Econ 399 Chapter4a

8/6/2019 Econ 399 Chapter4a

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4. Multiple Regression

Analysis: Estimation-Most econometric regressions are motivatedby a question

-ie: Do Canadian Heritage commercials

have a positive impact on Canadian identity?-Once a regression has been run, hypothesistests work to both refine the regression and

answer the question-To do this, we assume that the error isnormally distributed

-Hypothesis tests also assume no statistical

issues in the regression


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4. Multiple Regression Analysis:

Inference4.1 Sampling Distributions of the OLS

Estimators

4.2 Testing Hypotheses about a Single

Population Parameter: The t test

4.3 Confidence Intervals

4.4 Testing Hypothesis about a Single Linear

Combination of the Parameters4.5 Testing Multiple Linear Restrictions: The

F test

4.6 Reporting Regression Results


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4.1 Sampling Distributions of OLS

-In chapter 3, we formed assumptions that

make OLS unbiased and covered the issueof omitted variable bias

-In chapter 3 we also obtained estimates forOLS variance and showed it was smallest ofall linear unbiased estimators

-Expected value and variance are just thefirst two moments of Bjhat, its distribution

can still have any shape


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4.1 Sampling Distributions of OLS

-From our OLS estimate formulas, the

sample distributions of OLS estimatorsdepends on the underlying distribution ofthe errors

-In order to conduct hypothesis tests, we

assume that the error is normallydistributed in the population

-This is the NORMALITY ASSUMPTION:


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Assumption MLR. 6(Normality)

The population error u is independentof the explanatory variables x1, x2,,xkand is normally distributed with zeromean and variance 2:

),0(~ 2WNu


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Assumption MLR. 6 Notes

MLR. 6 is much stronger than any of our previousassumptions as it implies:

MLR. 4: E(u|X)=E(u)=0MLR. 5: Var(u|X)=Var(u)=2

Assumptions MLR. 1 through MRL. 6 are theCLASSICAL LINEAR MODEL (CLM) ASSUMPTIONSused to produce the CLASSICAL LINEAR MODEL

-CLM assumptions are all the Gauss-Markov

assumptions plus a normally distributed error


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4.1 CLM AssumptionsUnder the CLM assumptions, the efficiency of

OLSs estimators is enhanced

-OLS estimators are now the MINIMUMVARIANCE UNBIASED ESTIMATORS

-the linear requirement has been dropped andOLS is now BUE

-the population assumptions of CLM can be

summarised as:),...(| 222110 WFFFF kkxxxNXy

-conditional on x, y has a normal distribution with

mean linear in X and a constant variance


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4.1 CLM AssumptionsThe normal distribution of errors assumption is

driven by the following:

1) u is the sum of many unobserved factors thataffect y

2) By the CENTRAL LIMIT THEOREM (CLT), uhas an approximately normal distribution (see

appendix C)


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4.1 Normality Assumption ProblemsThis normality assumption has difficulties:

1) Factors affecting u can have widely differentdistributions

-this assumption becomes worse dependingon the number of factors in u and howdifferent their distributions are

2) The assumption assumes that u factors affecty in SEPARATE, additive fashions

-if u factors affect y in a complicated fashion,

CLT doesnt apply


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4.1 Normality Assumption ProblemsIn general, the normality of u is an empirical (not

theoretical) matter

-if empirical evidence shows that a distribution isNOT normal, we can practically ask if it isCLOSE to normal

-often applying a transformation (such as logs)

can make a non-normal distribution normal

-Consequences of nonnormality are covered in

Chapter 5


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4.1 Nonnormality-In some cases, MLR. 6 is clearly false

-Take the regression:

usin210 ! gFlosBrushingtsDentalVisi FFF

-since dental visits have only a few values formost people, our dependent variable is far fromnormal

-we will see that nonnormality is not a difficultyin large samples

-for now, we simply assume normality

-error normality extends to the OLS estimators:


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Theorem 4.1(Normal Sampling Distributions)

Under the CLM assumptions MLR.1 through MLR.6, conditional on the sample values of theindependent variable,

(3.3))]Var(,N[ j1j FFF

Where Var(Bjhat) was given in Chapter 3[equation 3.51]. Therefore,

N(0,1)-)cd(/)-( FFF


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Theorem 4.1 Proof

r

w

w

ijij

ijj

j

ij

SSR

u

!

