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Unformatted text preview: 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 restrictions, or df r – df ur n – k – 1 = df ur Fall 2008 under Econometrics Prof. Keunkwan Ryu 30 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 ~ F q,nk1 , where q is referred to as the numerator degrees of freedom and n – k – 1 as the denominator degrees of freedom Fall 2008 under Econometrics Prof. Keunkwan Ryu 31 c α (1 α29 f( F ) F The F statistic (cont) reject fail to reject Reject H at α significance level if F > c Fall 2008 under Econometrics Prof. Keunkwan Ryu 32 The R 2 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 – R 2 ) for any regression, so can substitute in for SSR u and SSR ur ( 29 ( 29 ( 29 ed unrestrict is ur and restricted is r again where , 1 1 2 2 2 ≡ k n R q R R F ur r ur Fall 2008 under Econometrics Prof. Keunkwan Ryu 33 Overall Significance A special case of exclusion restrictions is to test H : β 1 = β 2 =…= β k = 0 Since the R 2 from a model with only an intercept will be zero, the F statistic is simply ( 29 ( 29 1 1 2 2 = k n R k R F Fall 2008 under Econometrics Prof. Keunkwan Ryu 34 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 Fall 2008 under Econometrics Prof. Keunkwan Ryu 35 Example: Use same voting model as before Model is voteA = β + β 1 log( expendA ) + β 2 log( expendB ) + β 3 prtystrA + u now null is H : β 1 = 1, β 3 = 0 Substituting in the restrictions: voteA = β + log( expendA ) + β 2 log( expendB ) + u , so Use voteA  log( expendA ) = β + β 2 log( expendB ) + u as restricted model Fall 2008 under Econometrics Prof. Keunkwan Ryu 36 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 = t 2 , and the pvalues will be the same Eg. Salarypension tradeoff for teachers Log(totcomp)=f(productivity characteristics, other factors) Totcomp=salary+benefits=salary(1+benefits/salary) Log (salary) = β + β 1 (benefits/salary)+other factors Fall 2008 under Econometrics Prof. Keunkwan Ryu 37 Fall 2008 under Econometrics Prof. Keunkwan Ryu 38...
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 Fall '10
 H.Bierens
 Econometrics, Statistical hypothesis testing, Prof. Keunkwan Ryu

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