), then its p.d.f is
fX(x) = ϕ(x)= 1√2π
e− 12
x2 , −∞ < X < +∞
if X is standard normally distributed, then denoted as X ∼ N(0, 1)
The standard normal cumulative distribution function is denoted
Φ(z) = P(Z ≤ z)
where z is a standardize r.v. thus z = x−µXσX
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 52 / 100
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A Short Review of Probability Theory Famous Distributions
The Standard Normal Distribution
A special case of the normal distribution where the mean is zero(µ = 0) and the variance is one (σ2 = σ = 1), then its p.d.f is
fX(x) = ϕ(x)= 1√2π
e− 12
x2 , −∞ < X < +∞
if X is standard normally distributed, then denoted as X ∼ N(0, 1)
The standard normal cumulative distribution function is denoted
Φ(z) = P(Z ≤ z)
where z is a standardize r.v. thus z = x−µXσX
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 52 / 100
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A Short Review of Probability Theory Famous Distributions
The Standard Normal Distribution
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 53 / 100
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A Short Review of Probability Theory Famous Distributions
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 54 / 100
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A Short Review of Probability Theory Famous Distributions
The Chi-Square DistributionLet Zi(i = 1, 2, ...,m) be independent random variables, eachdistributed as standard normal. Then a new random variable can bedefined as the sum of the squares of Zi :
X =
m∑i=1
Z2i
Then X has a chi-squared distribution with m degrees of freedomThe form of the distribution varies with the number of degrees offreedom, i.e. the number of standard normal random variables Ziincluded in X.The distribution has a long tail, or is skewed, to the right. As thedegrees of freedom m gets larger, however, the distribution becomesmore symmetric and “bell-shaped”. In fact, as m gets larger, thechi-square distribution converges to, and essentially becomes, anormal distribution.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 55 / 100
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A Short Review of Probability Theory Famous Distributions
The Chi-Square DistributionLet Zi(i = 1, 2, ...,m) be independent random variables, eachdistributed as standard normal. Then a new random variable can bedefined as the sum of the squares of Zi :
X =
m∑i=1
Z2i
Then X has a chi-squared distribution with m degrees of freedomThe form of the distribution varies with the number of degrees offreedom, i.e. the number of standard normal random variables Ziincluded in X.The distribution has a long tail, or is skewed, to the right. As thedegrees of freedom m gets larger, however, the distribution becomesmore symmetric and “bell-shaped”. In fact, as m gets larger, thechi-square distribution converges to, and essentially becomes, anormal distribution.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 55 / 100
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A Short Review of Probability Theory Famous Distributions
The Chi-Square DistributionLet Zi(i = 1, 2, ...,m) be independent random variables, eachdistributed as standard normal. Then a new random variable can bedefined as the sum of the squares of Zi :
X =
m∑i=1
Z2i
Then X has a chi-squared distribution with m degrees of freedomThe form of the distribution varies with the number of degrees offreedom, i.e. the number of standard normal random variables Ziincluded in X.The distribution has a long tail, or is skewed, to the right. As thedegrees of freedom m gets larger, however, the distribution becomesmore symmetric and “bell-shaped”. In fact, as m gets larger, thechi-square distribution converges to, and essentially becomes, anormal distribution.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 55 / 100
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A Short Review of Probability Theory Famous Distributions
The Chi-Square Distribution
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 56 / 100
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A Short Review of Probability Theory Famous Distributions
The Student t Distribution
The Student t distribution can be obtained from a standard normaland a chi-square random variable.Let Z have a standard normal distribution, let X have a chi-squaredistribution with m degrees of freedom and assume that Z and X areindependent. Then the random variable
T =Z√X/n
has has a t-distribution with m degrees of freedom, denoted asT ∼ tn.The shape of the t-distribution is similar to that of a normaldistribution, except that the t-distribution has more probability massin the tails.As the degrees of freedom get large, the t-distribution approaches thestandard normal distribution.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 57 / 100
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A Short Review of Probability Theory Famous Distributions
The Student t Distribution
The Student t distribution can be obtained from a standard normaland a chi-square random variable.Let Z have a standard normal distribution, let X have a chi-squaredistribution with m degrees of freedom and assume that Z and X areindependent. Then the random variable
T =Z√X/n
has has a t-distribution with m degrees of freedom, denoted asT ∼ tn.The shape of the t-distribution is similar to that of a normaldistribution, except that the t-distribution has more probability massin the tails.As the degrees of freedom get large, the t-distribution approaches thestandard normal distribution.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 57 / 100
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A Short Review of Probability Theory Famous Distributions
The Student t Distribution
The Student t distribution can be obtained from a standard normaland a chi-square random variable.Let Z have a standard normal distribution, let X have a chi-squaredistribution with m degrees of freedom and assume that Z and X areindependent. Then the random variable
T =Z√X/n
has has a t-distribution with m degrees of freedom, denoted asT ∼ tn.The shape of the t-distribution is similar to that of a normaldistribution, except that the t-distribution has more probability massin the tails.As the degrees of freedom get large, the t-distribution approaches thestandard normal distribution.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 57 / 100
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A Short Review of Probability Theory Famous Distributions
The Student t Distribution
The Student t distribution can be obtained from a standard normaland a chi-square random variable.Let Z have a standard normal distribution, let X have a chi-squaredistribution with m degrees of freedom and assume that Z and X areindependent. Then the random variable
T =Z√X/n
has has a t-distribution with m degrees of freedom, denoted asT ∼ tn.The shape of the t-distribution is similar to that of a normaldistribution, except that the t-distribution has more probability massin the tails.As the degrees of freedom get large, the t-distribution approaches thestandard normal distribution.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 57 / 100
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A Short Review of Probability Theory Famous Distributions
The Student t Distribution
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 58 / 100
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A Short Review of Probability Theory Famous Distributions
The F Distribution
Let X1 ∼ χ2mand X2 ∼ χ2
n, and assume that X1 and X2 areindependent,
Z =X1
mX2
n∼ Fm,n
thus Z has an F-distribution with (m, n) degrees of freedom.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 59 / 100
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A Short Review of Probability Theory Famous Distributions
The F Distribution
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 60 / 100
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Causal Inference in Social Science
Causal Inference in Social Science
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 61 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
The Purposes of Empirical Work
To prove or disprove a theory(a relations)“The objective of science is the discovery of the relations”—Lord Kelvin
In most cases,we often want to explore the relationship betweentwo variables in one study.
eg. education and wageThen, in simplicity, there are two relationships between twovariables.
Correlation(相关）V.S. Causality（因果）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 62 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
The Purposes of Empirical Work
To prove or disprove a theory(a relations)“The objective of science is the discovery of the relations”—Lord Kelvin
In most cases,we often want to explore the relationship betweentwo variables in one study.
eg. education and wageThen, in simplicity, there are two relationships between twovariables.
Correlation(相关）V.S. Causality（因果）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 62 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
The Purposes of Empirical Work
To prove or disprove a theory(a relations)“The objective of science is the discovery of the relations”—Lord Kelvin
In most cases,we often want to explore the relationship betweentwo variables in one study.
eg. education and wageThen, in simplicity, there are two relationships between twovariables.
Correlation(相关）V.S. Causality（因果）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 62 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
The Purposes of Empirical Work
To prove or disprove a theory(a relations)“The objective of science is the discovery of the relations”—Lord Kelvin
In most cases,we often want to explore the relationship betweentwo variables in one study.
eg. education and wageThen, in simplicity, there are two relationships between twovariables.
Correlation(相关）V.S. Causality（因果）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 62 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
The Purposes of Empirical Work
To prove or disprove a theory(a relations)“The objective of science is the discovery of the relations”—Lord Kelvin
In most cases,we often want to explore the relationship betweentwo variables in one study.
eg. education and wageThen, in simplicity, there are two relationships between twovariables.
Correlation(相关）V.S. Causality（因果）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 62 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
The Purposes of Empirical Work
To prove or disprove a theory(a relations)“The objective of science is the discovery of the relations”—Lord Kelvin
In most cases,we often want to explore the relationship betweentwo variables in one study.
eg. education and wageThen, in simplicity, there are two relationships between twovariables.
Correlation(相关）V.S. Causality（因果）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 62 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
A Classical Example: Hemline Index（裙边指数）
George Taylor, an economist in the United States, made up thephrase it in the 1920s. The phrase is derived from the idea thathemlines on skirts are shorter or longer depending on theeconomy.
Before 1930s, fashion women favored middle skirts most.In 1929, long skirts became popular. While the Dow Jones IndustrialIndex(DJII) plunged from about 400 to 200 and to 40 two years later.In 1960s, DJII rushed to 1000. At the same time, short skirts showedup.In 1970s, DJII fell to 590 and women began to wear long skirts again.In 1990s, mini skirt debuted, DJII rushed to 10000.In 2000s, bikini became a nice choice for girls, DJII was high up to13000.So what is about now? Long skirt is resorting?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 63 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
A Classical Example: Hemline Index（裙边指数）
George Taylor, an economist in the United States, made up thephrase it in the 1920s. The phrase is derived from the idea thathemlines on skirts are shorter or longer depending on theeconomy.
Before 1930s, fashion women favored middle skirts most.In 1929, long skirts became popular. While the Dow Jones IndustrialIndex(DJII) plunged from about 400 to 200 and to 40 two years later.In 1960s, DJII rushed to 1000. At the same time, short skirts showedup.In 1970s, DJII fell to 590 and women began to wear long skirts again.In 1990s, mini skirt debuted, DJII rushed to 10000.In 2000s, bikini became a nice choice for girls, DJII was high up to13000.So what is about now? Long skirt is resorting?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 63 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
A Classical Example: Hemline Index（裙边指数）
George Taylor, an economist in the United States, made up thephrase it in the 1920s. The phrase is derived from the idea thathemlines on skirts are shorter or longer depending on theeconomy.
