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Econometrics

This article discusses hypothesis testing in econometrics, specifically in the context of general linear hypothesis tests. It covers the analytical framework, forming test statistics, testing procedures, and the distribution under the null hypothesis. The application section focuses on testing cost function price homogeneity and the fundamentals of test size and power.

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Econometrics

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  1. Chengyuan Yin School of Mathematics Econometrics

  2. Econometrics 9. Hypothesis Tests: Analytics and an Application

  3. General Linear Hypothesis Hypothesis Testing Analytical framework: y = X +  Hypothesis: R - q = 0, J linear restrictions

  4. 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 - q on the regression must degrade the fit (e'e or R2). Some degradation is simple algebraic. Is the loss of fit 'large?' In both cases, if the hypothesis is true, the answer will be no.

  5. 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: [(R2 - R*2)/J] F = ----------------------------- [(1 - R2)/(n - K)]

  6. 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.

  7. Distribution Under the Null

  8. Particular Cases Some particular cases: 1. 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.] = t2[d.f.] 2. 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 k wk bk Variance is kl wkwl Cov[bk,bl] This is the Wald statistic. Also the square of the somewhat familiar t statistic. 3. Several linear functions. Use Wald or F. Loss of fit measures may be easier to compute.

  9. Application: Cost Function

  10. Price Homogeneity

  11. Imposing the Restrictions

  12. Homotheticity

  13. Testing Fundamentals - I • SIZE of a test = Probability it will incorrectly reject a “true” null hypothesis. • This is the probability of a Type I error. Under the null hypothesis, F(3,100) has an F distribution with (3,100) degrees of freedom. Even if the null is true, F will be larger than the 5% critical value of 2.7 about 5% of the time.

  14. Testing Fundamentals - II • POWER of a test = the probability that it will correctly reject a “false null” hypothesis • This is 1 – the probability of a Type II error. • The power of a test depends on the specific null.

  15. Power of a Test Null: Mean = 0. Reject if observed mean < -1.96 or > +1.96.Prob(Reject null|mean=0) = 0.05 Prob(Reject null|mean=.5)=0.07902 Prob(Reject null|mean=1)=0.170066. Increases as the (alternative) mean rises.

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