! FF

Where rjhat and SSRj come from the regressionof xj on all other xs

-Therefore w is non random

-Bjhat is therefore a linear combination of theerror terms (u)

-MLR. 6 and MLR. 2 make the errorsindependent, normally distributed randomvariables


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Theorem 4.1 Proof

Any linear combination of independent normalrandom variables is itself normally distributed(Appendix B)

This proves the first equation in the proof

-The second equation comes from the fact thatstandardizing a normal random variable (bysubtracting its mean and dividing by itsstandard deviation) gives us a standardnormal random variable (statistical theory)


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4.1 Normality

Theorem 4.1 allows us to do simple hypothesistests by assigning a normal distribution to OLSestimators

Furthermore, since any linear combination of OLSestimators has a normal distribution, and anysubset of OLS estimators has a joint normal

distribution, more complicated tests can bedone


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4.2 Single Hypothesis Tests: t tests-this section covers testing hypotheses about

single parameters from the populationregression function

-Consider the population model:

(4.2)ux...xx kk22110 ! FFFFy

-we assume this model satisfies CLM assumptions

-we know that OLSs estimate of Bj is unbiased-the true Bj is unknown, and in order to

hypothesis about Bjs true value, we usestatistical inference and the following theorem:


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Theorem 4.2(t Distribution for the

Standardized Estimators)Under the CLM assumptions MLR.1 through

MLR. 6, (4.3)t~)se()-

(

1-k-n

j

jj

FFF

where k+1 is the number of unknown

parameters in the population model(4.2)ux...xx kk22110 ! FFFFy

(k slope parameters and the intercept B0).


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Theorem 4.2 Notes

Theorem 4.2 differs from theorem 4.1:

-4.1 deals with a normal distribution andstandard deviation

-4.2 deals with a t distribution and standard error

-replacing with hat causes this

-see section B.5 for more details


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4.2 Null Hypothesis-In order to perform a hypothesis test, we first

need a NULL HYPOTHESIS of the form:

(4.4)0: j0 !FH

-which examines the idea that xj has no partialeffect on y

-For example, given the regression and nullhypothesis:

0:H

uCarnationsRosestlowerEffec

10

210

!

!

F

FFFF

-We examine the idea that roses dont make no

impression (good or bad) in a bouquet


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4.2 Hypothesis Tests-Hypothesis tests are easy, the hard part is

calculating the needed values in the regression-our T STATISTIC or S RATIO is calculated as:

(4.5))se(

j

j

F

FF !jt

-therefore, given an OLS estimate for Bj and its

standard error, we can calculate a t statistic-note that our t stat will have the same sign as

Bjhat

-note also that larger Bjhats cause larger t stats


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4.2 Hypothesis Tests-Note that B

j

hat will never EXACTLY equal zero

-instead we ask: How far is Bjhat from zero?

-a Bj

hat far from zero provides evidence that Bjisnt zero

-but the sampling error (standard deviation) mustalso be taken into account

-Hence the t statistic

-t measures how many estimated standard

deviations Bjhat is from zero


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4.2 Hypothesis Tests-values of t significantly far from zero cause the

null hypothesis to be rejected

-to determine how far t must be from zero, weselect a SIGNIFICANCE LEVEL a probability ofrejecting H0 when it is true

-we know that the sample distribution of t is

tn-k-1, which is key

-note that we are testing the population

parameters (Bj) not the estimates (Bjhat)

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Econ 399 Chapter4a

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