9252012 p kolm 70 general linear restrictions the

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Unformatted text preview: can be tricky – will likely have to redefine variables again VER. 9/25/2012. © P. KOLM 71 F-Statistic Summary Just as with t -statistics, p-values can be calculated by determining the percentile in the appropriate F -distribution How are the F- and t-statistic related? o Answer: If only one exclusion is being tested, then F = t 2 , and the p- values will be the same VER. 9/25/2012. © P. KOLM 72 Testing a Linear Combination (1/3) Suppose instead of testing whether b1 is equal to a constant, we want to test if it is equal to another parameter, that is H 0 : b1 = b2 Note that this could be done with an F-test (For you: How?) We could also consider forming ˆ ˆ b1 - b2 t= ˆ -b ) ˆ se (b 1 VER. 9/25/2012. © P. KOLM 2 73 Testing a Linear Combination (2/3) Since ˆ ˆ ˆ ˆ se (b1 - b2 ) = Var (b1 - b2 ) , then ˆ ˆ ˆ ˆ ˆˆ Var (b1 - b2 ) = Var (b1 ) +Var (b2 ) - 2Cov (b1, b2 ) 2 2 ï ï ˆ - b ) = ìése (b )ù + ése (b )ù - 2s ü ˆ ˆ ˆ se (b1 íêë 2 1ú 2ú 12 ý êë û û ï ï ï ï î þ 1 2 ˆˆ where s12 is an estimate of Cov(b1, b2 ) VER. 9/25/2012. © P. KOLM 74 Testing a Linear Combination (3/3) So, to use formula, need s12 , which standard output does not have Many software packages will have an option to get it, or will just perform the test for us However, we can always restate the problem to get the test you want (this is a nice trick!) VER. 9/25/2012. © P. KOLM 75 Example (1/2) Model: log(wage) = b0 + b1 jc + b2univ + b3exper + u We want to ask the question whether the returns to junior college (jc) are the same as the returns to university (univ), i.e. H 0 : b1 = b2 , or H 0 : q1 = b1 - b2 = 0 b1 = q1 + b2 , so substitute in and rearrange gives us log(wage) = b0 + q1 jc + b2 ( jc + univ) + b3exper + u VER. 9/25/2012. © P. KOLM 76 Example (2/2) The estimated regression equation is log(wage) = 1.472 - 0.0102 jc + 0.0769(jc + univ) + 0.0049exper From the regression ˆ ˆ t = q / SE (q) = -0.0102 / 0.0069 = -1.48 This is the same model as originally, but now we obtain the standard error for b1 - b2 = q1 directly from the regression output! Any linear combination of parameters could be tested in a similar manner VER. 9/25/2012. © P. KOLM 77 Congrats for Making It to the Last Slide! Source: VER. 9/25/2012. © P. KOLM 78 References Wooldridge, J. M. (2009). Introductory Econometrics: A Modern Approach, South-Western Pub. Footnotes 1 2 Mathematically, we would say that “we assume the x’s to be linearly independent.” That is, it is the most efficient unbiased estimator amongst all unbiased estimators, linear or nonlinear. 3 See “3A.4 General Omitted Variable Bias” in Wooldridge (2009). 4 ˆ Recall that bj is normally distributed because it is a linear combination of the errors u . VER. 9/25/2012. © P. KOLM 79...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.

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