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 GaussMarkov 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
