Reverse regression continued. This and the next exercise continue the analysis of Exercise 4. In Exercise 4, interest centered on a particular dummy variable in which the regressors were accurately measured, here we consider the case in which the crucial regressor in the model is measured with error. The paper by Kamlich and Polachek (1982) is directed toward this issue. Consider the simple errors in the 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 σ2∗, σ2u , and σ2ε, and zero covariances. Obtain the probability limits of the least squares estimators of α and β.
b. As an alternative, consider regressing x on a constant and y, and then computing the reciprocal of the estimate. Obtain the probability limit of this estimator.
c. Do the “direct” and “reverse” estimators bound the true coefficient?
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