Before 1930s, fashion women favored middle skirts most.In 1929, long skirts became popular. While the Dow Jones IndustrialIndex(DJII) plunged from about 400 to 200 and to 40 two years later.In 1960s, DJII rushed to 1000. At the same time, short skirts showedup.In 1970s, DJII fell to 590 and women began to wear long skirts again.In 1990s, mini skirt debuted, DJII rushed to 10000.In 2000s, bikini became a nice choice for girls, DJII was high up to13000.So what is about now? Long skirt is resorting?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 63 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
A Classical Example: Hemline Index（裙边指数）
George Taylor, an economist in the United States, made up thephrase it in the 1920s. The phrase is derived from the idea thathemlines on skirts are shorter or longer depending on theeconomy.
Before 1930s, fashion women favored middle skirts most.In 1929, long skirts became popular. While the Dow Jones IndustrialIndex(DJII) plunged from about 400 to 200 and to 40 two years later.In 1960s, DJII rushed to 1000. At the same time, short skirts showedup.In 1970s, DJII fell to 590 and women began to wear long skirts again.In 1990s, mini skirt debuted, DJII rushed to 10000.In 2000s, bikini became a nice choice for girls, DJII was high up to13000.So what is about now? Long skirt is resorting?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 63 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
A Classical Example: Hemline Index（裙边指数）
George Taylor, an economist in the United States, made up thephrase it in the 1920s. The phrase is derived from the idea thathemlines on skirts are shorter or longer depending on theeconomy.
Before 1930s, fashion women favored middle skirts most.In 1929, long skirts became popular. While the Dow Jones IndustrialIndex(DJII) plunged from about 400 to 200 and to 40 two years later.In 1960s, DJII rushed to 1000. At the same time, short skirts showedup.In 1970s, DJII fell to 590 and women began to wear long skirts again.In 1990s, mini skirt debuted, DJII rushed to 10000.In 2000s, bikini became a nice choice for girls, DJII was high up to13000.So what is about now? Long skirt is resorting?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 63 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
A Classical Example: Hemline Index（裙边指数）
George Taylor, an economist in the United States, made up thephrase it in the 1920s. The phrase is derived from the idea thathemlines on skirts are shorter or longer depending on theeconomy.
Before 1930s, fashion women favored middle skirts most.In 1929, long skirts became popular. While the Dow Jones IndustrialIndex(DJII) plunged from about 400 to 200 and to 40 two years later.In 1960s, DJII rushed to 1000. At the same time, short skirts showedup.In 1970s, DJII fell to 590 and women began to wear long skirts again.In 1990s, mini skirt debuted, DJII rushed to 10000.In 2000s, bikini became a nice choice for girls, DJII was high up to13000.So what is about now? Long skirt is resorting?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 63 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
A Classical Example: Hemline Index（裙边指数）
George Taylor, an economist in the United States, made up thephrase it in the 1920s. The phrase is derived from the idea thathemlines on skirts are shorter or longer depending on theeconomy.
Before 1930s, fashion women favored middle skirts most.In 1929, long skirts became popular. While the Dow Jones IndustrialIndex(DJII) plunged from about 400 to 200 and to 40 two years later.In 1960s, DJII rushed to 1000. At the same time, short skirts showedup.In 1970s, DJII fell to 590 and women began to wear long skirts again.In 1990s, mini skirt debuted, DJII rushed to 10000.In 2000s, bikini became a nice choice for girls, DJII was high up to13000.So what is about now? Long skirt is resorting?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 63 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
A Classical Example: Hemline Index（裙边指数）
George Taylor, an economist in the United States, made up thephrase it in the 1920s. The phrase is derived from the idea thathemlines on skirts are shorter or longer depending on theeconomy.
Before 1930s, fashion women favored middle skirts most.In 1929, long skirts became popular. While the Dow Jones IndustrialIndex(DJII) plunged from about 400 to 200 and to 40 two years later.In 1960s, DJII rushed to 1000. At the same time, short skirts showedup.In 1970s, DJII fell to 590 and women began to wear long skirts again.In 1990s, mini skirt debuted, DJII rushed to 10000.In 2000s, bikini became a nice choice for girls, DJII was high up to13000.So what is about now? Long skirt is resorting?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 63 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Hemline Index:1920s-2010s
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 64 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Some Big Data researchers think causality is not important anymore in our times..“Look at correlations. Look at the ’what’ rather than the
’why’, because that is often good enough.”-ViktorMayer-Schonberger(2013)
Most empirical economists think that correlation only tell us thesuperficial, even false relationship while causal relationship canprovide solid evidence to make interference to the realrelationship.
Today, empirical economists care more about the causalrelationship of their interests than ever before.“the most interesting and challenging research in social
science is about cause and effect”——Angrist andLavy(2008)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 65 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Some Big Data researchers think causality is not important anymore in our times..“Look at correlations. Look at the ’what’ rather than the
’why’, because that is often good enough.”-ViktorMayer-Schonberger(2013)
Most empirical economists think that correlation only tell us thesuperficial, even false relationship while causal relationship canprovide solid evidence to make interference to the realrelationship.
Today, empirical economists care more about the causalrelationship of their interests than ever before.“the most interesting and challenging research in social
science is about cause and effect”——Angrist andLavy(2008)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 65 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Some Big Data researchers think causality is not important anymore in our times..“Look at correlations. Look at the ’what’ rather than the
’why’, because that is often good enough.”-ViktorMayer-Schonberger(2013)
Most empirical economists think that correlation only tell us thesuperficial, even false relationship while causal relationship canprovide solid evidence to make interference to the realrelationship.
Today, empirical economists care more about the causalrelationship of their interests than ever before.“the most interesting and challenging research in social
science is about cause and effect”——Angrist andLavy(2008)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 65 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Some Big Data researchers think causality is not important anymore in our times..“Look at correlations. Look at the ’what’ rather than the
’why’, because that is often good enough.”-ViktorMayer-Schonberger(2013)
Most empirical economists think that correlation only tell us thesuperficial, even false relationship while causal relationship canprovide solid evidence to make interference to the realrelationship.
Today, empirical economists care more about the causalrelationship of their interests than ever before.“the most interesting and challenging research in social
science is about cause and effect”——Angrist andLavy(2008)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 65 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Some Big Data researchers think causality is not important anymore in our times..“Look at correlations. Look at the ’what’ rather than the
’why’, because that is often good enough.”-ViktorMayer-Schonberger(2013)
Most empirical economists think that correlation only tell us thesuperficial, even false relationship while causal relationship canprovide solid evidence to make interference to the realrelationship.
Today, empirical economists care more about the causalrelationship of their interests than ever before.“the most interesting and challenging research in social
science is about cause and effect”——Angrist andLavy(2008)
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 65 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Machine learning is a set of data-driven algorithms that usedata to predict or classify some variable Y as a function of othervariables X.
There are many machine learning algorithm. The bestmethods vary with the particular data application
Machine learning is mostly about prediction.Having a good prediction does work sometimes but doesNOT mean understanding causality.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 66 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Machine learning is a set of data-driven algorithms that usedata to predict or classify some variable Y as a function of othervariables X.
There are many machine learning algorithm. The bestmethods vary with the particular data application
Machine learning is mostly about prediction.Having a good prediction does work sometimes but doesNOT mean understanding causality.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 66 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Machine learning is a set of data-driven algorithms that usedata to predict or classify some variable Y as a function of othervariables X.
There are many machine learning algorithm. The bestmethods vary with the particular data application
Machine learning is mostly about prediction.Having a good prediction does work sometimes but doesNOT mean understanding causality.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 66 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Machine learning is a set of data-driven algorithms that usedata to predict or classify some variable Y as a function of othervariables X.
There are many machine learning algorithm. The bestmethods vary with the particular data application
Machine learning is mostly about prediction.Having a good prediction does work sometimes but doesNOT mean understanding causality.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 66 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Even though forecasting need not involve causal relationships,economic theory suggests patterns and relationships that mightbe useful for forecasting.
Econometric analysis(times series) allows us to quantifyhistorical relationships suggested by economic theory, tocheck whether those relationships have been stable overtime, to make quantitative forecasts about the future, and toassess the accuracy of those forecasts.
The biggest difference between machine learning andeconometrics(or causal inference).
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 67 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Even though forecasting need not involve causal relationships,economic theory suggests patterns and relationships that mightbe useful for forecasting.
Econometric analysis(times series) allows us to quantifyhistorical relationships suggested by economic theory, tocheck whether those relationships have been stable overtime, to make quantitative forecasts about the future, and toassess the accuracy of those forecasts.
The biggest difference between machine learning andeconometrics(or causal inference).
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 67 / 100
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Causal Inference in Social Science The Core of Empirical Studies: Causal Inference
Causality v.s. Forecasting
Even though forecasting need not involve causal relationships,economic theory suggests patterns and relationships that mightbe useful for forecasting.
Econometric analysis(times series) allows us to quantifyhistorical relationships suggested by economic theory, tocheck whether those relationships have been stable overtime, to make quantitative forecasts about the future, and toassess the accuracy of those forecasts.
The biggest difference between machine learning andeconometrics(or causal inference).
