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8/21/2019 Application of Econometrics in Economics
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Application of Econometrics inManagerial Economics
Dr . Manoj Kumar DashM.A; M .Phil; M BA; Ph.D
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What is Econometrics?
• Simply, Econometrics, means “economic
measurement”.
• Some formal definitions:
Econometrics is concerned with the empirical
determination/quantification of theoretical
postulations – economics, management, political
science etc.
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Econometrics is “defined as the social science in
which the tools of economic theory, mathematics,
and statistical inference are applied to theanalysis of economic phenomena (Goldberger,
1964).”
Econometrics “….. consists of the application of
mathematical statistics to economic data to lend
empirical support to the models constructed bymathematical economics and to obtain numerical
results (Gerhard Tintner, 1968).”
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Econometrics renders “a positive help in trying
to dispel the poor public image of economics …… as a subject in which empty boxes are opened by
assuming the existence of can-openers to reveal
contents which any ten economists will interpretin 11 ways.”
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Examples
• Consumption expenditure = F (Income, Wealth
etc)
• Hourly earnings = F (Education, Labour
productivity etc)
• Quantity Demanded = F (Price, Income of
consumer, Prices of relative commodities etc)
• Election Outcome = F (Economic performance,
Populism etc)
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• Debt/Equity = F (Corporate tax rate, Capital
gains tax rate, Inflation rate etc)
• Sales = F (Price, Advertising expenses etc)
• Crop yield = F (Temperature, Rainfall, Fertilizer
use etc.)
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Why Study Econometrics?
(1)To provide empirical support to theories:
• Theories make statements or hypotheses that aremostly qualitative in nature.
• They do not provide numerical measure of the proposed ideas.
• Example: Law of Demand in Economics
• In a given situation how to quantify this law?
Econometrics comes to the rescue.
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(2)To provide empirical support to mathematical
models:
• Many theories are expressed in mathematical
form/models (Common in Economics, Finance)
• These theoretical models are developed without
regard to their measurability
• Econometrics helps to put these models intoempirical testing (e.g. Tax competition models).
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(3) For academicians:
• Research in economics, finance, management,marketing etc is becoming increasingly
quantitative [Show examples].
• Hence, knowledge about econometrics is very
important to conduct empirical research.
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(4) For students:
• For students a command over econometrics will be of great use in their employment.
• Example: forecasting of sales, consumer behaviour (KPOs) – corporate research.
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Steps in Econometric Analysis
1. Identify research issue
2. Select Variables
3. Check Data Availability
4. Specify Econometric Model
5. Data Collection
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6. Model Estimation
7. Hypothesis Testing
8.Using Estimated Model for
Forecasting/Prediction
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Step 1: Identify research issue
• You should have a research problem to probe(How to identify a problem?)
• Example from Labour Economics: Effect of
economic conditions on people‟s willingness to
work (or LFPR).
• Two hypothesises/arguments are postulated.
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• Argument 1 (Discouraged worker hypothesis): As
economic conditions worsen, unemployed may
drop out of labour force (falling LFPR) as theyloose hope of finding a job.
• Argument 2 (Added worker hypothesis): As
economic conditions worsen, unemployed may
join labour force (rising LFPR) if the main breadwinner in the family loses job.
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• In this context, our research objective could be
to test relative strength/validity of these two
contrasting claims.
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Step 2: Select Variables
• For running a regression, we have to select aDEPENDENT variable and an INDEPENDENT variable(s).
• A model with one dependent variable and oneindependent variable is called simple or twovariable regression model (TWRM).
• A model with one dependent variable and morethan one independent variable is called multipleregression model (MRM).
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• Example:
LFPR = f (Unemployment Rate) - TWRM
LFPR = f (Unemployment Rate; Hourly
Earnings; Family Wealth) – MRM
• Generally, MRM is preferred because it
enhances credibility of research findings.
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• How many independent variables do we include?
• Strictly, guiding principle should be underlying
theory or prior literature or nature of problem.
• Even then, it may not always possible to include
every possible variable
• Why?
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Non-availability of data
Inherent randomness in human behaviour, which cannot be captured easily (Ex: Being unemployed due toattitude problem).
Inability to properly quantify certain variables (Ex.Interest groups)
Due to ignorance we may miss relevant variables
Anyway, ultimate purpose of econometrics is not tocapture complete reality
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• How to rectify this problem?
Add a term – called error term – which could takecare of influence of all omitted variables.
In other words, error term captures all those
forces that affect the dependent variable but are
not explicitly included in the model.
Error term will also be useful when we use proxyindependent variables (e.g. interest groups)
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If the error term is small it implies that the
combined influence of omitted variables is
small/negligible.
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Step 3: Check Data Availability
• Before proceeding further, check data availabilityfor relevant variables.
• Three types of data are generally available for
empirical analysis – Cross-sectional, Time-series, Pooled Data/Panel Data.
• In cross-section data, values of one or morevariables are collected for several sample units,or entities at the same point in time (e.g. GDPfigure of countries for a given year).
