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【正文】 , you can always restate the problem to get the test you want 22 Example: Suppose you are interested in the effect of campaign expenditures on outes Model is voteA = b0 + b1log(expendA) + b2log(expendB) + b3prtystrA + u H0: b1 = b2, or H0: q1 = b1 + b2 = 0 b1 = q1 – b2, so substitute in and rearrange ? voteA = b0 + q1log(expendA) + b2log(expendB expendA) + b3prtystrA + u 23 Example (cont): This is the same model as originally, but now you get a standard error for b1 – b2 = q1 directly from the basic regression Any linear bination of parameters could be tested in a similar manner Other examples of hypotheses about a single linear bination of parameters: ? b1 = 1 + b2 。 b1 = 5b2 。 etc 24 Multiple Linear Restrictions Everything we’ve done so far has involved testing a single linear restriction, (. b1 = 0 or b1 = b2 ) However, we may want to jointly test multiple hypotheses about our parameters A typical example is testing “exclusion restrictions” – we want to know if a group of parameters are all equal to zero 25 Testing Exclusion Restrictions Now the null hypothesis might be something like H0: bkq+1 = 0, ... , bk = 0 The alternative is just H1: H0 is not true Can’t just check each t statistic separately, because we want to know if the q parameters are jointly significant at a given level – it is possible for none to be individually significant at that level 26 Exclusion Restrictions (cont) To do the test we need to estimate the “restricted model” without xkq+1, …, xk 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 xkq+1, …, xk ? ?? ?edun r e s t r i c t isur a nd r e s t r i c t e d isr w he r e,1????knSSRqSSRSSRFururr27 The F statistic The F statistic is always positive, since the SSR from the restricted model can’t be less than the SSR from the unrestricted Essentially the F statistic is measuring the relative increase in SSR when moving from the unrestricted to restricted model q = number of r
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