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【正文】 estrictions, or dfr – dfur n – k – 1 = dfur 28 The F statistic (cont) 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 our F stat Not surprisingly, F ~ Fq,nk1, where q is referred to as the numerator degrees of freedom and n – k – 1 as the denominator degrees of freedom 29 0 c a ?1 ? a? f(F) F The F statistic (cont) reject fail to reject Reject H0 at a significance level if F c 30 The R2 form of the F statistic Because the SSR’s may be large and unwieldy, an alternative form of the formula is useful We use the fact that SSR = SST(1 – R2) for any regression, so can substitute in for SSRu and SSRur ? ?? ? ? ?edu n r e s t r i c t isu r a n d r e s t r i c t e d isr a g a i n w h e r e,11222?????knRqRRFurrur31 Overall Significance A special case of exclusion restrictions is to test H0: b1 = b2 =…= bk = 0 Since the R2 from a model with only an intercept will be zero, the F statistic is simply ? ? ? ?11 22????knRkRF32 General Linear Restrictions The basic form of the F statistic will work for any set of linear restrictions First estimate the unrestricted model and then estimate the restricted model In each case, make note of the SSR Imposing the restrictions can be tricky – will likely have to redefine variables again 33 Example: Use same voting model as before Model is voteA = b0 + b1log(expendA) + b2log(expendB) + b3prtystrA + u now null is H0: b1 = 1, b3 = 0 Substituting in the restrictions: voteA = b0 + log(expendA) + b2log(expendB) + u, so Use voteA log(expendA) = b0 + b2log(expendB) + u as restricted model 34 F Statistic Summary Just as with t statistics, pvalues can be calculated by looking up the percentile in the appropriate F distribution Stata will do this by entering: display fprob(q, n – k – 1, F), where the appropriate values of F, q,and n – k – 1 are used If only one exclusion is being tested, then F = t2, and the pvalues will be the same
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