C S ti l d t
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Cross-Sectional data
States Tax Revenue in 2000-01 (in crores)
Andhra Pradesh 10551.92
Bihar 2934.75
Haryana 4311.48
Karnataka 9042.67
Kerala 5870.26
Maharashtra 19724.28
Orissa 2184.03
Punjab 4895.22
Rajasthan 5299.97
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• In time series data we observe the values of oneor more variables over a period of time – daily
stock prices or annual GDP figures
• In panel data, same cross-sectional unit (say afirm or state or country) is surveyed over time
• Thus, in panel data we have elements of bothtime series and cross-sectional data.
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Time Series Data
Year Tax Revenue of Andhra Pradesh
1990-91 2647.25
1991-92 3054.96
1992-93 3388.72
1993-94 3832.93
1994-95 4227.43
1995-96 4120.44
1996-97 4881.83
1997-98 7113.55
1998-99 7961.4
1999-00 9008.6
2000-01 10551.92
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Panel Data: Tax Revenues of States
Year AP KAR KER TN 1990-91 2647.25 2332.12 1340.35 3124.05 1991-92 3054.96 2900.2 1673.93 3734.11 1992-93 3388.72 3097.81 1886.97 4162.06 1993-94 3832.93 3812.34 2344.87 4801.37 1994-95 4227.43 4289.31 2799.1 5833.76 1995-96 4120.44 5273.93 3382.68 7151.2 1996-97 4881.83 5767.84 3898.5 7983.45 1997-98 7113.55 6411.87 4501.05 8682.64 1998-99 7961.4 6943.1 4649.5 9625.3 1999-00 9008.6 7744.37 5193.51 10918.93 2000-01
10551.92
9042.67
5870.26
12282.25
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Step 4: Specify Econometric Model
• LFPR = B1 + B2UR +u
LFPR – Labour force participation rate
UR – Unemployment rate (a proxy for economic
condition. Other option could be GDP)
B 1 - Intercept term. Gives value of LFPR (dependent
variable) when UR (value of independent variable) iszero
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B 2 - Slope term. Measures the rate of change in LFPR
for a unit change in UR. Together B1 and B2 are
known as the parameters of the regression modelu – Error term [LFPR – ( B1 + B2UR)]
• This is a linear regression model - LFPR is linearly
related to UR
• Our objective is to explain the behaviour of dependent
variable in relation to the explanatory variable.
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Step 5: Data Collection
• For empirical estimation, we need to collect dataon the variables used in the econometric model.
• Data can be obtained either from primary sources[HR] or from secondary sources [Finance,
Economics]
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Step 6: Model Estimation
• By estimation we mean estimating the parameters(B1 & B2) of the chosen model.
• Estimation is carried out using the technique of
regression analysis.
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• What is regression analysis?
“Regression analysis is concerned with thestudy of the dependence of one variable, the
dependent variable, on one or more other
variables, the explanatory var iables, with a view
to estimating and/or predicting the (population)
mean or average value of the former in terms of
the known or fixed (in repeated sampling)
values of the latter ”
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• Assume that after estimation we get the followingresult:
LFPR = 50.63 - 0.45UR
B2 = – 0.45. Implies that if UR goes up by 1%
point, ceteris paribus, LFPR is expected todecrease on the average by about 0.45% points – Discouraged worker hypothesis finds support.
B1 = 50.63. Implies that average value of LFRPwill be about 50.63% if the UR were zero (i.e.full employment).
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Sometimes/often, intercept term has no
particular economic meaning.
But, in the present example it has meaning
(How?)
We say “on the average” because the presence
of error term is likely to make the relationship
somewhat imprecise.
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Step 7: Hypothesis Testing
• Objective here is to test certain hypothesessuggested by theory and/or prior empirical
experience on parameters of the model.
• Specifically, we are interested to verify how
close the estimated parameter is to a pre-
supposed value of that parameter
(e.g. )45.0ˆ;1: 110 B B H
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• Hypothesis testing helps to verify whether the
results obtained through regression analysis
conform to the underlying theory
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Step 8: Using Estimated Model for
Forecasting/Prediction
• LFPR = 50.6333 - 0.4486UR
• If we want to predict LFPR in some future
periods for a given value of UR (say 5) we can
obtain it (How?)
• Substitute UR value (5) in the above equation
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• When data on LFPR (for a given UR ) for
future period is out, we can compare the
predicted value with the actual value.
• The discrepancy between the two is called the
prediction error.
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Regression Vs. Correlation
Regression Correlation
Estimate/predict average value of
one variable on the basis of the
fixed values of other variables
Measure the strength or degree of
linear association between two
variables
The dependent variable is treated
as stochastic or random
The explanatory variables are
assumed to have fixed values
Both the variables are treated as
random
O I di t T k
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Our Immediate Task
• Identification of research issue, selection of
variables, checking of data availability,specification of econometric model and data
collection are not big tasks
• Hence, our focus would be on
Model Estimation
Hypothesis Testing
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Criticisms
• By and large, events can be explained without
econometric analysis
• Data mining – Results are created!
• Problem with intercept term
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Two-Variable Regression Model
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• Term linear can be interpreted in 2 ways:
Linearity in the variables and linearity in the parameters.
• “Linear” regression always means a regressionthat is linear in the parameters. It may or may not
be linear in the explanatory variables
Example:Y = B1 + B2X +u (or)
Y = B1 + B2X2 +u
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Linear Regression Models
Model linear in parameters?
Model linear in variables?
