Wooldridge PPT ch4

– 1 df and call this c the critical value we can

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Unformatted text preview: – 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 Fall 2008 under Econometrics Prof. Keunkwan Ryu 11 y i = β + β 1 x i1 + … + β k x ik + u i H : β j = 0 H 1 : β j > 0 c α (1 -α29 One-Sided Alternatives (cont) Fail to reject reject Fall 2008 under Econometrics Prof. Keunkwan Ryu 12 One-sided vs Two-sided Because the t distribution is symmetric, testing H 1 : β j < 0 is straightforward. The critical value is just the negative of before We can reject the null if the t statistic < – c , and if the t statistic > than – c then we fail to reject the null For a two-sided test, we set the critical value based on α /2 and reject H 1 : β j ≠ 0 if the absolute value of the t statistic > c Fall 2008 under Econometrics Prof. Keunkwan Ryu 13 y i = β + β 1 X i1 + … + β k X ik + u i H : β j = 0 H 1 : β j > 0 c α/2 (1 -α29-c α/ 2 Two-Sided Alternatives reject reject fail to reject Fall 2008 under Econometrics Prof. Keunkwan Ryu 14 Summary for H : β j = 0 Unless otherwise stated, the alternative is assumed to be two-sided If we reject the null, we typically say “ x j is statistically significant at the α % level” If we fail to reject the null, we typically say “ x j is statistically insignificant at the α % level” Fall 2008 under Econometrics Prof. Keunkwan Ryu 15 Testing other hypotheses about β j A more general form of the t statistic recognizes that we may want to test something like H : β j = a j In this case, the appropriate t statistic is ( 29 ( 29 test standard for the where , ˆ ˆ =- = j j j j a se a t β β Eg. Campus crime and enrollment Fall 2008 under Econometrics Prof. Keunkwan Ryu 16 Log(crime) = β + β 1 log(enroll)+ u Log(price) = β + β 1 log(nox)+ β 2 log(dist) + β 3 rooms+ β 4 stratio + u Fall 2008 under Econometrics Prof. Keunkwan Ryu 17 Eg. Housing prices and air pollution Fall 2008 under Econometrics Prof. Keunkwan Ryu 18 Computing p-values for t tests An alternative to the classical approach is to ask, “what is the smallest significance level at which the null would be rejected?” So, compute the t statistic, and then look up what percentile it is in the appropriate t distribution – this is the p-value p-value is the probability we would observe the t statistic we did, if the null were true Fall 2008 under Econometrics Prof. Keunkwan Ryu 19 Fall 2008 under Econometrics Prof. Keunkwan Ryu 20 Confidence Intervals Another way to use classical statistical testing is to construct a confidence interval using the same critical value as was used for a two-sided test A (1 - α ) % confidence interval is defined as ( 29 on distributi a in percentile 2- 1 the is c where , ˆ ˆ 1-- • ± k n j j t se c α β β Fall 2008 under Econometrics Prof. Keunkwan Ryu 21 Testing a Linear Combination Suppose instead of testing whether β 1 is equal to a constant, you want to test if it is equal to another...
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– 1 df and call this c the critical value We can reject...

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