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 67 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(I)
A simple example: Do hospitals make people healthier? (Q:Dependent variable and Independent variable?)A naive solution: compare the health status of those who havebeen to the hospital to the health of those who have not.Two key questions are documented by the questionnaires（问卷）from The National Health Interview Survey(NHIS)
1“During the past 12 months, was the respondent a patient ina hospital overnight?”
2“Would you say your health in general is excellent, verygood, good ,fair and poor”and scale it from the number“1”to “5”respectively.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 68 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(I)
A simple example: Do hospitals make people healthier? (Q:Dependent variable and Independent variable?)A naive solution: compare the health status of those who havebeen to the hospital to the health of those who have not.Two key questions are documented by the questionnaires（问卷）from The National Health Interview Survey(NHIS)
1“During the past 12 months, was the respondent a patient ina hospital overnight?”
2“Would you say your health in general is excellent, verygood, good ,fair and poor”and scale it from the number“1”to “5”respectively.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 68 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(I)
A simple example: Do hospitals make people healthier? (Q:Dependent variable and Independent variable?)A naive solution: compare the health status of those who havebeen to the hospital to the health of those who have not.Two key questions are documented by the questionnaires（问卷）from The National Health Interview Survey(NHIS)
1“During the past 12 months, was the respondent a patient ina hospital overnight?”
2“Would you say your health in general is excellent, verygood, good ,fair and poor”and scale it from the number“1”to “5”respectively.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 68 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(I)
A simple example: Do hospitals make people healthier? (Q:Dependent variable and Independent variable?)A naive solution: compare the health status of those who havebeen to the hospital to the health of those who have not.Two key questions are documented by the questionnaires（问卷）from The National Health Interview Survey(NHIS)
1“During the past 12 months, was the respondent a patient ina hospital overnight?”
2“Would you say your health in general is excellent, verygood, good ,fair and poor”and scale it from the number“1”to “5”respectively.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 68 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(I)
A simple example: Do hospitals make people healthier? (Q:Dependent variable and Independent variable?)A naive solution: compare the health status of those who havebeen to the hospital to the health of those who have not.Two key questions are documented by the questionnaires（问卷）from The National Health Interview Survey(NHIS)
1“During the past 12 months, was the respondent a patient ina hospital overnight?”
2“Would you say your health in general is excellent, verygood, good ,fair and poor”and scale it from the number“1”to “5”respectively.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 68 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(II)Hospital v.s. No Hospital
Group Sample Size Mean Health Status Std.DevHospital 7774 2.79 0.014
No Hospital 90049 2.07 0.003
In favor of the non-hospitalized, WHY?Hospitals not only cure but also hurt people.
1 hospitals are full of other sick people who might infect us2 dangerous machines and chemicals that might hurt us.
More important : people having worse health tends to visithospitals.
This simple case exhibits that it is not easy to answer an causalquestion, so let us formalize an model to show where the problemis.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 69 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(II)Hospital v.s. No Hospital
Group Sample Size Mean Health Status Std.DevHospital 7774 2.79 0.014
No Hospital 90049 2.07 0.003
In favor of the non-hospitalized, WHY?Hospitals not only cure but also hurt people.
1 hospitals are full of other sick people who might infect us2 dangerous machines and chemicals that might hurt us.
More important : people having worse health tends to visithospitals.
This simple case exhibits that it is not easy to answer an causalquestion, so let us formalize an model to show where the problemis.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 69 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(II)Hospital v.s. No Hospital
Group Sample Size Mean Health Status Std.DevHospital 7774 2.79 0.014
No Hospital 90049 2.07 0.003
In favor of the non-hospitalized, WHY?Hospitals not only cure but also hurt people.
1 hospitals are full of other sick people who might infect us2 dangerous machines and chemicals that might hurt us.
More important : people having worse health tends to visithospitals.
This simple case exhibits that it is not easy to answer an causalquestion, so let us formalize an model to show where the problemis.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 69 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(II)Hospital v.s. No Hospital
Group Sample Size Mean Health Status Std.DevHospital 7774 2.79 0.014
No Hospital 90049 2.07 0.003
In favor of the non-hospitalized, WHY?Hospitals not only cure but also hurt people.
1 hospitals are full of other sick people who might infect us2 dangerous machines and chemicals that might hurt us.
More important : people having worse health tends to visithospitals.
This simple case exhibits that it is not easy to answer an causalquestion, so let us formalize an model to show where the problemis.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 69 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(II)Hospital v.s. No Hospital
Group Sample Size Mean Health Status Std.DevHospital 7774 2.79 0.014
No Hospital 90049 2.07 0.003
In favor of the non-hospitalized, WHY?Hospitals not only cure but also hurt people.
1 hospitals are full of other sick people who might infect us2 dangerous machines and chemicals that might hurt us.
More important : people having worse health tends to visithospitals.
This simple case exhibits that it is not easy to answer an causalquestion, so let us formalize an model to show where the problemis.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 69 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(II)Hospital v.s. No Hospital
Group Sample Size Mean Health Status Std.DevHospital 7774 2.79 0.014
No Hospital 90049 2.07 0.003
In favor of the non-hospitalized, WHY?Hospitals not only cure but also hurt people.
1 hospitals are full of other sick people who might infect us2 dangerous machines and chemicals that might hurt us.
More important : people having worse health tends to visithospitals.
This simple case exhibits that it is not easy to answer an causalquestion, so let us formalize an model to show where the problemis.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 69 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(II)Hospital v.s. No Hospital
Group Sample Size Mean Health Status Std.DevHospital 7774 2.79 0.014
No Hospital 90049 2.07 0.003
In favor of the non-hospitalized, WHY?Hospitals not only cure but also hurt people.
1 hospitals are full of other sick people who might infect us2 dangerous machines and chemicals that might hurt us.
More important : people having worse health tends to visithospitals.
This simple case exhibits that it is not easy to answer an causalquestion, so let us formalize an model to show where the problemis.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 69 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(III)
So A right way to answer a causal questions is construct acounterfactual world, thus “What If ....then”, Such asAn example: How much wage premium you can get from collegeattendance(上大学使工资增加多少？）
For any worker, we want to compareWage if he have a college degree (上了大学后的工资）Wage if he had not a college degree （假设没上大学，工作的工资）
Then make a difference. This is the right answer to ourquestion.
Difficulty in Identification: only one state can be observed
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 70 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(III)
So A right way to answer a causal questions is construct acounterfactual world, thus “What If ....then”, Such asAn example: How much wage premium you can get from collegeattendance(上大学使工资增加多少？）
For any worker, we want to compareWage if he have a college degree (上了大学后的工资）Wage if he had not a college degree （假设没上大学，工作的工资）
Then make a difference. This is the right answer to ourquestion.
Difficulty in Identification: only one state can be observed
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 70 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(III)
So A right way to answer a causal questions is construct acounterfactual world, thus “What If ....then”, Such asAn example: How much wage premium you can get from collegeattendance(上大学使工资增加多少？）
For any worker, we want to compareWage if he have a college degree (上了大学后的工资）Wage if he had not a college degree （假设没上大学，工作的工资）
Then make a difference. This is the right answer to ourquestion.
Difficulty in Identification: only one state can be observed
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 70 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(III)
So A right way to answer a causal questions is construct acounterfactual world, thus “What If ....then”, Such asAn example: How much wage premium you can get from collegeattendance(上大学使工资增加多少？）
For any worker, we want to compareWage if he have a college degree (上了大学后的工资）Wage if he had not a college degree （假设没上大学，工作的工资）
Then make a difference. This is the right answer to ourquestion.
Difficulty in Identification: only one state can be observed
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 70 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(III)
So A right way to answer a causal questions is construct acounterfactual world, thus “What If ....then”, Such asAn example: How much wage premium you can get from collegeattendance(上大学使工资增加多少？）
For any worker, we want to compareWage if he have a college degree (上了大学后的工资）Wage if he had not a college degree （假设没上大学，工作的工资）
Then make a difference. This is the right answer to ourquestion.
Difficulty in Identification: only one state can be observed
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 70 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(III)
So A right way to answer a causal questions is construct acounterfactual world, thus “What If ....then”, Such asAn example: How much wage premium you can get from collegeattendance(上大学使工资增加多少？）
For any worker, we want to compareWage if he have a college degree (上了大学后的工资）Wage if he had not a college degree （假设没上大学，工作的工资）
Then make a difference. This is the right answer to ourquestion.
Difficulty in Identification: only one state can be observed
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 70 / 100
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Causal Inference in Social Science The Central Question of Causality
The Central Question of Causality(III)
So A right way to answer a causal questions is construct acounterfactual world, thus “What If ....then”, Such asAn example: How much wage premium you can get from collegeattendance(上大学使工资增加多少？）
For any worker, we want to compareWage if he have a college degree (上了大学后的工资）Wage if he had not a college degree （假设没上大学，工作的工资）
Then make a difference. This is the right answer to ourquestion.
Difficulty in Identification: only one state can be observed
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 70 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Treatment : Di is a dummy that indicate whether individual ireceive treatment or not
Di =
1 if individual i received the treatment0 otherwise
Examples:Go to college or notHave health insurance or notJoin a training program or notMake an online-advertisement or not....
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 71 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Treatment : Di is a dummy that indicate whether individual ireceive treatment or not
Di =
1 if individual i received the treatment0 otherwise
Examples:Go to college or notHave health insurance or notJoin a training program or notMake an online-advertisement or not....