Yes No
Yes LRM LRM
No NLRM NLRM
P l ti R i F ti (PRF)
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Population Regression Function (PRF)
• Let us consider an example of the law of demand.
The demand schedule for commodity x Price
(X)
Quantity Demanded (Y) No. of
Consumers
Average
demand
1 45,46,47,48,49,50,51 7 48
2 44,45,46,47,48 5 46
3 40,42,44,46,48 5 44
4 35,38,42,44,46,47 6 42
5 36,39,40,42,43 5 40
6 32,35,37,38,39,42,43 7 38
7 32,34,36,38,40 5 36
8 31,32,33,34,35,36,37 7 34
9 28,30,32,34,36 5 32
10 29,30,31 3 30
Total 55
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Figure 1: Population Regression Line
0
10
20
30
40
50
60
0 1 2 3 4 5 6 7 8 9 10 11 12
Price
Q u a n t i t y D e m a n d e d
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• The impression we get from scattergram is that
demand (Y) decreases as price (X) increases, and
vice versa.
• The downward slopping line is called Population
Regression Line (PRL).
• It is nothing but the locus of conditional means of
the dependent variable for the fixed values of the
explanatory variable(s)
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• Thus, PRL gives the average value of the
dependent variable corresponding to each value
of the independent variable.
• The point on the PRL represents expected or
population mean value of Y corresponding to
the various Xs.
• The adjective population comes as our example
deals with entire population of 55 consumers.
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• PRL can be expressed in following functional
form:
E(Y/Xi) = B1+B2Xi (1)
where i is ith subpopulation
• Eq (1) gives average value of Y corresponding to
each value of X and is called Population
Regression Function (PRF) or non-stochastic
PRF.
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• In regression analysis our interest is in
estimating the PRFs (i.e. B1 and B2) on the basis
of observations on Y and X.
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Stochastic Specification of the PRF
• How to explain the demands of the individualconsumer in relation to price?
• The best we can do is to say that any
individual‟s demand is equal to the average for
that group plus or minus some quantity.
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D e m a n d
O
2 7X
36
u
32
Price
u
46
Y
48
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• We can express the deviation of an individual Yi
around its expected value as follows:
Yi = E(Y/Xi) + ui (2)
• In Eq. (2), ui is an unobservable random variable
– called stochastic error term - taking positive or
negative values.
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• Now, by substituting Eq. (1) in (2), we get
Yi = B1+B2Xi+ui (3)
• Eq. (3) is called stochastic PRF (SPRF), whereas
Eq. (1) is called non-stochastic PRF (NPRF).
• NPRF represents means of various Y values
corresponding to specified prices.
• SPRF tells us how individual demands vary around
their mean values due to presence of stochastic error
term, u .
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• How to interpret SPRF (Eq.3)?
• We can say that demand of an individual consumer
(say i) corresponding to a specific price can be
expressed as sum of following 2 components:
(i) Systematic/deterministic component:
B1+B2Xi (Nothing but average quantity demanded
by all the consumers at a given price level Xi)
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(ii) Nonsystematic/random component: ui
(Determined by factors other than price) [See
Figure]
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D e m a n d
O
2 7X
36
u
32
Price
u
46
Y
48
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Sample Regression Function (SRF)
• If we have data on whole population (like in
following Table) arriving at PRF is an
easy/straightforward exercise
• That is, find conditional means of Ycorresponding to each X and then join these
means
• Unfortunately, in practice, we rarely have entire population at our disposal.
Th d d h d l f di
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The demand schedule for commodity x
Price
(X)
Quantity Demanded (Y) No. of
Consumers
Average
demand
1 45,46,47,48,49,50,51 7 48
2 44,45,46,47,48 5 46
3 40,42,44,46,48 5 44
4 35,38,42,44,46,47 6 42
5 36,39,40,42,43 5 40
6 32,35,37,38,39,42,43 7 38
7 32,34,36,38,40 5 36
8 31,32,33,34,35,36,37 7 34
9 28,30,32,34,36 5 32
10 29,30,31 3 30
Total 55
W l h l f h l i
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• We only have a sample from the population.
• The following is an example from our case
Sample 1
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p
Y (Demand) X (Price)
49 145 2
44 3
39 438 5
37 6
34 7
33 8
30 9
29 10
H t k i t ti t th PRF th b i f
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• Hence, our task is to estimate the PRF on the basis of
sample information
[OR]
Task is to estimate average quantity demanded in the
population as a whole corresponding to each X (price)from sample data such as above
• But, we may not be able to estimate the PRF accurately
because of sampling error
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• To see this clearly,
suppose another random
sample (Sample 2) isdrawn from the population
of above Table.
• If we plot the data of these
two samples, and entire
population we may obtain
corresponding SRLs and
PRL as follows
Y (Demand) X (Price)
51
1
47 2
46 3
42 4
40 5
37 6
36 7
35 8
32 9
30 10
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D e m a n d
Y
X
SRL1
SRL2
PRL
Price
N th ti i hi h f th t SRL
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• Now the question is: which of the two SRLs
represents the “true” PRL?
