By Lord Greene
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Additional resources for Econometric Analysis. SOLUTIONS
Under the null hypothesis of equal variances, the ratio F = [e1′e1/(n1 - 2)]/[e2′e2/(n2 - 2)] (or vice versa for the subscripts) is the ratio of two independent chi-squared variables each divided by their respective degrees of freedom. Although it might seem so from the discussion in the text (and the literature) there is nothing in the test which requires that the coefficient vectors be assumed equal across groups. 8 to the F table. 61, so the hypothesis of equal variances is rejected. ∧ The two step estimator is β = [(1/s12)X1′X1 + (1/s22)X2′X2]-1[(1/s12)X1′y1 + (1/s22)X2′y2].
60, respectively. 99, so the second and third are statistically significant while the first is not. The disturbance variance underlying these data is, in fact, σi2 = σ2(1 + γXi22) so the Goldfeld-Quandt and Glejser tests have given the right diagnosis. For the Glejser test, the finding that the linear model is inappropriate makes sense since X2 takes negative values. 44 6. Using the data of Exercise 5, reestimate the parameters, using a two step feasible GLS estimator. Try (12-19), (12-20), and (12-21).
Consider the simple errors in variables model, y = α + βx* + ε, x = x* + u, where u and ε are uncorrelated, and x is the erroneously measured, observed counterpart to x*. (a) Assume that x*, u, and ε are all normally distributed with means µ*, 0, and 0, variances σ*2, σu2, 2 and σε and zero covariances. Obtain the probability limits of the least squares estimates of α and β. (b) As an alternative, consider regressing x on a constant and y, then computing the reciprocal of the estimate. Obtain the probability limit of this estimate.