《计量经济学》课程教学资源（PPT课件讲稿，英文版）ch04 Multiple regression analysis

Multiple regression analysis y=Bo B Bx+ Bx +.Bkk+u ◆2. Inference Economics 20- Prof anderson

Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = b0 + b1 x1 + b2 x2 + . . . bk xk + u 2. Inference

Assumptions of the Classical Linear Model(CLm) So far, we know that given the Gauss Markov assumptions, ols iS BLUE e In order to do classical hypothesis testing we need to add another assumption(beyond the Gauss-Markov assumptions) ◆ Assume that u is independent ofxI,x2,…,xk and u is normally distributed with zero mean and variance 02: u- normal(0, 0) Economics 20- Prof anderson

Economics 20 - Prof. Anderson 2 Assumptions of the Classical Linear Model (CLM) 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) Assume that u is independent of x1 , x2 ,…, xk and u is normally distributed with zero mean and variance s 2 : u ~ Normal(0,s 2 )

CLM ASsumptions(cont) o Under CLM, OLS is not only blue, but is the minimum variance unbiased estimator o We can summarize the population assumptions of ClM as follows ◆yx~ Normal(B0+Bx1+…+Bxha) e While for now we just assume normality clear that sometimes not the case e Large samples will let us drop normality Economics 20- Prof anderson

Economics 20 - Prof. Anderson 3 CLM Assumptions (cont) Under CLM, OLS is not only BLUE, but is the minimum variance unbiased estimator We can summarize the population assumptions of CLM as follows y|x ~ Normal(b0 + b1 x1 +…+ bk xk , s 2 ) 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 fylx E(x)=Bo+ Bx t Normal distributions x Economics 20- Prof anderson 4

Economics 20 - Prof. Anderson 4 . . x1 x2 The homoskedastic normal distribution with a single explanatory variable E(y|x) = b0 + b1x y f(y|x) Normal distributions

Normal Sampling distributions Under the clm assumption s conditiona l on the sample values of the independen t variable s B - NO orma 1 B, vare, l so that B 16) normal l(0,4) B is distribute d normally because it is a linear combinatio n of the errors Economics 20- Prof anderson 5

Economics 20 - Prof. Anderson 5 Normal Sampling Distributions  ( ) ( ) ( ) ( ) is a linear combinatio n of the errors is distribute d normally because it ˆ ~ Normal 0,1 ˆ ˆ ,so that ˆ ~ Normal , ˆ the sample values of the independen t variable s Under the CLM assumption s, conditiona l on b j b b b b b b j j j j j j sd Var −

The t test Under the Clm assumption s BB) se Note this is at distributi on( vS norma because we have to estimate o by o2 Note the degrees of freedom: n-k-1 Economics 20- Prof anderson 6

Economics 20 - Prof. Anderson 6 The t Test ( ) ( ) Note the degrees of freedom : 1 because we have to estimate by ˆ Note this is a distributi on (vs normal) ~ ˆ ˆ Under the CLM assumption s 2 2 1 j − − − − − n k t t se n k j j s s b b b

The t Test(cont) o Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis ◆ For example,H:B=0 o If accept null, then accept that x has no effect on y, controlling for other xs Economics 20- Prof anderson 7

Economics 20 - Prof. Anderson 7 The t Test (cont) Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis For example, H0 : bj=0 If accept null, then accept that xj has no effect on y, controlling for other x’s

The t Test(cont) To perform our test w e first need to form the"t statistic for B: to sev Ve will then use our t statistic along with We a rejection rule to determine whether to accept the null hypothesis, Ho Economics 20- Prof anderson 8

Economics 20 - Prof. Anderson 8 The t Test (cont) ( ) 0 j ˆ accept the null hypothesis , H a rejection rule to determine whether t o We will then use our statistic along with ˆ ˆ : ˆ "the" statistic for To perform our test w e first need to form t se t t j j j b b b b 

t Test: One-Sided Alternatives o Besides our null. h. we need an alternative hypothesis, HI, and a significance level Himay be one-sided, or two-Sided ◆H1:B>0andH1:B<C <O are one-sided H: Bi*0 is a two-sided alternative e If we want to have only a 5% probability of rejecting Ho if it is really true, then we say our significance level is 5% Economics 20- Prof anderson 9

Economics 20 - Prof. Anderson 9 t Test: One-Sided Alternatives Besides our null, H0 , we need an alternative hypothesis, H1 , and a significance level H1 may be one-sided, or two-sided H1 : bj > 0 and H1 : bj < 0 are one-sided H1 : bj  0 is a two-sided alternative If we want to have only a 5% probability of rejecting H0 if it is really true, then we say our significance level is 5%

One-Sided Alternatives(cont) e Having picked a significance level, a,we look up the(1-a)th percentile in a t distribution with n-k-1 df and call this c the critical value o We can reject the null hypothesis if the t statistic is greater than the critical value o If the t statistic is less than the critical value then we fail to reject the null Economics 20- Prof anderson 10

Economics 20 - Prof. Anderson 10 One-Sided Alternatives (cont) Having picked a significance level, a, we look up the (1 – a) th percentile in a t distribution with n – k – 1 df and call this c, the critical value We can reject the null hypothesis if the t statistic is greater than the critical value If the t statistic is less than the critical value then we fail to reject the null