LinearRegression3

9252012 p kolm 56 testing exclusion restrictions 12

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Unformatted text preview: f the parameters is not zero in the population Note, we can’t just check each t -statistic separately, because we want to know if the q parameters are jointly significant at a given level o It is possible for none to be individually significant at that level (For you: Can you think about an example of this situation?) VER. 9/25/2012. © P. KOLM 57 Testing Exclusion Restrictions (2/2) To do the test we need to estimate the “restricted model” without x k -q +1,..., x k included, as well as the “unrestricted model” with all x ’s included Intuitively, we want to know if the change in SSR is big enough to warrant inclusion of x k -q +1,...,x k Fº (SSR r - SSRur ) q SSRur (n - k - 1) , where “r ” and “ur ” represent the restricted and unrestricted models, respectively F is called the F-statistic (or the F-ratio) VER. 9/25/2012. © P. KOLM 58 The F-Statistic (1/2) The F statistic is always positive, since the SSR from the restricted model can’t be less than the SSR from the restricted Essentially the F statistic is measuring the relative increase in SSR when moving from the unrestricted to restricted model q = number of restrictions, or dfr - dfur n - k - 1 = dfur VER. 9/25/2012. © P. KOLM 59 The F-statistic (2/2) To decide if the increase in SSR when we move to a restricted model is “big enough” to reject the exclusions, we need to know about the sampling distribution of the F -statistic The result we use is that F Fq ,n -k -1 where q is referred to as the numerator degrees of freedom and n - k - 1 as the denominator degrees of freedom VER. 9/25/2012. © P. KOLM 60 The F statistic (cont) f (F ) Reject H 0 at a Significance level If F > c Fail to reject Reject a (1 - a) 0 VER. 9/25/2012. © P. KOLM c F 61 Computing p-values, t tests, etc. Using Software Most computer packages will compute the p-value for you, assuming a two- sided test If you really want a one-sided alternative, just divide the two-sided p-value by 2 Refer to the Matlab and Excel examples for more details VER. 9/25/2012. © P. KOLM 62 Example (Problem 4.10 in Wooldridge) In this example, we test whether the market efficiently uses information in valuing stocks. For this purpose, we recall: The efficient markets hypothesis (EMH) says that stock returns should not be systematically related to public information For example, if firm characteristics help predict stock returns, then we would have to reject the EMH (and as a consolation prize make some money!) Consider the following multivariate linear regression specification: return = b0 + b1 dkr + b2 eps + b3 netinc + b4 salary + u where return: total stock return over the four-year period 1990-1994 dkr: a firm’s debt to capital ratio eps: earnings per share netinc: net income salary: total compensation for the CEO VER. 9/25/2012. © P. KOLM 63 The Test: We test whether the explanatory variables are jointly and/or individually significant at the 5% level Hypothesis (“jointly”): H0: b1 = b2 = b3 = b4 = 0 at the 5% level H1: H0 is not true This is an F-t...
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