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 71 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Treatment : Di is a dummy that indicate whether individual ireceive treatment or not
Di =
1 if individual i received the treatment0 otherwise
Examples:Go to college or notHave health insurance or notJoin a training program or notMake an online-advertisement or not....
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 71 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Treatment : Di is a dummy that indicate whether individual ireceive treatment or not
Di =
1 if individual i received the treatment0 otherwise
Examples:Go to college or notHave health insurance or notJoin a training program or notMake an online-advertisement or not....
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 71 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Treatment : Di is a dummy that indicate whether individual ireceive treatment or not
Di =
1 if individual i received the treatment0 otherwise
Examples:Go to college or notHave health insurance or notJoin a training program or notMake an online-advertisement or not....
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 71 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Treatment : Di is a dummy that indicate whether individual ireceive treatment or not
Di =
1 if individual i received the treatment0 otherwise
Examples:Go to college or notHave health insurance or notJoin a training program or notMake an online-advertisement or not....
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 71 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Treatment : Di is a dummy that indicate whether individual ireceive treatment or not
Di =
1 if individual i received the treatment0 otherwise
Examples:Go to college or notHave health insurance or notJoin a training program or notMake an online-advertisement or not....
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 71 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Treatment
Treatment : Di can be a multiple valued(continues) variable
Di = s
Examples:Schooling yearsNumber of ChildrenNumber of advertisementsMoney Supply
For simplicity, we assume treatment variable Di is just a dummy.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 72 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Treatment
Treatment : Di can be a multiple valued(continues) variable
Di = s
Examples:Schooling yearsNumber of ChildrenNumber of advertisementsMoney Supply
For simplicity, we assume treatment variable Di is just a dummy.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 72 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Treatment
Treatment : Di can be a multiple valued(continues) variable
Di = s
Examples:Schooling yearsNumber of ChildrenNumber of advertisementsMoney Supply
For simplicity, we assume treatment variable Di is just a dummy.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 72 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Treatment
Treatment : Di can be a multiple valued(continues) variable
Di = s
Examples:Schooling yearsNumber of ChildrenNumber of advertisementsMoney Supply
For simplicity, we assume treatment variable Di is just a dummy.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 72 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Treatment
Treatment : Di can be a multiple valued(continues) variable
Di = s
Examples:Schooling yearsNumber of ChildrenNumber of advertisementsMoney Supply
For simplicity, we assume treatment variable Di is just a dummy.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 72 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Treatment
Treatment : Di can be a multiple valued(continues) variable
Di = s
Examples:Schooling yearsNumber of ChildrenNumber of advertisementsMoney Supply
For simplicity, we assume treatment variable Di is just a dummy.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 72 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Treatment
Treatment : Di can be a multiple valued(continues) variable
Di = s
Examples:Schooling yearsNumber of ChildrenNumber of advertisementsMoney Supply
For simplicity, we assume treatment variable Di is just a dummy.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 72 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Potential Outcomes
A potential outcome is the outcome that would be realized if theindividual received a specific value of the treatment.
Annual earnings if attending to collegeAnnual earnings if not attending to college
For each individual, we has two potential outcomes,Y1i and Y0i,one for each value of the treatment
Y1i : Potential outcome for an individual i with treatment.Y0i : Potential outcome for an individual i with treatment.
Potential Outcomes =
Y1i if Di = 1
Y0i if Di = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 73 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Potential Outcomes
A potential outcome is the outcome that would be realized if theindividual received a specific value of the treatment.
Annual earnings if attending to collegeAnnual earnings if not attending to college
For each individual, we has two potential outcomes,Y1i and Y0i,one for each value of the treatment
Y1i : Potential outcome for an individual i with treatment.Y0i : Potential outcome for an individual i with treatment.
Potential Outcomes =
Y1i if Di = 1
Y0i if Di = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 73 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Potential Outcomes
A potential outcome is the outcome that would be realized if theindividual received a specific value of the treatment.
Annual earnings if attending to collegeAnnual earnings if not attending to college
For each individual, we has two potential outcomes,Y1i and Y0i,one for each value of the treatment
Y1i : Potential outcome for an individual i with treatment.Y0i : Potential outcome for an individual i with treatment.
Potential Outcomes =
Y1i if Di = 1
Y0i if Di = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 73 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Potential Outcomes
A potential outcome is the outcome that would be realized if theindividual received a specific value of the treatment.
Annual earnings if attending to collegeAnnual earnings if not attending to college
For each individual, we has two potential outcomes,Y1i and Y0i,one for each value of the treatment
Y1i : Potential outcome for an individual i with treatment.Y0i : Potential outcome for an individual i with treatment.
Potential Outcomes =
Y1i if Di = 1
Y0i if Di = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 73 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Potential Outcomes
A potential outcome is the outcome that would be realized if theindividual received a specific value of the treatment.
Annual earnings if attending to collegeAnnual earnings if not attending to college
For each individual, we has two potential outcomes,Y1i and Y0i,one for each value of the treatment
Y1i : Potential outcome for an individual i with treatment.Y0i : Potential outcome for an individual i with treatment.
Potential Outcomes =
Y1i if Di = 1
Y0i if Di = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 73 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Potential Outcomes
A potential outcome is the outcome that would be realized if theindividual received a specific value of the treatment.
Annual earnings if attending to collegeAnnual earnings if not attending to college
For each individual, we has two potential outcomes,Y1i and Y0i,one for each value of the treatment
Y1i : Potential outcome for an individual i with treatment.Y0i : Potential outcome for an individual i with treatment.
Potential Outcomes =
Y1i if Di = 1
Y0i if Di = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 73 / 100
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Causal Inference in Social Science Rubin Causal Model
Stable Unit Treatment Value Assumption (SUTVA)
Observed outcomes are realized as
Yi = Y1iDi + Y0i(1− Di)
Implies that potential outcomes for an individual i are unaffectedby the treatment status of other individual jIndividual j ’s potential outcomes are only affected by his/herown treatment.Rules out possible treatment effect from other individuals(spillover effect/externality)
ContagionDisplacement
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 74 / 100
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Causal Inference in Social Science Rubin Causal Model
Stable Unit Treatment Value Assumption (SUTVA)
Observed outcomes are realized as
Yi = Y1iDi + Y0i(1− Di)
Implies that potential outcomes for an individual i are unaffectedby the treatment status of other individual jIndividual j ’s potential outcomes are only affected by his/herown treatment.Rules out possible treatment effect from other individuals(spillover effect/externality)
ContagionDisplacement
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 74 / 100
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Causal Inference in Social Science Rubin Causal Model
Stable Unit Treatment Value Assumption (SUTVA)
Observed outcomes are realized as
Yi = Y1iDi + Y0i(1− Di)
Implies that potential outcomes for an individual i are unaffectedby the treatment status of other individual jIndividual j ’s potential outcomes are only affected by his/herown treatment.Rules out possible treatment effect from other individuals(spillover effect/externality)
ContagionDisplacement
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 74 / 100
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Causal Inference in Social Science Rubin Causal Model
Stable Unit Treatment Value Assumption (SUTVA)
Observed outcomes are realized as
Yi = Y1iDi + Y0i(1− Di)
Implies that potential outcomes for an individual i are unaffectedby the treatment status of other individual jIndividual j ’s potential outcomes are only affected by his/herown treatment.Rules out possible treatment effect from other individuals(spillover effect/externality)
ContagionDisplacement
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 74 / 100
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Causal Inference in Social Science Rubin Causal Model
Stable Unit Treatment Value Assumption (SUTVA)
Observed outcomes are realized as
Yi = Y1iDi + Y0i(1− Di)
Implies that potential outcomes for an individual i are unaffectedby the treatment status of other individual jIndividual j ’s potential outcomes are only affected by his/herown treatment.Rules out possible treatment effect from other individuals(spillover effect/externality)
ContagionDisplacement
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 74 / 100
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Causal Inference in Social Science Rubin Causal Model
Stable Unit Treatment Value Assumption (SUTVA)
Observed outcomes are realized as
Yi = Y1iDi + Y0i(1− Di)
Implies that potential outcomes for an individual i are unaffectedby the treatment status of other individual jIndividual j ’s potential outcomes are only affected by his/herown treatment.Rules out possible treatment effect from other individuals(spillover effect/externality)
ContagionDisplacement
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 74 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal effect for an Individual
To know the difference between Y1i and Y0i, which can be said tobe the causal effect of going to college for individual i. (Do youagree with it?)
DefinitionCausal inference is the process of estimating a comparison ofcounterfactuals under different treatment conditions on the same setof units. It also call Individual Treatment Effect(ICE)
δi = Y1i − Y0i
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 75 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal effect for an Individual
To know the difference between Y1i and Y0i, which can be said tobe the causal effect of going to college for individual i. (Do youagree with it?)
DefinitionCausal inference is the process of estimating a comparison ofcounterfactuals under different treatment conditions on the same setof units. It also call Individual Treatment Effect(ICE)
δi = Y1i − Y0i
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 75 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Estimate ICE
Due to unobserved counterfactual outcome, we need to makestrong assumptions to estimate ICE.
Rule out that the ICE differs across individuals(heterogeneity effect)
Knowing individual effect is not our final goal. As a socialscientist, we would like more to know the average effect as asocial pattern.So it make us focus on the average wage for a group of people.
How can we get the average wage premium for collegeattendance?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 76 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Estimate ICE
Due to unobserved counterfactual outcome, we need to makestrong assumptions to estimate ICE.
Rule out that the ICE differs across individuals(heterogeneity effect)
Knowing individual effect is not our final goal. As a socialscientist, we would like more to know the average effect as asocial pattern.So it make us focus on the average wage for a group of people.
