Econometrics-I-9 - Applied Econometrics William Greene...

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Unformatted text preview: Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 9. Hypothesis Tests: Analytics and an Application General Linear Hypothesis Hypothesis Testing Analytical framework: y = X β + ε Hypothesis: R β - q = , J linear restrictions Procedures Classical procedures based on two algebraically equivalent frameworks Distance measure: Is Rb - q = m 'far' from zero? (It cannot be identically zero.) Fit measure: Imposing R β - q on the regression must degrade the fit ( e'e or R 2 ). Some degradation is simple algebraic. Is the loss of fit 'large?' In both cases, if the hypothesis is true, the answer will be no. Test Statistics Forming test statistics: For distance measures use Wald type of distance measure, W = (1/J) m ′ [Est.Var( m )] m For the fit measures, use a normalized measure of the loss of fit: [(R 2 - R* 2 )/J] F = ----------------------------- [(1 - R 2 )/(n - K)] Testing Procedures How to determine if the statistic is 'large.' Need a 'null distribution.' Logic of the Neyman-Pearson methodology. If the hypothesis is true, then the statistic will have a certain distribution. This tells you how likely certain values are, and in particular, if the hypothesis is true, then 'large values' will be unlikely. If the observed value is too large, conclude that the assumed distribution must be incorrect and the hypothesis should be rejected. For the linear regression model, the distribution of the statistic is F with J and n-K degrees of freedom. Distribution Under the Null Density of F[3,100] X .250 .500 .750 .000 1 2 3 4 FDENSITY Particular Cases Some particular cases: One coefficient equals a particular value: F = [(b - value) / Standard error of b ] 2 = square of familiar t ratio. Relationship is F [ 1, d.f.] = t 2 [d.f.] A linear function of coefficients equals a particular value (linear function of coefficients - value) 2 F = ---------------------------------------------------- Variance of linear function Note square of distance in numerator Suppose linear function is Σ...
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This note was uploaded on 06/20/2011 for the course ECON 803 taught by Professor Pp during the Spring '11 term at Thammasat University.

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Econometrics-I-9 - Applied Econometrics William Greene...

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