econ 399 chapter4a
Post on 08-Apr-2018
221 Views
Preview:
TRANSCRIPT
-
8/6/2019 Econ 399 Chapter4a
1/22
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
-
8/6/2019 Econ 399 Chapter4a
2/22
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
-
8/6/2019 Econ 399 Chapter4a
3/22
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
-
8/6/2019 Econ 399 Chapter4a
4/22
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:
-
8/6/2019 Econ 399 Chapter4a
5/22
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
-
8/6/2019 Econ 399 Chapter4a
6/22
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
-
8/6/2019 Econ 399 Chapter4a
7/22
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
-
8/6/2019 Econ 399 Chapter4a
8/22
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)
-
8/6/2019 Econ 399 Chapter4a
9/22
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
-
8/6/2019 Econ 399 Chapter4a
10/22
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
-
8/6/2019 Econ 399 Chapter4a
11/22
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:
-
8/6/2019 Econ 399 Chapter4a
12/22
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
-
8/6/2019 Econ 399 Chapter4a
13/22
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
-
8/6/2019 Econ 399 Chapter4a
14/22
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)
-
8/6/2019 Econ 399 Chapter4a
15/22
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
-
8/6/2019 Econ 399 Chapter4a
16/22
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:
-
8/6/2019 Econ 399 Chapter4a
17/22
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).
-
8/6/2019 Econ 399 Chapter4a
18/22
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
-
8/6/2019 Econ 399 Chapter4a
19/22
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
-
8/6/2019 Econ 399 Chapter4a
20/22
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
-
8/6/2019 Econ 399 Chapter4a
21/22
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
-
8/6/2019 Econ 399 Chapter4a
22/22
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)
top related