How can we get the average wage premium for collegeattendance?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 76 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Estimate ICE
Due to unobserved counterfactual outcome, we need to makestrong assumptions to estimate ICE.
Rule out that the ICE differs across individuals(heterogeneity effect)
Knowing individual effect is not our final goal. As a socialscientist, we would like more to know the average effect as asocial pattern.So it make us focus on the average wage for a group of people.
How can we get the average wage premium for collegeattendance?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 76 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Estimate ICE
Due to unobserved counterfactual outcome, we need to makestrong assumptions to estimate ICE.
Rule out that the ICE differs across individuals(heterogeneity effect)
Knowing individual effect is not our final goal. As a socialscientist, we would like more to know the average effect as asocial pattern.So it make us focus on the average wage for a group of people.
How can we get the average wage premium for collegeattendance?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 76 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Estimate ICE
Due to unobserved counterfactual outcome, we need to makestrong assumptions to estimate ICE.
Rule out that the ICE differs across individuals(heterogeneity effect)
Knowing individual effect is not our final goal. As a socialscientist, we would like more to know the average effect as asocial pattern.So it make us focus on the average wage for a group of people.
How can we get the average wage premium for collegeattendance?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 76 / 100
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Causal Inference in Social Science Rubin Causal Model
Conditional Expectation
Expectation: We usually use E[Yi] (the expectation of avariable Yi) to denote population average of Yi
Suppose we have a population with N individuals
E[Yi] =1
NΣNi=1Yi
Conditional Expectation:The average wage for those who attend college: E[Yi|Di = 1]The average wage for those who did not attend college: E[Yi|Di = 0]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 77 / 100
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Causal Inference in Social Science Rubin Causal Model
Conditional Expectation
Expectation: We usually use E[Yi] (the expectation of avariable Yi) to denote population average of Yi
Suppose we have a population with N individuals
E[Yi] =1
NΣNi=1Yi
Conditional Expectation:The average wage for those who attend college: E[Yi|Di = 1]The average wage for those who did not attend college: E[Yi|Di = 0]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 77 / 100
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Causal Inference in Social Science Rubin Causal Model
Conditional Expectation
Expectation: We usually use E[Yi] (the expectation of avariable Yi) to denote population average of Yi
Suppose we have a population with N individuals
E[Yi] =1
NΣNi=1Yi
Conditional Expectation:The average wage for those who attend college: E[Yi|Di = 1]The average wage for those who did not attend college: E[Yi|Di = 0]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 77 / 100
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Causal Inference in Social Science Rubin Causal Model
Conditional Expectation
Expectation: We usually use E[Yi] (the expectation of avariable Yi) to denote population average of Yi
Suppose we have a population with N individuals
E[Yi] =1
NΣNi=1Yi
Conditional Expectation:The average wage for those who attend college: E[Yi|Di = 1]The average wage for those who did not attend college: E[Yi|Di = 0]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 77 / 100
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Causal Inference in Social Science Rubin Causal Model
Conditional Expectation
Expectation: We usually use E[Yi] (the expectation of avariable Yi) to denote population average of Yi
Suppose we have a population with N individuals
E[Yi] =1
NΣNi=1Yi
Conditional Expectation:The average wage for those who attend college: E[Yi|Di = 1]The average wage for those who did not attend college: E[Yi|Di = 0]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 77 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Causal Effects
Average Treatment Effect (ATE)
αATE = E[δi] = E[Y1i − Y0i]
It is average of ICEs over the population.
Average treatment effect on the treated(ATT)
αATT = E[δi|Di = 1] = E[Y1i − Y0i|Di = 1]
Average of ICEs over the treated population
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 78 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Causal Effects
Average Treatment Effect (ATE)
αATE = E[δi] = E[Y1i − Y0i]
It is average of ICEs over the population.
Average treatment effect on the treated(ATT)
αATT = E[δi|Di = 1] = E[Y1i − Y0i|Di = 1]
Average of ICEs over the treated population
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 78 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Causal Effects
Average Treatment Effect (ATE)
αATE = E[δi] = E[Y1i − Y0i]
It is average of ICEs over the population.
Average treatment effect on the treated(ATT)
αATT = E[δi|Di = 1] = E[Y1i − Y0i|Di = 1]
Average of ICEs over the treated population
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 78 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Causal Effects
Average Treatment Effect (ATE)
αATE = E[δi] = E[Y1i − Y0i]
It is average of ICEs over the population.
Average treatment effect on the treated(ATT)
αATT = E[δi|Di = 1] = E[Y1i − Y0i|Di = 1]
Average of ICEs over the treated population
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 78 / 100
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Causal Inference in Social Science Rubin Causal Model
Fundamental Problem of Causal Inference
We can never directly observe causal effects (ICE, ATE or ATT)Because we can never observe both potential outcomes (Y0i,Y1i)for any individual.We need to compare potential outcomes, but we only haveobserved outcomesSo by this view, causal inference is a missing data problem.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 79 / 100
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Causal Inference in Social Science Rubin Causal Model
Fundamental Problem of Causal Inference
We can never directly observe causal effects (ICE, ATE or ATT)Because we can never observe both potential outcomes (Y0i,Y1i)for any individual.We need to compare potential outcomes, but we only haveobserved outcomesSo by this view, causal inference is a missing data problem.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 79 / 100
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Causal Inference in Social Science Rubin Causal Model
Fundamental Problem of Causal Inference
We can never directly observe causal effects (ICE, ATE or ATT)Because we can never observe both potential outcomes (Y0i,Y1i)for any individual.We need to compare potential outcomes, but we only haveobserved outcomesSo by this view, causal inference is a missing data problem.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 79 / 100
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Causal Inference in Social Science Rubin Causal Model
Fundamental Problem of Causal Inference
We can never directly observe causal effects (ICE, ATE or ATT)Because we can never observe both potential outcomes (Y0i,Y1i)for any individual.We need to compare potential outcomes, but we only haveobserved outcomesSo by this view, causal inference is a missing data problem.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 79 / 100
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Causal Inference in Social Science Rubin Causal Model
Fundamental Problem of Causal Inference
Imagine a population with 4 people
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 ? 3 1 ?Jerry 2 ? 2 1 ?
Scarlett ? 1 1 0 ?Nicole ? 1 1 0 ?
What is Individual causal effect (ICE) of attending college forTom? for Nicole?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 80 / 100
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Causal Inference in Social Science Rubin Causal Model
Fundamental Problem of Causal Inference
Imagine a population with 4 people
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 ? 3 1 ?Jerry 2 ? 2 1 ?
Scarlett ? 1 1 0 ?Nicole ? 1 1 0 ?
What is Individual causal effect (ICE) of attending college forTom? for Nicole?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 80 / 100
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Causal Inference in Social Science Rubin Causal Model
Individual Causal Effect
Suppose we can observe counterfactual outcomes
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 2 3 1 1Jerry 2 1 2 1 1
Scarlett 1 1 1 0 0Nicole 1 1 1 0 0
The ICE for TomδTom = 3− 2 = 11
The ICE for NicoleδNicole = 1− 1 = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 81 / 100
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Causal Inference in Social Science Rubin Causal Model
Individual Causal Effect
Suppose we can observe counterfactual outcomes
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 2 3 1 1Jerry 2 1 2 1 1
Scarlett 1 1 1 0 0Nicole 1 1 1 0 0
The ICE for TomδTom = 3− 2 = 11
The ICE for NicoleδNicole = 1− 1 = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 81 / 100
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Causal Inference in Social Science Rubin Causal Model
Individual Causal Effect
Suppose we can observe counterfactual outcomes
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 2 3 1 1Jerry 2 1 2 1 1
Scarlett 1 1 1 0 0Nicole 1 1 1 0 0
The ICE for TomδTom = 3− 2 = 11
The ICE for NicoleδNicole = 1− 1 = 0
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 81 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Treatment Effect(ATE)Missing data problem also arises when we estimate ATE
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 ? 3 1 ?Jerry 2 ? 2 1 ?
Scarlett ? 1 1 0 ?Nicole ? 1 1 0 ?E[Y1i] ?E[Y0i] ?
E[Y1i − Y0i] ?
What is the effect of attending college on average wage ofpopulation(ATE)
αATE = E[δi] = E[Y1i − Y0i]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 82 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Treatment Effect(ATE)Missing data problem also arises when we estimate ATE
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 ? 3 1 ?Jerry 2 ? 2 1 ?
Scarlett ? 1 1 0 ?Nicole ? 1 1 0 ?E[Y1i] ?E[Y0i] ?
E[Y1i − Y0i] ?
What is the effect of attending college on average wage ofpopulation(ATE)
αATE = E[δi] = E[Y1i − Y0i]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 82 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Treatment Effect(ATE)Missing data problem also arises when we estimate ATE
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 2 3 1 1Jerry 2 1 2 1 1
Scarlett 1 1 1 0 0Nicole 1 1 1 0 0E[Y1i]
3+2+1+14 = 1.75
E[Y0i]2+1+1+1
4 = 1.25
E[Y1i − Y0i] 0.5
What is the effect of attending college on average wage of thepopulation(ATE)
αATE = E[δi] = E[Y1i − Y0i] =1 + 1 + 0 + 0
4= 0.5
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 83 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Treatment Effect(ATE)Missing data problem also arises when we estimate ATE
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 2 3 1 1Jerry 2 1 2 1 1
Scarlett 1 1 1 0 0Nicole 1 1 1 0 0E[Y1i]
3+2+1+14 = 1.75
E[Y0i]2+1+1+1
4 = 1.25
E[Y1i − Y0i] 0.5
What is the effect of attending college on average wage of thepopulation(ATE)
αATE = E[δi] = E[Y1i − Y0i] =1 + 1 + 0 + 0
4= 0.5
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 83 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Treatment Effect on the Treated(ATT)Missing data problem arises when we estimate ATT
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 ? 3 1 ?Jerry 2 ? 2 1 ?
