Fall 2008 under econometrics prof keunkwan ryu 11 y i

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Fall 2008 under Econometrics Prof. Keunkwan Ryu 11 y i = β 0 + β 1 x i1 + … + β k x ik + u i H 0 : β j = 0 H 1 : β j > 0 c 0 α (1 -α29 One-Sided Alternatives (cont) Fail to reject reject
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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
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 13 y i = β 0 + β 1 X i1 + … + β k X ik + u i H 0 : β j = 0 H 1 : β j > 0 c 0 α/2 (1 -α29 -c α/ 2 Two-Sided Alternatives reject reject fail to reject
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 14 Summary for H 0 : 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”
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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 0 : β j = a j In this case, the appropriate t statistic is ( 29 ( 29 test standard for the 0 where , ˆ ˆ = - = j j j j a se a t β β
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Eg. Campus crime and enrollment Fall 2008 under Econometrics Prof. Keunkwan Ryu 16 Log(crime) = β 0 + β 1 log(enroll)+ u
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Log(price) = β 0 + 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
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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
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 19
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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
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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 parameter, that is H 0 : β 1 = β 2 Use same basic procedure for forming a t statistic ( 29 2 1 2 1 ˆ ˆ ˆ ˆ β β β β - - = se t
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 22 Testing Linear Combo (cont) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 [ ] { } ( 29 2 1 12 2 1 12 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 ˆ , ˆ of estimate an is where 2 ˆ ˆ ˆ ˆ ˆ , ˆ 2 ˆ ˆ ˆ ˆ then , ˆ ˆ ˆ ˆ Since β β β β β β β β β β β β β β β β Cov s s se se se Cov Var
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  • Fall '10
  • H.Bierens
  • Econometrics, Statistical hypothesis testing, Prof. Keunkwan Ryu

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