• If we avoid temptation of looking at above
figure, which represents the PRL, there is no way
we can be sure that either of the SRLs represents
the true PRL
• In general, we get K different SRLs for K
different samples and all these SRLs are notlikely to be the same
N l t PRF th t d li th PRL
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• Now, analogous to PRF that underlies the PRL,
we can develop SRF to represent SRL as
follows:
= estimator of E(Y/Xi) b1 = estimator of B1
b2 = estimator of B2
Where is read as “Y-hat” or “Y-cap” Y
)4(ˆ21 ii
X bbY
iY ˆ
A ti t i l f l th t i di t
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• An estimator is a rule or formula that indicates
how to estimate the population parameter at
hand.
• A particular numerical value obtained by the
estimator is an estimate.
• Now, we can express SRF (Eq.4) in its
stochastic form as follows:
b + b X + e (5)Y
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= b1 + b2Xi + ei (5)
Where ei is the estimator of ui
• It is analogous to ui and is introduced for same
reasons as ui was introduced in the PRF
iY
T i bj i i i
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• To sum up, our primary objective in regression
analysis is to estimate the stochastic PRF (Eq.3)
on the basis of stochastic SRF (Eq.5) becausemore often than not our analysis is based on a
single sample from some population.
• But, because of sampling variation, our estimate
of PRF based on SRF is only approximate.
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D e m a n d
Y
X
PRF
SRF
Price
Error
Error
• Granted that SRF is only an approximation of the PRF
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• Granted that SRF is only an approximation of the PRF
the question is:
• Can we find a method or a procedure that will make
this approximation as “close” as possible?
[OR]
How should we construct the SRF so that b1 is as close
as possible to B1 and b2 is as close as possible to B2?
• This can be done by adopting the method of Ordinary
Least Squares (OLS).
What is OLS method?
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• It chooses SRF or b1 and b2 in such a way that that
the sum of the squares of the residuals is as smallas possible.
• Symbolically,
Minimize
Where Yi = actual Y value
= estimated/predicted Y value
• In this way, SRF is made as close as possible to
PRF
2
2 ˆiii Y Y e
iY ˆ
• Why and not ?2
ie ie
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Why and not ?
• Due to two reasons. Reason 1: To give different weightage to
residuals according to the extent of their
closeness to SRF
Example: e1 = 10, e2 = -2, e3 = +2 and e4 = -10
By squaring ei we can give more weightage to
large errors (10) comparing others (2)
i i
R 2 Thi d id th bl f
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Reason 2: This procedure avoids the problem of
sign of the residuals which can be positive as
well as negative, and therefore can add to zero.
How to select b1 and b2 values to minimise ?2
ie
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ow to se ect b1 a d b2 va ues to se ?ie
• From Eq. (5), we know that
• Which is nothing but
• Hence, = f (b1, b2)
• Hence, for any given set of data, choosingdifferent values of b1 and b2 will give different
ei‟s and hence different values of
22
i i iˆe Y Y
2
21 ii X bbY
2
ie
2
ie
• An Example:
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4 1 2.929 1.071 1.147 4 0 0
5 4 7.000 -2.000 4.000 7 -2 4
7 5 8.357 -1.357 1.841 8 -1 1
12 6 9.714 2.286 5.226 9 3 9
Sum:28 16 0 12.214 0 14
iY i X iY 1
ˆ ie1ˆ
ie 12
ˆiY 2
ˆie2
ˆie 2
2ˆ
ˆ
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Notes:
• Now which sets of estimated b (parameter)
values should we choose?
• Since b values of 1st experiment gives us lower
than that obtained from b values of 2nd experiment,
we say b‟s of first experiment are “best” values.
357.1;572.1357.1572.1ˆ211
bb X Y ii
)1;3(13ˆ212 bb X Y ii
)ˆ(ˆ 11 iii Y Y e
iii Y Y e 22
ˆˆ
2
ie
B h d i hi ?
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• But how do we ascertain this?
• We still can choose many more values for b‟s that gives us the least possible value of
• However, in doing so we must be sure that we
have considered all the conceivable values of b1 and b2
• If we have infinite time and patience we can do
this exercise
2
ie
• But fortunately OLS method chooses b and b
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• But, fortunately, OLS method chooses b1 and b2
in such a manner that, for a given set of data,
is as small as possible.
[OR]
For a given sample, OLS method provides us
with unique estimates of b1 and b2 that give the
smallest possible values of
2
ie
2
ie
How do we accomplish this?
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• This is a straight-forward exercise in differential calculus.
• Values of b1 and b2 that minimize are obtained by
solving the following two simultaneous equations:
• These simultaneous equations are known as least squares
normal equations.
