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Chapter 4: Statistical Hypothesis Testing

Christophe Hurlin

November 20, 2015

Christophe Hurlin () Advanced Econometrics - Master ESA November 20, 2015 1 / 225

Section 1

Introduction

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 2 / 225

1. Introduction

The outline of this chapter is the following:

Section 2. Statistical hypothesis testing

Section 3. Tests in the multiple linear regression model

Subsection 3.1. The Student test

Subsection 3.2. The Fisher test

Section 4. MLE and Inference

Subsection 4.1. The Likelihood Ratio (LR) test

Subsection 4.2.The Wald test

Subsection 4.3. The Lagrange Multiplier (LM) test

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 3 / 225

1. Introduction

References

Amemiya T. (1985), Advanced Econometrics. Harvard University Press.

Greene W. (2007), Econometric Analysis, sixth edition, Pearson - Prentice Hil (recommended)

Ruud P., (2000) An introduction to Classical Econometric Theory, Oxford University Press.

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 4 / 225

1. Introduction

Notations: In this chapter, I will (try to...) follow some conventions of notation.

fY (y) probability density or mass function

FY (y) cumulative distribution function

Pr () probability

y vector

Y matrix

Be careful: in this chapter, I dont distinguish between a random vector (matrix) and a vector (matrix) of deterministic elements (except in section 2). For more appropriate notations, see:

Abadir and Magnus (2002), Notation in econometrics: a proposal for a standard, Econometrics Journal.

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 5 / 225

Section 2

Statistical hypothesis testing

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 6 / 225

2. Statistical hypothesis testing Objectives

The objective of this section is to dene the following concepts:

1 Null and alternative hypotheses

2 One-sided and two-sided tests

3 Rejection region, test statistic and critical value

4 Size, power and power function

5 Uniformly most powerful (UMP) test

6 Neyman Pearson lemma

7 Consistent test and unbiased test

8 p-value

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 7 / 225

2. Statistical hypothesis testing

Introduction

1 A statistical hypothesis test is a method of making decisions or a rule of decision (as concerned a statement about a population parameter) using the data of sample.

2 Statistical hypothesis tests dene a procedure that controls (xes) the probability of incorrectly deciding that a default position (null hypothesis) is incorrect based on how likely it would be for a set of observations to occur if the null hypothesis were true.

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 8 / 225

2. Statistical hypothesis testing

Introduction (contd)

In general we distinguish two types of tests:

1 The parametric tests assume that the data have come from a type of probability distribution and makes inferences about the parameters of the distribution

2 The non-parametric tests refer to tests that do not assume the data or population have any characteristic structure or parameters.

In this course, we only consider the parametric tests.

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 9 / 225

2. Statistical hypothesis testing

Introduction (contd)

A statistical test is based on three elements:

1 A null hypothesis and an alternative hypothesis

2 A rejection region based on a test statistic and a critical value

3 A type I error and a type II error

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 10 / 225

2. Statistical hypothesis testing

Introduction (contd)

A statistical test is based on three elements:

1 A null hypothesis and an alternative hypothesis

2 A rejection region based on a test statistic and a critical value

3 A type I error and a type II error

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 11 / 225

2. Statistical hypothesis testing

Denition (Hypothesis) A hypothesis is a statement about a population parameter. The formal testing procedure involves a statement of the hypothesis, usually in terms of a nullor maintained hypothesis and an alternative, conventionally denoted H0 and H1, respectively.

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 12 / 225

2. Statistical hypothesis testing

Introduction

1 The null hypothesis refers to a general or default position: that there is no relationship between two measured phenomena or that a potential medical treatment has no e¤ect.

2 The costs associated to the violation of the null must be higher than the cost of a violation of the alternative.

Example (Choice of the null hypothesis) In a credit scoring problem, in general we have: H0 : the client is not risky(acceptance of the loan) versus H1 : the client is risky (refusal of the loan).

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 13 / 225

2. Statistical hypothesis testing

Denition (Simple and composite hypotheses) A simple hypothesis species the population distribution completely. A composite hypothesis does not specify the population distribution completely.

Example (Simple and composite hypotheses)

If X � t (θ) , H0 : θ = θ0 is a simple hypothesis. H1 : θ > θ0, H1 : θ < θ0, and H1 : θ 6= θ0 are composite hypotheses.

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 14 / 225

2. Statistical hypothesis testing

Denition (One-sided test) A one-sided test has the general form:

H0 : θ = θ0 or H0 : θ = θ0 H1 : θ < θ0 H1 : θ > θ0

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 15 / 225

2. Statistical hypothesis testing

Denition (Two-sided test) A two-sided test has the general form:

H0 : θ = θ0 H1 : θ 6= θ0

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 16 / 225

2. Statistical hypothesis testing

Introduction (contd)

A statistical test is based on three elements:

1 A null hypothesis and an alternative hypothesis

2 A rejection region based on a test statistic and a critical value

3 A type I error and a type II error

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 17 / 225

2. Statistical hypothesis testing

Denition (Rejection region)

The rejection region is the set of values of the test statistic (or equivalently the set of samples) for which the null hypothesis is rejected. The rejection region is denoted W. For example, a standard rejection region W is of the form:

W = fx : T (x) > cg

or equivalently W = fx1, .., xN : T (x1, .., xN ) > cg

where x denotes a sample fx1, .., xNg , T (x) the realisation of a test statistic and c the critical value.

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 18 / 225

2. Statistical hypothesis testing

Remarks

1 A (hypothesis) test is thus a rule that species:

1 For which sample values the decision is made to fail to reject H0as true;

2 For which sample values the decision is made to reject H0.

3 Never say "Accept H1", "fail to reject H1" etc..

2 The complement of the rejection region is the non-rejection region.

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 19 / 225

2. Statistical hypothesis testing

Remark

The rejection region is dened as to be:

W = fx : T (x)| {z } test statistic

7 c|{z}g critical value

T (x) is the realisation of the statistic (random variable):

T (X ) = T (X1, ..,XN )

The test statistic T (X ) has an exact or an asymptotic distribution D under the null H0.

T (X ) � H0 D or T (X )

d! H0 D

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master ESA November 20, 2015 20 / 225

2. Statistical hypothesis testing

Introduction (contd)

A statistical test is based on three elements:

1 A null hypothesis and an alternative hypothesis

2 A rejection region based on a test statistic and a critical value

3 A type I error and a type II error

Christophe Hurlin (University of Orléans) Advanced Econometrics - Master