Scarlett ? 1 1 0 ?Nicole ? 1 1 0 ?
E[Y1i|Di = 1] ?E[Y0i|Di = 1] ?
E[Y1i − Y0i|Di = 1] ?
What is the effect of attending college on average wage for thosewho attend college(ATT)
αATE = E[δi] = E[Y1i − Y0i|Di = 1]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 84 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Treatment Effect on the Treated(ATT)Missing data problem arises when we estimate ATT
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 ? 3 1 ?Jerry 2 ? 2 1 ?
Scarlett ? 1 1 0 ?Nicole ? 1 1 0 ?
E[Y1i|Di = 1] ?E[Y0i|Di = 1] ?
E[Y1i − Y0i|Di = 1] ?
What is the effect of attending college on average wage for thosewho attend college(ATT)
αATE = E[δi] = E[Y1i − Y0i|Di = 1]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 84 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Treatment Effect on the Treated(ATT)Missing data problem also arises when we estimate ATE
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 2 3 1 1Jerry 2 1 2 1 1
Scarlett 1 1 1 0 0Nicole 1 1 1 0 0
E[Y1i|Di = 1] 3+22 = 2.5
E[Y0i|Di = 1] 2+12 = 1.5
E[Y1i − Y0i|Di = 1] 1
The effect of attending college on average wage for those whoattend college(ATT)
αATE = E[Y1i − Y0i|Di = 1] =1 + 1
2= 1
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 85 / 100
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Causal Inference in Social Science Rubin Causal Model
Average Treatment Effect on the Treated(ATT)Missing data problem also arises when we estimate ATE
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 2 3 1 1Jerry 2 1 2 1 1
Scarlett 1 1 1 0 0Nicole 1 1 1 0 0
E[Y1i|Di = 1] 3+22 = 2.5
E[Y0i|Di = 1] 2+12 = 1.5
E[Y1i − Y0i|Di = 1] 1
The effect of attending college on average wage for those whoattend college(ATT)
αATE = E[Y1i − Y0i|Di = 1] =1 + 1
2= 1
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 85 / 100
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Causal Inference in Social Science Rubin Causal Model
Observed Association and Selection Bias
Causality is defined by potential outcomes, not by realized(observed) outcomes.In fact, we can not observe all potential outcomes .Therefore, wecan not estimate the above causal effects without furtherassumptions.By using observed data, we can only establish association(correlation), which is the observed difference in averageoutcome between those getting treatment and those not gettingtreatment.
αcorr = E[Y1i|Di = 1]− E[Y0i|Di = 0]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 86 / 100
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Causal Inference in Social Science Rubin Causal Model
Observed Association and Selection Bias
Causality is defined by potential outcomes, not by realized(observed) outcomes.In fact, we can not observe all potential outcomes .Therefore, wecan not estimate the above causal effects without furtherassumptions.By using observed data, we can only establish association(correlation), which is the observed difference in averageoutcome between those getting treatment and those not gettingtreatment.
αcorr = E[Y1i|Di = 1]− E[Y0i|Di = 0]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 86 / 100
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Causal Inference in Social Science Rubin Causal Model
Observed Association and Selection Bias
Causality is defined by potential outcomes, not by realized(observed) outcomes.In fact, we can not observe all potential outcomes .Therefore, wecan not estimate the above causal effects without furtherassumptions.By using observed data, we can only establish association(correlation), which is the observed difference in averageoutcome between those getting treatment and those not gettingtreatment.
αcorr = E[Y1i|Di = 1]− E[Y0i|Di = 0]
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 86 / 100
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Causal Inference in Social Science Rubin Causal Model
College vs Non-College Wage Differentials:
Comparing the average wage in labor market who went to collegeand did not go.
College vs Non-College Wage Differentials:
=E[Y1i|Di = 1]− E[Y0i|Di = 0]
=E[Y1i|Di = 1]−E[Y0i|Di = 1]+ E[Y0i|Di = 1]− E[Y0i|Di = 0]
Question 1: Which one defines the causal effect of collegeattendance?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 87 / 100
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Causal Inference in Social Science Rubin Causal Model
College vs Non-College Wage Differentials:
Comparing the average wage in labor market who went to collegeand did not go.
College vs Non-College Wage Differentials:
=E[Y1i|Di = 1]− E[Y0i|Di = 0]
=E[Y1i|Di = 1]−E[Y0i|Di = 1]+ E[Y0i|Di = 1]− E[Y0i|Di = 0]
Question 1: Which one defines the causal effect of collegeattendance?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 87 / 100
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Causal Inference in Social Science Rubin Causal Model
College vs Non-College Wage Differentials:
Comparing the average wage in labor market who went to collegeand did not go.
College vs Non-College Wage Differentials:
=E[Y1i|Di = 1]− E[Y0i|Di = 0]
=E[Y1i|Di = 1]−E[Y0i|Di = 1]+ E[Y0i|Di = 1]− E[Y0i|Di = 0]
Question 1: Which one defines the causal effect of collegeattendance?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 87 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Selection Bias(SB) implies the potential outcomes oftreatment and control groups are different even if both groupsreceive the same treatment
E[Y0i|Di = 1]− E[Y0i|Di = 0]
Question 2: Selection Bias is positive or negative in the case?This means two groups could be quite different in otherdimensions: other things are not equal.Observed association is neither necessary nor sufficient forcausality.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 88 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Selection Bias(SB) implies the potential outcomes oftreatment and control groups are different even if both groupsreceive the same treatment
E[Y0i|Di = 1]− E[Y0i|Di = 0]
Question 2: Selection Bias is positive or negative in the case?This means two groups could be quite different in otherdimensions: other things are not equal.Observed association is neither necessary nor sufficient forcausality.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 88 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Selection Bias(SB) implies the potential outcomes oftreatment and control groups are different even if both groupsreceive the same treatment
E[Y0i|Di = 1]− E[Y0i|Di = 0]
Question 2: Selection Bias is positive or negative in the case?This means two groups could be quite different in otherdimensions: other things are not equal.Observed association is neither necessary nor sufficient forcausality.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 88 / 100
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Causal Inference in Social Science Rubin Causal Model
Formalization: Rubin Causal Model
Selection Bias(SB) implies the potential outcomes oftreatment and control groups are different even if both groupsreceive the same treatment
E[Y0i|Di = 1]− E[Y0i|Di = 0]
Question 2: Selection Bias is positive or negative in the case?This means two groups could be quite different in otherdimensions: other things are not equal.Observed association is neither necessary nor sufficient forcausality.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 88 / 100
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Causal Inference in Social Science Rubin Causal Model
Observed Association:College vs Non-CollegeMissing data problem also arises when we estimate ATE
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 ? 3 1 ?Jerry 2 ? 2 1 ?
Scarlett ? 1 1 0 ?Nicole ? 1 1 0 ?
E[Y1i|Di = 1] 3+22 = 2.5
E[Y0i|Di = 0] 1+12 = 1
E[Y1i|Di = 1]− E[Y0i|Di = 0] 1.5
The Observed Association of attending college on average wageαcorr = 2.5− 1 = 1.5
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 89 / 100
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Causal Inference in Social Science Rubin Causal Model
Observed Association:College vs Non-CollegeMissing data problem also arises when we estimate ATE
i Yi1 Y0i Yi Di Yi1 − Y0iTom 3 ? 3 1 ?Jerry 2 ? 2 1 ?
Scarlett ? 1 1 0 ?Nicole ? 1 1 0 ?
E[Y1i|Di = 1] 3+22 = 2.5
E[Y0i|Di = 0] 1+12 = 1
E[Y1i|Di = 1]− E[Y0i|Di = 0] 1.5
The Observed Association of attending college on average wageαcorr = 2.5− 1 = 1.5
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 89 / 100
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Causal Inference in Social Science Rubin Causal Model
Observed Association and Selection Bias
But we are interested in causal effect, here is ATT
αATT = E[δi|Di = 1] = E[Y1i − Y0i|Di = 1] = 1
So the selection bias
E[Y0i|Di = 1]− E[Y0i|Di = 0] = 0.5
The Selection Bias is positive: Those who attend college couldbe more intelligent so they can earn more even if they did notattend college.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 90 / 100
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Causal Inference in Social Science Rubin Causal Model
Observed Association and Selection Bias
But we are interested in causal effect, here is ATT
αATT = E[δi|Di = 1] = E[Y1i − Y0i|Di = 1] = 1
So the selection bias
E[Y0i|Di = 1]− E[Y0i|Di = 0] = 0.5
The Selection Bias is positive: Those who attend college couldbe more intelligent so they can earn more even if they did notattend college.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 90 / 100
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Causal Inference in Social Science Rubin Causal Model
Observed Association and Selection Bias
But we are interested in causal effect, here is ATT
αATT = E[δi|Di = 1] = E[Y1i − Y0i|Di = 1] = 1
So the selection bias
E[Y0i|Di = 1]− E[Y0i|Di = 0] = 0.5
The Selection Bias is positive: Those who attend college couldbe more intelligent so they can earn more even if they did notattend college.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 90 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal Effect and Identification Strategy
Many Many Other examplesthe effect of job training program on worker’s earningsthe effect of class size on students performance....
