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Wooldridge PPT ch4

Var var var se fall 2008 under econometrics prof

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Var Var Var se - + = - - + = - - = -
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 23 Testing a Linear Combo (cont) So, to use formula, need s 12 , which standard output does not have Many packages will have an option to get it, or will just perform the test for you In Stata, after reg y x1 x2 … xk you would type test x1 = x2 to get a p -value for the test More generally, you can always restate the problem to get the test you want
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 24 Example: Suppose you are interested in the effect of campaign expenditures on outcomes Model is voteA = β 0 + β 1 log( expendA ) + β 2 log( expendB ) + β 3 prtystrA + u H 0 : β 1 = - β 2 , or H 0 : θ 1 = β 1 + β 2 = 0 β 1 = θ 1 β 2 , so substitute in and rearrange voteA = β 0 + θ 1 log( expendA ) + β 2 log( expendB - expendA ) + β 3 prtystrA + u
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 25 Example (cont): This is the same model as originally, but now you get a standard error for β 1 β 2 = θ 1 directly from the basic regression Any linear combination of parameters could be tested in a similar manner Other examples of hypotheses about a single linear combination of parameters: β 1 = 1 + β 2 ; β 1 = 5 β 2 ; β 1 = -1/2 β 2 ; etc
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 26 Multiple Linear Restrictions Everything we’ve done so far has involved testing a single linear restriction, (e.g. β 1 = 0 or β 1 = β 2 ) 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
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 27 Testing Exclusion Restrictions Now the null hypothesis might be something like H 0 : β k-q+1 = 0, ... , β k = 0 The alternative is just H 1 : H 0 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
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 28 Exclusion Restrictions (cont) 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 ( 29 ( 29 ed unrestrict is ur and restricted is r where , 1 - - - k n SSR q SSR SSR F ur ur r
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 29 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 restrictions, or df r df ur n – k – 1 = df ur
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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,n-k-1 , where q is referred to as the numerator degrees of freedom and n – k – 1 as the denominator degrees of freedom
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Fall 2008 under Econometrics Prof. Keunkwan Ryu 31 0
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