2
ie
ii X bnbY 21
2
21 iiii X b X b X Y
• In above equations n is sample size
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• In above equations n is sample size
• Unknowns are bs. Knowns are quantities
involving sums, sum of squares, and sums of
cross products of the variables Y and X
• The knowns can be obtained from sample athand
• Solving these equations simultaneously, we
obtain following solutions for b1 and b2:
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Where
- Mean of Y variable- Mean of X variable
- Deviation from sample mean values
- OLS estimators
X bY b 21
22
X X
Y Y X X b
i
ii
Y X
Y Y X X ii ,
21,bb
Example:
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Y X
Y X
49
1
-4.5
11.2
20.25
-50.4
45 2 -3.5 7.2 12.25 -25.2
44 3 -2.5 6.2 6.25 -15.5
39 4 -1.5 1.2 2.25 -1.8
38
5
-0.5
0.2
0.25
-0.1
37 6 0.5 -0.8 0.25 -0.4
34 7 1.5 -3.8 2.25 -5.7
33 8 2.5 -4.8 6.25 -12
30 9 3.5 -7.8 12.25 -27.3 29 10 4.5 -8.8 20.25 -39.6
Mean = 37.8 Mean = 5.5 82.5 -178
X X i Y Y i
2)( X X i Y Y X X ii
• Using the above formulas we obtain estimates
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Using the above formulas we obtain estimates
of b1and b2 as follows
b2 = -2.1576
b1 = 49.667
• The results can be obtained using regression
packages without much effort (Demonstration)
Properties of OLS estimators
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• OLS estimators b1 and b2 satisfy the BLUE –
Best Linear Unbiased Estimator – property
Linearity:
• Estimators are a linear function of the dependent
and independent variables
• A linear estimator is much easier to deal with
than a nonlinear estimator
Unbiasedness:
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• Average or expected value of the estimators isequal to true/population value
Minimum variance/Efficiency:
• It has minimum variance in the class of all suchlinear unbiased estimators
• Smaller the variance of b1 or b2, the closer theywill be to true B1 or B2
• Implication: Regression coefficient estimated byOLS on average coincides with population/truevalue
Assumptions Underlying OLS
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Assumptions Underlying OLS
Assumption 1:
• The regression model is linear in parameters (Bs)
Assumption 2:
• Explanatory variables (Xs) are fixed in repeated
sampling
• Implication: Changes in Y is conditional on thegiven values of X
• Example:
The demand schedule for commodity x
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The demand schedule for commodity x
Price (X) Quantity Demanded (Y)
1
45,46,47,48,49,50,51
2 44,45,46,47,48
3 40,42,44,46,48
4 35,38,42,44,46,47
5 36,39,40,42,43
6 32,35,37,38,39,42,43
7 32,34,36,38,40
8 31,32,33,34,35,36,37
9 28,30,32,34,36
10 29,30,31
Assumption 3:
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Assumption 3:
• Mean value of disturbance ui is zero
• Because positive ui values cancel out negative ui
values (see figure)
• Implication: factors not explicitly included in themodel (i.e. ui) don‟t systematically affect mean
of Y (or) ui‟s average effect on Y is zero
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X
X1 X2 X3
PRF
+ ui
- ui
Mean
. .
. .
.
.
Assumption 4:
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p
• The variance of ui is same for all observations
(Xs) [Homoscedasticity or equal (homo) spread
(scedasticity)]
• Implication: Variation around regression line ofindividual Y values remains same regardless of
values taken by Xs; it neither increases or
decreases as X varies
• Hence all Y values corresponding to the various
Xs are equally important.
• Violation of this assumption (i.e. increase in variation
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around regression line of Y values as X increases) is
called heteroscedasticity
• Example for homoscedasticity: Richer families on
average consume more than poorer families, but there
is no/not much variability in consumption pattern between richer and poorer families
• Example for heteroscedasticity: There is greater
variability in consumption pattern of richer families
compared to poorer ones becoz. as income grows
people have more consumption choice.
Assumption 5:
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• No autocorrelation between error terms
(Implications for time series data)
• Example for autocorrelation:Yt = B1 +B2Xt +ut, where ut and ut-1 are positively
correlated. Here, Yt depends not only on Xt, but
also on ut-1 for ut-1 to some extent determined ut
• By invoking no autocorrelation assumption, we
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y g p ,
consider only the effect on Xt on Yt and not
worry about other influences that might act on Yas a result of possible inter-correlations among
u‟s
Assumption 6:
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Assumption 6:
• No correlation between u and explanatory
variables (Xs)
• Implication: If X and u are correlated, it is not
possible to assess their individual effects on Y
• If X and u are positively correlated, X increases
when u increases and it decreases when u
decreases.
Assumption 7:
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• Number of observations n must be greater than the
number of parameters to be estimated or explanatory
variables.
Assumption 8:
• X values in given sample must not all be the same(Applies to Y as well)
• If so, Xi = and hence denominator of estimator b2
will be zero.
• This makes it impossible to estimate b2 and b1
X
Assumption 9:
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• The regression model is correctly specified
• Omission of important variables or inclusion of wrongvariables undermines the validity of regression exercise
• Theory should be the guiding principle in building
econometric model
• If theory is not clear, we have to use some judgment in
choosing the model and interpreting the results (e.g. taxcompetition)
• But, “data mining” should be avoided.
Assumption 10:
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• There are no perfect linear relationships among Xs
[No multicollinearity]
• Important with respect to multiple regression models
• Implication: In the presence of multicollinearity, wecannot assess the separate influence of Xs on Y.
All these assumptions pertain to PRF only and not
SRF. This means that SRF may not always duplicateall these assumptions (Example: presence of
autocorrelation and multicollinearity problems).
Coefficient of Determination (r2)
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• This is a measure of “goodness of fit” of the (sample)
regression line to a given set of data
[OR]
It is a summary measure that tells how well SRF fits
given data
• r 2 measures % of total variation in Y explained by
regression model or X(s).
• A “perfect” fit of regression line is rarely the case
• Generally, there will be some positive and
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negative errors
• Our goal is to minimize the errors as far as
possible
• See Figure: If all the observations were to lie on
the regression line, we would obtain a “perfect”
fit.
