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Unformatted text preview: constant, you want to test if it is equal to another parameter, that is H : β 1 = β 2 Use same basic procedure for forming a t statistic ( 29 2 1 2 1 ˆ ˆ ˆ ˆ β β β β = se t Fall 2008 under Econometrics Prof. Keunkwan Ryu 22 Testing Linear Combo (cont) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 [ ] { } ( 29 2 1 12 2 1 12 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 ˆ , ˆ of estimate an is where 2 ˆ ˆ ˆ ˆ ˆ , ˆ 2 ˆ ˆ ˆ ˆ then , ˆ ˆ ˆ ˆ Since β β β β β β β β β β β β β β β β Cov s s se se se Cov Var Var Var Var se + = + = = 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 pvalue for the test More generally, you can always restate the problem to get the test you want 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 + β 1 log( expendA ) + β 2 log( expendB ) + β 3 prtystrA + u H : β 1 =  β 2 , or H : θ 1 = β 1 + β 2 = 0 β 1 = θ 1 – β 2 , so substitute in and rearrange ⇒ voteA = β + θ 1 log( expendA ) + β 2 log( expendB  expendA ) + β 3 prtystrA + u 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 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 = 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 Fall 2008 under Econometrics Prof. Keunkwan Ryu 27 Testing Exclusion Restrictions Now the null hypothesis might be something like H : β kq+1 = 0, ... , β k = 0 The alternative is just H 1 : H 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 Fall 2008 under Econometrics Prof. Keunkwan Ryu 28 Exclusion Restrictions (cont) To do the test we need to estimate the “restricted model” without x kq+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 kq+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 Fall 2008 under Econometrics Prof. Keunkwan Ryu 29 The F statistic The F statistic is always positive, since the...
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 Fall '10
 H.Bierens
 Econometrics, Statistical hypothesis testing, Prof. Keunkwan Ryu

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