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Unformatted text preview: 8 The One-Sided Alternative: b j < 0 Because the t -distribution is symmetric, testing H 1 : b j < 0 is straightforward. The critical value is just the negative of the one before (-c) We reject the null if the t -statistic < -c , and if the t -statistic ³ -c then we fail to reject the null VER. 9/25/2012. © P. KOLM 49 The Two-Sided Alternative: b j ¹ 0 For a two-sided test, we set the critical value based on a / 2 We reject the null
H 0 : bj = 0 if the absolute value of the t -statistic > c , i.e. | t |> c VER. 9/25/2012. © P. KOLM 50 Two-Sided Alternatives: b j ¹ 0 yi = b0 +b1x il +¼+bk x ik + ui H 0 : b j = 0 H 1 : bj ¹ 0 Fail to reject (1 - a) Reject Reject a/2 a/2
‐ VER. 9/25/2012. © P. KOLM c 0 c 51 Summary for H 0 : b 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 a % level” If we fail to reject the null, we typically say “ x j is statistically insignificant at the a % level” VER. 9/25/2012. © P. KOLM 52 Testing Other Hypotheses A more general form of the t -statistic recognizes that we may want to test something like H 0 : b j = b j0 In this case, the appropriate -statistic is
bj - b 0
se (b j ) , → The procedure for doing this tests is exactly the same as the ones we just discussed VER. 9/25/2012. © P. KOLM 53 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 - a ) % confidence interval is defined as ˆ
bj c ⋅ se (bj ) ,
where c is the ç1- ÷ percentile in a tn-k -1 -distribution
ø How do we interpret or explain what a 95% confidence interval means?
o “We are 95% confident the true parameter lies in the interval” (loose)
o “In repeated samples the true parameter will lie in the interval 95% of the time” (better) VER. 9/25/2012. © P. KOLM 54 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 calculated, if the null was true Key points to remember:
o Smaller p-values mean a “more significant” regressor
o p <0.05 means reject at 5% significance level VER. 9/25/2012. © P. KOLM 55 Multiple Linear Restrictions We may want to jointly test multiple hypotheses about our parameters A typical example is testing “exclusion restrictions”
o That is, we may want to know if a group of parameters are all equal to zero If we fail to reject then those associated explanatory variables should be excluded from the model VER. 9/25/2012. © P. KOLM 56 Testing Exclusion Restrictions (1/2) The null hypothesis might be something like
H 0 : bk -q -1 = 0,..., bk = 0 The alternative is just
H 1 : H 0 is not true This means that at least one o...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
- Fall '14