Y
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X
X 1 X 2 X 3 X 4
SRFY i
1u
2u
3u
4u
i 1 2 iY X
• The concept of coefficient of determination can
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be explained using the following diagram
Y
Y
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O
Xi X
Yi
Y
Total= (Yi - )Y
SRF
ei = Due to residual
( - ) = Due to
regression/Explained by X
(Why?)
iY ˆ Y
iY ˆ
• iiii Y Y Y Y Y Y ˆˆ
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- Mean of the sample data
- Predicted value of Y for a given X (Point on
SRF)
- An individual sample observation
• Numerical proof (Consider following
example)
iiii
iY
Y
iY
Y X
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49 1
45 2
44 3
39 4
38 5
37 6
34 7
33 8
30
9
29 10
= 37.8
Y
• For the above data b2 = -2.1576; b1 = 49.667
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2 1
• Hence = 49.667 – 2.1576Xi
• For Xi = 1, = 47.509
• Now applying for Yi = 49 weget
• 49-37.8 = (47.509-37.8) + (49-47.509)
• 11.2 = 11.2, i.e. LHS = RHS
iY ˆ
iY ˆ
iiii Y Y Y Y Y Y ˆˆ
• By squaring the above identity and summing
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By squaring the above identity and summing
them we get,
- Called TSS (Total variation in Y)
- Called ESS (Variation due to X)
- Called RSS (Variation due to error)
222 ˆˆ
iiii Y Y Y Y Y Y
2)( Y Y i
2ˆ Y Y i
2
2ˆ
iii eor Y Y
• Thus, TSS = ESS + RSS
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• If all actual Ys lie on fitted SRF, RSS=0 and hence
ESS=TSS (Polar cases)
• If X explains no variation in Y, ESS=0 and hence
RSS=TSS (Polar cases)
• If ESS is relatively larger than RSS, then the chosen
SRF fits the data well
• If RSS is relatively larger than ESS, then the chosen
SRF fits the data poorly
• Now, r 2 is defined as
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,
r 2 =
r 2 =
This is nothing but portion of variation in Y (TSS)
explained by X (ESS)
TSS
ESS
2
2ˆ
Y Y
Y Y
i
i
Properties of r2
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• It is a nonnegative quantity (Why?)
• Its limits are 0 r 21
• An r 2
of 1 means a perfect fit, that is, foreach i.
• An r 2 of zero means that there is no relationship
between Y and X
ii Y Y ˆ
• Example: In a regression with quantity
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demanded as dependent variable and price
independent variable an r 2
value of 0.975implies that price variable explains about 98%
of variation in quantity demanded . In this case,
we can say that sample regression gives an
excellent fit.
Coefficient of correlation (r)
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• It measures degree of linear association
between Y and X
• It is nothing but
• In practice r is of little importance
• The more meaningful quantity is r 2 (Why?)
• r is also called simple correlation coefficient orcorrelation coefficient of zero order
2r
• Interpretation of r: r 12 means correlation
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p 12
between variable 1 (say Y) and variable 2 (say
X2)
Standard Error (SE) of Regression Coefficients
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• We know that least-squares estimates (b1 and b2)
are estimated using sample data
• But since data are likely to change from sample
to sample, the estimates will change as well
• Therefore, what is needed is some measure of
“reliability” of OLS estimators
• The precision of an estimate or regression
coefficient is measured by SE
• SE is the standard deviation (positive square rootof variance) of sampling distribution (SD) of the
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of variance) of sampling distribution (SD) of theestimator (say b2)
• SD of an estimator is a distribution of set ofvalues of estimator obtained from all possible
samples of same size from a given population.
• Thus, SE of an estimator is the amount it varies
across samples.
• SEs of OLS estimates can be obtained as
f ll
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follows:
• Where var = variance; se = standard error and
is homoscedastic variance of ui (Assumption 4)
2
2
2 )var(i x
b
2
2 )(
i x
b se
2
2
2
1 )var(
i
i
xn
X b
2
2
1)(i
i
xn
X b se
2
2
ˆ
ˆ
2
2
n
ui
• Here the variance of b2 is inversely proportional to 2
i x
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• That is, given , larger the variation in X values, the
smaller the variance of b2 and hence greater the
precision with which b2 can be estimated.
• In short, if there is substantial variation in Xs (recallAssumption 8), b2 can be measured more accurately
than when Xs do not vary substantially.
2
• Hence, what is a „big‟ SE of regression
coefficients and what is a „small‟ SE depends on
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coefficients and what is a small SE depends on
the context (i.e. variation in Xs)
• A more standardized statistic, which also gives a
measure of the „goodness of fit‟ of estimated
equation is R 2
• SEs of regression coefficients can be used for
hypothesis testing and constructing confidenceintervals (discussed later)
Standard Error of Regression/Residuals
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• SE of regression is the standard deviation
(Positive square root of variance) of individual Yvalues about the estimated regression line or
error term
• If SE of residuals is high, then deviation will also
be high and hence fitness will be poor
• SE of residuals can be obtained using the
f ll i f l
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following formula
2
ˆ
ˆ
22
2
2
n
xb y ii
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Multiple Regression Analysis
Meaning
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• In single/two variable model there is only one
explanatory variable
• In practice, most problems can‟t be explained by
this model
• Example: Apart from prices, demand is a function
of many other factors
• Hence, we use multiple regression models whichcontain more than one Xs
How the model looks like?
