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Midterm07

# Midterm07 - Department of Economics ECONOMETRICS I Fall...

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2 . Suppose (y,x) have a bivariate normal distribution in which E[y] = 0, E[x] = 0, Var[y] = 1, Var[x] = 1, Cov[x,y] = ρ . We have a random sample (y i ,x i ),i = 1,…,n. We are interested in estimating ρ which is the one unknown parameter in this distribution. [5] a. By virtue of the law of large numbers, the sample covariance between x and y can be used to estimate ρ consistently. Explain. [10] b. An alternative approach: We know that E[y|x] = α + β x where β = Cov[x,y]/Var[x] = ρ and α = E[y] - β E[x] = 0. So, y = ρ x + ε . Thus, linear regression of y on x without a constant term consistently estimates ρ . Correct? Explain. What is the asymptotic distribution of this estimator? Explain. How does this estimator differ from the one in part a? [5] c. An alert observer of part b notices that the equation there implies that x = (1/ ρ )y - (1/ ρ ) ε = δ y + δε He therefore suggests that we can also regress x on y to estimate δ , then, by virtue of the Slutsky theorem, obtain a consistent estimator of ρ by taking the reciprocal of the estimator of δ . True or false. Explain.
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Midterm07 - Department of Economics ECONOMETRICS I Fall...

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