Identification strategy tells us what we can learn about a causaleffect from the available data.The main goal of identification strategy is to eliminate theselection bias.Identification depends on assumptions, not on estimationstrategies.“What’s your identification strategy?”= what are theassumptions that allow you to claim you’ve estimated a causaleffect?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 91 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal Effect and Identification Strategy
Many Many Other examplesthe effect of job training program on worker’s earningsthe effect of class size on students performance....
Identification strategy tells us what we can learn about a causaleffect from the available data.The main goal of identification strategy is to eliminate theselection bias.Identification depends on assumptions, not on estimationstrategies.“What’s your identification strategy?”= what are theassumptions that allow you to claim you’ve estimated a causaleffect?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 91 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal Effect and Identification Strategy
Many Many Other examplesthe effect of job training program on worker’s earningsthe effect of class size on students performance....
Identification strategy tells us what we can learn about a causaleffect from the available data.The main goal of identification strategy is to eliminate theselection bias.Identification depends on assumptions, not on estimationstrategies.“What’s your identification strategy?”= what are theassumptions that allow you to claim you’ve estimated a causaleffect?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 91 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal Effect and Identification Strategy
Many Many Other examplesthe effect of job training program on worker’s earningsthe effect of class size on students performance....
Identification strategy tells us what we can learn about a causaleffect from the available data.The main goal of identification strategy is to eliminate theselection bias.Identification depends on assumptions, not on estimationstrategies.“What’s your identification strategy?”= what are theassumptions that allow you to claim you’ve estimated a causaleffect?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 91 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal Effect and Identification Strategy
Many Many Other examplesthe effect of job training program on worker’s earningsthe effect of class size on students performance....
Identification strategy tells us what we can learn about a causaleffect from the available data.The main goal of identification strategy is to eliminate theselection bias.Identification depends on assumptions, not on estimationstrategies.“What’s your identification strategy?”= what are theassumptions that allow you to claim you’ve estimated a causaleffect?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 91 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal Effect and Identification Strategy
Many Many Other examplesthe effect of job training program on worker’s earningsthe effect of class size on students performance....
Identification strategy tells us what we can learn about a causaleffect from the available data.The main goal of identification strategy is to eliminate theselection bias.Identification depends on assumptions, not on estimationstrategies.“What’s your identification strategy?”= what are theassumptions that allow you to claim you’ve estimated a causaleffect?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 91 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal Effect and Identification Strategy
Many Many Other examplesthe effect of job training program on worker’s earningsthe effect of class size on students performance....
Identification strategy tells us what we can learn about a causaleffect from the available data.The main goal of identification strategy is to eliminate theselection bias.Identification depends on assumptions, not on estimationstrategies.“What’s your identification strategy?”= what are theassumptions that allow you to claim you’ve estimated a causaleffect?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 91 / 100
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Causal Inference in Social Science Rubin Causal Model
Causal Effect and Identification Strategy
Many Many Other examplesthe effect of job training program on worker’s earningsthe effect of class size on students performance....
Identification strategy tells us what we can learn about a causaleffect from the available data.The main goal of identification strategy is to eliminate theselection bias.Identification depends on assumptions, not on estimationstrategies.“What’s your identification strategy?”= what are theassumptions that allow you to claim you’ve estimated a causaleffect?
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 91 / 100
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Experimental Design as a Benchmark
Experimental Design as a Benchmark
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 92 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Randomized Controlled Trial
A randomized controlled trial (RCT) is a form of investigation inwhich units of observation (e.g. individuals, households, schools,states) are randomly assigned to treatment and control groups.RCT has two features that can help us hold other things equal andthen eliminates selection bias
Random assign treatment:Randomly assign treatment (such as a coin flip) ensures that everyobservation has the same probability of being assigned to the treatmentgroup.Therefore, the probability of receiving treatment is unrelated to anyother confounding factors.
Sufficient large sampleLarge sample size can ensure that the group differences in individualcharacteristics wash out
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 93 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Randomized Controlled Trial
A randomized controlled trial (RCT) is a form of investigation inwhich units of observation (e.g. individuals, households, schools,states) are randomly assigned to treatment and control groups.RCT has two features that can help us hold other things equal andthen eliminates selection bias
Random assign treatment:Randomly assign treatment (such as a coin flip) ensures that everyobservation has the same probability of being assigned to the treatmentgroup.Therefore, the probability of receiving treatment is unrelated to anyother confounding factors.
Sufficient large sampleLarge sample size can ensure that the group differences in individualcharacteristics wash out
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 93 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Randomized Controlled Trial
A randomized controlled trial (RCT) is a form of investigation inwhich units of observation (e.g. individuals, households, schools,states) are randomly assigned to treatment and control groups.RCT has two features that can help us hold other things equal andthen eliminates selection bias
Random assign treatment:Randomly assign treatment (such as a coin flip) ensures that everyobservation has the same probability of being assigned to the treatmentgroup.Therefore, the probability of receiving treatment is unrelated to anyother confounding factors.
Sufficient large sampleLarge sample size can ensure that the group differences in individualcharacteristics wash out
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 93 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Randomized Controlled Trial
A randomized controlled trial (RCT) is a form of investigation inwhich units of observation (e.g. individuals, households, schools,states) are randomly assigned to treatment and control groups.RCT has two features that can help us hold other things equal andthen eliminates selection bias
Random assign treatment:Randomly assign treatment (such as a coin flip) ensures that everyobservation has the same probability of being assigned to the treatmentgroup.Therefore, the probability of receiving treatment is unrelated to anyother confounding factors.
Sufficient large sampleLarge sample size can ensure that the group differences in individualcharacteristics wash out
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 93 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Randomized Controlled Trial
A randomized controlled trial (RCT) is a form of investigation inwhich units of observation (e.g. individuals, households, schools,states) are randomly assigned to treatment and control groups.RCT has two features that can help us hold other things equal andthen eliminates selection bias
Random assign treatment:Randomly assign treatment (such as a coin flip) ensures that everyobservation has the same probability of being assigned to the treatmentgroup.Therefore, the probability of receiving treatment is unrelated to anyother confounding factors.
Sufficient large sampleLarge sample size can ensure that the group differences in individualcharacteristics wash out
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 93 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Randomized Controlled Trial
A randomized controlled trial (RCT) is a form of investigation inwhich units of observation (e.g. individuals, households, schools,states) are randomly assigned to treatment and control groups.RCT has two features that can help us hold other things equal andthen eliminates selection bias
Random assign treatment:Randomly assign treatment (such as a coin flip) ensures that everyobservation has the same probability of being assigned to the treatmentgroup.Therefore, the probability of receiving treatment is unrelated to anyother confounding factors.
Sufficient large sampleLarge sample size can ensure that the group differences in individualcharacteristics wash out
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 93 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Randomized Controlled Trial
A randomized controlled trial (RCT) is a form of investigation inwhich units of observation (e.g. individuals, households, schools,states) are randomly assigned to treatment and control groups.RCT has two features that can help us hold other things equal andthen eliminates selection bias
Random assign treatment:Randomly assign treatment (such as a coin flip) ensures that everyobservation has the same probability of being assigned to the treatmentgroup.Therefore, the probability of receiving treatment is unrelated to anyother confounding factors.
Sufficient large sampleLarge sample size can ensure that the group differences in individualcharacteristics wash out
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 93 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
How to Solves the Selection Problem
Random assignment of treatment Di can eliminates selectionbias. It means that the treated group is a random sample fromthe population.Being a random sample, we know that those included in thesample are the same, on average, as those not included in thesample on any measure.Mathematically ,it makes Di independent of potentialoutcomes, thus
Di ⊥ (Y0i,Y1i)
Independence: Two variables are said to be independent ifknowing the outcome of one provides no useful information aboutthe outcome of the other.
Knowing outcome of Di(0, 1) does not help us understand whatpotential outcomes of (Y0i,Y1i) will be
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 94 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
How to Solves the Selection Problem
Random assignment of treatment Di can eliminates selectionbias. It means that the treated group is a random sample fromthe population.Being a random sample, we know that those included in thesample are the same, on average, as those not included in thesample on any measure.Mathematically ,it makes Di independent of potentialoutcomes, thus
Di ⊥ (Y0i,Y1i)
Independence: Two variables are said to be independent ifknowing the outcome of one provides no useful information aboutthe outcome of the other.
Knowing outcome of Di(0, 1) does not help us understand whatpotential outcomes of (Y0i,Y1i) will be
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 94 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
How to Solves the Selection Problem
Random assignment of treatment Di can eliminates selectionbias. It means that the treated group is a random sample fromthe population.Being a random sample, we know that those included in thesample are the same, on average, as those not included in thesample on any measure.Mathematically ,it makes Di independent of potentialoutcomes, thus
Di ⊥ (Y0i,Y1i)
Independence: Two variables are said to be independent ifknowing the outcome of one provides no useful information aboutthe outcome of the other.
Knowing outcome of Di(0, 1) does not help us understand whatpotential outcomes of (Y0i,Y1i) will be
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 94 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
How to Solves the Selection Problem
Random assignment of treatment Di can eliminates selectionbias. It means that the treated group is a random sample fromthe population.Being a random sample, we know that those included in thesample are the same, on average, as those not included in thesample on any measure.Mathematically ,it makes Di independent of potentialoutcomes, thus
Di ⊥ (Y0i,Y1i)
Independence: Two variables are said to be independent ifknowing the outcome of one provides no useful information aboutthe outcome of the other.