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•
PRF for cross sectional data•
PRF for time-series data
• Any individual Y value can be expressed assum of 2 components
Deterministic component [E (Yi)]
Random component [ ]
iiii U X B X B BY 33221
t t t t U X B X B BY 33221
ii X B X B B 33221
iU
Assumptions
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p
All the assumptions of two-variable
model are applicable in the case of
multiple regression as well
Eq.
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• It gives conditional mean value of Y
conditional upon the given/fixed values of Xs
• Symbolically
• Thus, what we obtain is the mean value of Y for
the given values of Xs
iiiii X B X B B X X Y E
3322132 ,
(PRCs)
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• In multiple regression B2&B3 are called PRCs
• B2 measures change in the mean value of Y per
unit change in X2, holding value of X3 constant
[OR]
B2 gives “direct” or “net” effect of a unit change
in X2 on E(Y), net of any effect that X3 may have
on mean Y
• Similar explanation is applicable for B2 as well
Estimating PRCs
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• To estimate parameters of model we use OLS
method
• Let SRF corresponding to PRF described above
as:
• Where b1, b2 & b3 are estimators of unknown
population coefficients B1, B2 & B3 respectively
• ei is sample counterpart of residual term U
iiii e X b X bbY 33221
• OLS principle chooses values of unknown
t i h th t th RSS i2
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parameters in such a way that the RSS is as
small as possible
• Symbolically,
• Minimization of this involves differentiation
with respect to unknowns, setting resulting
expressions to zero, and solving themsimultaneously
2
ie
2
33221
2min iiii X b X bbY e
• This procedure generates following formulas for
arriving at numerical values of OLS estimators
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arriving at numerical values of OLS estimators
b1, b2 & b3
33221 X b X bY b
2
32
2
3
2
2
323322
2
iiii
iiiiiii
x x x x
x x x y x x yb
2
32
2
3
2
2
32222
3
3
iiii
iiiiiii
x x x x
x x x y x x yb
• In these formulas, lowercase letters denote
d i ti f l l
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deviations from sample mean values
Properties of OLS estimators
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The BLUE property continues to hold here aswell
(R 2)
E l i ti f i ti i Y l i d
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• Explains proportion of variation in Y explained by Xs jointly
• Conceptually, R 2 is akin to r 2
• As in two-variable case, R 2 is defined as
R 2 =
=
• R 2 lies between 0 and 1
TSS ESS
2
3322
i
iiii
y
x yb x yb
• If R 2=1, fitted regression line explains
100% of variation in Y
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100% of variation in Y.
• If R 2=0, model does not explain any of the
variation in Y
• The fit of regression model is said to be
“better”, closer is R 2 to 1
• By and large, as the number of Xs increasesR 2 value increases (Why? See below)
R 2 and Adjusted R 2
• One aspect of R2 is: As the no of Xs increases
) R( 2
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• One aspect of R 2 is: As the no. of Xs increases,
R 2 almost invariably increases
• Why? From elsewhere, we know
TSS
RSS
TSS
ESS
1
TSS
RSS R 21
TSS RSS R 12
2
2
1i
i
y
e
• Here depends on no. of Xs, but not
d i
2
ie
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denominator
• Hence, as Xs increase is likely to decrease
(or at least it will not increase). Hence R 2
increases
2
ie
Increasing R 2 • Is it desirable to increase R2 by adding more Xs?
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• Is it desirable to increase R by adding more Xs?
• We should adopt a cautious approach here
• Why?
(i) With larger Xs, R 2 gives an overly optimistic
picture of regression fit
(ii) R 2 does not take into account d.f.
(iii) We need to have a measure of goodness of fit thatis adjusted for no. of Xs added in the model
• Such a measure is known as [adjusted R 2]
which is defined as
2 R
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which is defined as
• Here k is no. of parameters in the model
including intercept term
• Term adjusted means adjusted for d.f.associated with sums of squares entering into
above identity
)1(
)(1
2
2
2
n y
k ne R
i
i
Features of Adjusted R 2
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• If k 1, R 2; i.e. as Xs increase becomes
increasingly less than R 2 or increases lessthan unadjusted R 2
• This means that, a penalty is involved in addingmore Xs in to a regression model
• can be negative, but not R 2 (Why?)
2 R
2 R
2 R
2 R
What we do in practice?
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• In practice, mainly R 2 is used to measure
“goodness” of fit
• is used in deciding inclusion of a newvariable
• If inclusion of a new variable increases , it isretained in the model
• When does increases? If value of thecoefficient of the added variable is larger than 1
2 R
2 R
2 R
t
Why maximizing opposed? 2 R
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• Our objective is not to obtain a high
• The researcher should be more concerned about
logical or theoretical relevance of the Xs to Yand their statistical significance
• If this process produces a high it is well andgood
2 R
2 R
Hypothesis Testing
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• The procedure is same as in two-variable case
• We can adopt both Confidence interval
approach and Test of significance approach
Testing under CIA
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• We construct a confidence interval and see
whether hypothesized value of population parameters (B1/ B2/ B3) lies inside this interval.
