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 GaussMarkov 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