上海交通大学：《计量经济学》教学资源_教学资料_Multiple Regression Analysis

Multiple Regression Analysis ◆y=B,+Bx1+Bx2+.·Bxk+u ◆Inference Econometrics 15-Zhuxi@SJTU 1

Econometrics 15 - Zhuxi@SJTU 1 Multiple Regression Analysis y = b0 + b1 x1 + b2 x2 + . . . bk xk + u Inference

Gauss-Markov assumptions A1.Population model is linear in parameters: y=Bo+Bx+B2x2+...+Bxk+u A2.We can use a random sample of size n, {...,)i=1,2,...n},from the population model ◆A3.E(ulx,x2…xx)=0,implying that all of the explanatory variables are exogenous Econometrics 15-Zhuxi@SJTU 2

Econometrics 15 - Zhuxi@SJTU 2 Gauss-Markov Assumptions A1. Population model is linear in parameters: y = b0 + b1 x1 + b2 x2 +…+ bk xk + u A2. We can use a random sample of size n, {(xi1, xi2,…, xik, yi ): i=1, 2, …, n}, from the population model A3. E(u|x1 , x2 ,… xk ) = 0, implying that all of the explanatory variables are exogenous

Gauss-Markov Assumptions(cont.) A4.None of the x's is constant,and there are no exact linear relationships among them A5.Assume Var(ux ,x2.....x)=o (Homoskedasticity) Given A1-A5,OLS estimator is BLUE B=(XX)XY Econometrics 15-Zhuxi@SJTU 3

Econometrics 15 - Zhuxi@SJTU 3 Gauss-Markov Assumptions(cont.) A4. None of the x’s is constant, and there are no exact linear relationships among them A5. Assume Var(u|x1 , x2 ,…, xk ) = s2 (Homoskedasticity) Given A1-A5, OLS estimator is BLUE.   1 b ˆ X X X Y    

3.1 Sampling distribution of OLS estimator So far,we know that given the Gauss- Markov assumptions,OLS is BLUE, In order to do classical hypothesis testing, we need to add another assumption(beyond the Gauss-Markov assumptions) A6.Assume that conditional on X,u is normally distributed with zero mean and variance o2:uX~Normal(0,o21) Econometrics 15-Zhuxi@SJTU 4

Econometrics 15 - Zhuxi@SJTU 4 3.1 Sampling distribution of OLS estimator So far, we know that given the Gauss￾Markov assumptions, OLS is BLUE, In order to do classical hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions) A6. Assume that conditional on X, u is normally distributed with zero mean and variance s2 : u|X ~ Normal(0,s2 I)

Classical Linear Model Assumptions Under CLM,OLS is not only BLUE,but also the minimum variance unbiased estimator ◆ We can summarize the population assumptions of CLM(A6)as follows yx Normal(Bo+Bx+...+Bo) While for now we just assume normality,clear that sometimes not the case Large samples will let us drop normality Econometrics 15-Zhuxi@SJTU 5

Econometrics 15 - Zhuxi@SJTU 5 Classical Linear Model Assumptions Under CLM, OLS is not only BLUE, but also the minimum variance unbiased estimator We can summarize the population assumptions of CLM (A6) as follows y|x ~ Normal(b0 + b1 x1 +…+ bk xk , s2 ) While for now we just assume normality, clear that sometimes not the case Large samples will let us drop normality

The homoskedastic normal distribution with a single explanatory variable y ↑fyx) E(ylx)=Bo+Bx {Normal distributions 1 X2 Econometrics 15-Zhuxi@SJTU 6

Econometrics 15 - Zhuxi@SJTU 6 . . x1 x2 The homoskedastic normal distribution with a single explanatory variable E(y|x) = b0 + b1 x y f(y|x) Normal distributions

Normal Sampling Distributions Theorem (normal sampling distribution): Under the CLM assumptions, 月1x~N[B,ar(a小 so that (B-e,)/sd(B,)1X~N(o,) -周s B is distributed normally because it is a linear combination of the errors Econometrics 15-Zhuxi@SJTU 7

Econometrics 15 - Zhuxi@SJTU 7 Normal Sampling Distributions             2 2 2 j Theorem (normal sampling distribution): Under the CLM assumptions, ˆ ˆ | ~ , , so that ˆ ˆ | ~ 0,1 . ˆ . 1 ˆ is distributed normally because it is a linear combination of j j j j j j j j j X N Var sd X N sd R SST b b b b b b s b b        the errors

Normal Sampling Distribution Proof: B=(XX)XY=B°+(XX)Xu =B°+Cu=B+∑Ca4, By assumption A6,we have N(,2) and u,4,u..is mutually independent, thus we have B-B°1X0N(0,o2(XX)） Econometrics 15-Zhuxi@SJTU 8

Econometrics 15 - Zhuxi@SJTU 8 Normal Sampling Distribution           1 1 0 0 0 2 1 2 3 1 0 2 Proof: ˆ . By assumption A6, we have | 0, , and , , ... is mutually independent, thus we have ˆ | 0, . ki i i i X X X Y X X X u C u C u u X N u u u X N X X b b b b s b b s                  

Two Types of Errors H:0=0 H:0≠0，or0>0,or0<0 Type I error:=P(Reject HoHo holds) Type II error:B=P(Accept Ho|H holds) Conventionally,there are trade-off between them, we control a for a test(significance level). and then try to minimize B. Econometrics 15-Zhuxi@SJTU 9

Econometrics 15 - Zhuxi@SJTU 9 Two Types of Errors     0 1 0 0 0 1 : 0 : 0, or 0, or 0 Type I error: = P Reject H | H holds Type II error: = P Accept H | H holds Conventionally, there are trade-off between them, we control for a test(significance level) H H      b      , and then try to minimize . b

3.2 The t Test Under the CLM assumptions (B-p,)/ e(a)- Note this is a t distribution (vs normal) because we have to estimate oby 2: j (-R)Ss7, Note the degrees of freedom:n-k-1 Econometrics 15-Zhuxi@SJTU 10

Econometrics 15 - Zhuxi@SJTU 10 3.2 The t Test         j 1 2 2 2 2 2 Under the CLM assumptions ˆ ~ ˆ Note this is a distribution (vs normal) because we have to estimate by : ˆ ˆ ˆ . 1 Note the degrees of freedom: 1 j n k j j j j t se t se R SST n k b b b s s s b       