• If it lies inside, we do not reject H0
• If it lies outside, we can reject H0
• The remaining procedure is same as in the caseof two-variable model
• The test (t ) statistic we use for this purpose is
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(For testing B1)
(For testing B2)
(For testing B3)
)( 1
11
b se
Bb
t
)( 2
22
b se
Bbt
)( 3
33
b se
Bbt
Testing under ToSA
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Step 1: Set H0 and H1 separately for each partial
regression coefficient
Examples: H0:B2 =0 and H1:B20H0:B3 =0 and H1:B30
Step 2: Compute a test (t ) statistic from sample
data (See above)
Step 3: Choose level of significance () (or) probability of committing Type 1 error
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(0.01 or 0.05 or 0.10)
Step 4: Find probability of obtaining computed test
(t ) statistic for certain d.f.
Note: d.f. is (n-k ), where n - no. of
observations, k- no. of Xs includingintercept term
Step 5: If this probability is less than the
prechosen reject H0. Otherwise, accept H0
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p j 0 p 0
(OR)
After Step 3 Use following rules to accept orreject H0
Null Hypothesis Alternative Critical region: Reject H0 if
A summary of the t test
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(H0) Hypothesis (H1)
Bx = 0 Bx 0 [One tailed] Calculated test statistic (t ) t, d.f.
(i.e. Table t value at level of
significance and certain degrees of
freedom)
Bx = 0 Bx 0 [One tailed] Calculated test statistic (t ) -t, d.f.
(i.e. Table t value at level of
significance and certain degrees of
freedom)
Bx = 0 Bx 0 [Two tailed] Calculated absolute value of test
statistic ( ) t/2, d.f. (i.e. Table t
value at /2 level of significance
and certain degrees of freedom)
t
ANOVA or F test – Relevance
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• This is a complementary way of hypothesis
testing
• Commonly used to test joint H0 in multiple
regression models
• But, can be used in two variable regression model
as well
What is a joint H0?
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• H0: B2 = B3 = 0
• Means that B2 and B3 are jointly equal to zero
(or) Xs have no influence on Y
• A test of joint H0 is called a test of the overallsignificance of estimated regression line
How to construct F test statistic?• From R 2 discussion, we know that
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TSS = ESS + RSS (Or)
• The d.f. associated with components of thisidentity is
TSS = n-1 because we lose 1 d.f. in computing
sample mean
ESS = 2 (k-1) because ESS is a function of B2 and B3 (where k is no Xs)
2
3322
2
iiiiii e x yb x yb y
Y
RSS = n-3 (n-k) because in computing RSS weneed to estimate B1 B2 and B3
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• In case of two-variable model the corresponding
d.f. are:
TSS = n-1ESS = 1
RSS = n-2
• In general, in a regression model with k
explanatory variables (incl. intercept), the d.f. are
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explanatory variables (incl. intercept), the d.f. are
as follows
TSS = n-1 (always)
ESS = k -1
RSS = n-k
• Now, by arranging sums of squares and d.f. we
get ANOVA table
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Source of variation Sum of squares (SS) d.f. Mean sum of Squares
(MSS) = SS/d.f.
Due to regression
(ESS) 2
Due to residuals
(RSS) n-3 n-3
Total (TSS) n-1
iiii x yb x yb 3322
2
ie
2
i y
2 i 2i 3 i 3i b y x b y x / 2
2
ie /
• Now, define F statistic as
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F = =
F =
..
..
f d RSS
f d ESS
)(
)1(
k n RSS
k ESS
)3(
2
2
3322
ne
x yb x yb
i
iiii
Using F-ratio for hypothesis testing
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• Set H0: B2 = B3 = 0 & H1: Not all Bs are
simultaneously zero
• Calculate F ratio using formula
• We reject H0, if F value computed from formulaexceeds critical/table F value at level ofsignificance and given d.f. in numerator anddenominator
• Otherwise, we do not reject H0:
• Alternatively, if the p value of computed F ratiois sufficiently low, we reject H0
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Intuitive Reasoning
.. f d ESS
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• In F =
Numerator explains variance of Y explained by
Xs
Denominator explains variance of Y not
explained by Xs
If numerator denominator, F1
Increasingly large F is an evidence against H0
.. f d RSS
f
Relationship between F and R 2
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• The null B2
= B3
= 0 is same as saying that H0
:
R 2=0 (why?)
• Thus, F test is also a test of significance of R 2
(i.e. whether R 2 is different from zero)
• The relationship between F ratio and R 2 is as
follows
• )/()1(
)1(2
2
knR
k R F
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• Here, when R 2=0, F=0
• The larger R 2 is, the greater the F value will be
• One advantage of this formula is the ease of
computation of F value. All we need to know isR 2 value
)/()1( k n R
Testing significance of R 2 using F test
S b tit t R2 l i d t F)1(
2kR
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• Substitute R 2 value in and compute F
ratio
• We reject H0: R 2=0, if F value computed from formulaexceeds the critical/table F value at level ofsignificance and given d.f. in numerator anddenominator
• Otherwise, we do not reject the null
)/()1(
)1(2
k n R
k R F
Usefulness of this statistic
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• In cross-sectional data involving severalobservations, one generally obtains low R 2
• This is due to diversity of the cross-sectional
units
• Here, the statistical significance of R 2 value can
be verified using)/()1(
)1(2
2
kR
k R F