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