Knowing outcome of Di(0, 1) does not help us understand whatpotential outcomes of (Y0i,Y1i) will be
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 94 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
How to Solves the Selection Problem
Random assignment of treatment Di can eliminates selectionbias. It means that the treated group is a random sample fromthe population.Being a random sample, we know that those included in thesample are the same, on average, as those not included in thesample on any measure.Mathematically ,it makes Di independent of potentialoutcomes, thus
Di ⊥ (Y0i,Y1i)
Independence: Two variables are said to be independent ifknowing the outcome of one provides no useful information aboutthe outcome of the other.
Knowing outcome of Di(0, 1) does not help us understand whatpotential outcomes of (Y0i,Y1i) will be
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 94 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Random Assignment Solves the Selection Problem
So we haveE[Y0i|Di = 1] = E[Y0i|Di = 0]
Thus the Selection Bias equals to ZERO.Then ATT equals Observed Association because the
E[Y1i|Di = 1]− E[Y0i|Di = 0] = E[Y1i|Di = 1]− E[Y0i|Di = 1]
=E[Y1i − Y0i|Di = 1]
No matter what assumptions we make about the distribution ofY , we can always estimate it with the difference in means.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 95 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Random Assignment Solves the Selection Problem
So we haveE[Y0i|Di = 1] = E[Y0i|Di = 0]
Thus the Selection Bias equals to ZERO.Then ATT equals Observed Association because the
E[Y1i|Di = 1]− E[Y0i|Di = 0] = E[Y1i|Di = 1]− E[Y0i|Di = 1]
=E[Y1i − Y0i|Di = 1]
No matter what assumptions we make about the distribution ofY , we can always estimate it with the difference in means.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 95 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Random Assignment Solves the Selection Problem
So we haveE[Y0i|Di = 1] = E[Y0i|Di = 0]
Thus the Selection Bias equals to ZERO.Then ATT equals Observed Association because the
E[Y1i|Di = 1]− E[Y0i|Di = 0] = E[Y1i|Di = 1]− E[Y0i|Di = 1]
=E[Y1i − Y0i|Di = 1]
No matter what assumptions we make about the distribution ofY , we can always estimate it with the difference in means.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 95 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Random Assignment Solves the Selection Problem
So we haveE[Y0i|Di = 1] = E[Y0i|Di = 0]
Thus the Selection Bias equals to ZERO.Then ATT equals Observed Association because the
E[Y1i|Di = 1]− E[Y0i|Di = 0] = E[Y1i|Di = 1]− E[Y0i|Di = 1]
=E[Y1i − Y0i|Di = 1]
No matter what assumptions we make about the distribution ofY , we can always estimate it with the difference in means.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 95 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Our Benchmark: Randomized Experiments
Think of causal effects in terms of comparing counterfactuals orpotential outcomes. However, we can never observe bothcounterfactuals —fundamental problem of causal inference.To construct the counterfactuals, we could use two broadcategories of empirical strategies.
Random Controlled Trials/Experiments:it can eliminates selection bias which is the mostimportant bias arises in empirical research. If we couldobserve the counterfactual directly, then there is noevaluation problem, just simply difference.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 96 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Our Benchmark: Randomized Experiments
Think of causal effects in terms of comparing counterfactuals orpotential outcomes. However, we can never observe bothcounterfactuals —fundamental problem of causal inference.To construct the counterfactuals, we could use two broadcategories of empirical strategies.
Random Controlled Trials/Experiments:it can eliminates selection bias which is the mostimportant bias arises in empirical research. If we couldobserve the counterfactual directly, then there is noevaluation problem, just simply difference.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 96 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Our Benchmark: Randomized Experiments
Think of causal effects in terms of comparing counterfactuals orpotential outcomes. However, we can never observe bothcounterfactuals —fundamental problem of causal inference.To construct the counterfactuals, we could use two broadcategories of empirical strategies.
Random Controlled Trials/Experiments:it can eliminates selection bias which is the mostimportant bias arises in empirical research. If we couldobserve the counterfactual directly, then there is noevaluation problem, just simply difference.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 96 / 100
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Experimental Design as a Benchmark RCTs Can Solve the Selection Bias
Our Benchmark: Randomized Experiments
Think of causal effects in terms of comparing counterfactuals orpotential outcomes. However, we can never observe bothcounterfactuals —fundamental problem of causal inference.To construct the counterfactuals, we could use two broadcategories of empirical strategies.
Random Controlled Trials/Experiments:it can eliminates selection bias which is the mostimportant bias arises in empirical research. If we couldobserve the counterfactual directly, then there is noevaluation problem, just simply difference.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 96 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
We can generate the data of our interest by controllingexperiments just as physical scientists or biologists do. But tooobviously, we face more difficult and controversy situation thanthose in any other sciences.The various approaches using naturally-occurring data providealternative methods of constructing the proper counterfactual
EconometricsCongratulation! We are working and studying in a more tough andintractable area than others including most science knowledge.
We should take the randomized experimental methods as ourbenchmark when we do empirical research whatever the methodswe apply.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 97 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
We can generate the data of our interest by controllingexperiments just as physical scientists or biologists do. But tooobviously, we face more difficult and controversy situation thanthose in any other sciences.The various approaches using naturally-occurring data providealternative methods of constructing the proper counterfactual
EconometricsCongratulation! We are working and studying in a more tough andintractable area than others including most science knowledge.
We should take the randomized experimental methods as ourbenchmark when we do empirical research whatever the methodswe apply.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 97 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
We can generate the data of our interest by controllingexperiments just as physical scientists or biologists do. But tooobviously, we face more difficult and controversy situation thanthose in any other sciences.The various approaches using naturally-occurring data providealternative methods of constructing the proper counterfactual
EconometricsCongratulation! We are working and studying in a more tough andintractable area than others including most science knowledge.
We should take the randomized experimental methods as ourbenchmark when we do empirical research whatever the methodswe apply.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 97 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
We can generate the data of our interest by controllingexperiments just as physical scientists or biologists do. But tooobviously, we face more difficult and controversy situation thanthose in any other sciences.The various approaches using naturally-occurring data providealternative methods of constructing the proper counterfactual
EconometricsCongratulation! We are working and studying in a more tough andintractable area than others including most science knowledge.
We should take the randomized experimental methods as ourbenchmark when we do empirical research whatever the methodswe apply.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 97 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
We can generate the data of our interest by controllingexperiments just as physical scientists or biologists do. But tooobviously, we face more difficult and controversy situation thanthose in any other sciences.The various approaches using naturally-occurring data providealternative methods of constructing the proper counterfactual
EconometricsCongratulation! We are working and studying in a more tough andintractable area than others including most science knowledge.
We should take the randomized experimental methods as ourbenchmark when we do empirical research whatever the methodswe apply.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 97 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
Question: How to do empirical research scientifically when wecan not do experiments? It means that we always have selectionbias in our data, or in term of endogeneity.Answer: Build a reasonable counterfactual world by naturallyoccurring data to find a proper control group is the core ofeconometrical methods.Here you Furious Seven Weapons in Applied Econometrics(七种盖世武器）
1 Random Controlled Trials(RCT)2 OLS(回归）3 Matching（匹配）4 Decomposition（分解）5 Instrumental Variable（工具变量）6 Regression Discontinuity（断点回归）7 Panel Data, Differences in Differences（双差分) ,Synthetic Control（合成控制法）
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 98 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
These Furious Seven are the most basic and popular methods inapplied econometrics and so powerful that
even if you just master one, you may finish your empiricalpaper and get a good score.if you master several ones, you could have opportunity topublish your paper.If you master all of them, you might to teach appliedeconometrics class just as what I am doing now.
We will introduce essentials of these methods in the class asmany as possible. Let’s start our journey together.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 99 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
These Furious Seven are the most basic and popular methods inapplied econometrics and so powerful that
even if you just master one, you may finish your empiricalpaper and get a good score.if you master several ones, you could have opportunity topublish your paper.If you master all of them, you might to teach appliedeconometrics class just as what I am doing now.
We will introduce essentials of these methods in the class asmany as possible. Let’s start our journey together.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 99 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
These Furious Seven are the most basic and popular methods inapplied econometrics and so powerful that
even if you just master one, you may finish your empiricalpaper and get a good score.if you master several ones, you could have opportunity topublish your paper.If you master all of them, you might to teach appliedeconometrics class just as what I am doing now.
We will introduce essentials of these methods in the class asmany as possible. Let’s start our journey together.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 99 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
These Furious Seven are the most basic and popular methods inapplied econometrics and so powerful that
even if you just master one, you may finish your empiricalpaper and get a good score.if you master several ones, you could have opportunity topublish your paper.If you master all of them, you might to teach appliedeconometrics class just as what I am doing now.
We will introduce essentials of these methods in the class asmany as possible. Let’s start our journey together.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 99 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
Program Evaluation Econometrics
These Furious Seven are the most basic and popular methods inapplied econometrics and so powerful that
even if you just master one, you may finish your empiricalpaper and get a good score.if you master several ones, you could have opportunity topublish your paper.If you master all of them, you might to teach appliedeconometrics class just as what I am doing now.
We will introduce essentials of these methods in the class asmany as possible. Let’s start our journey together.
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 99 / 100
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Experimental Design as a Benchmark Program Evaluation Econometrics
An Amazing But Tough Journey
Zhaopeng Qu (Nanjing University) Introduction to Econometrics Sep. 18, 2020 100 / 100
