Of the random variables y y x only the pair y x are

Question

Econometrics 710 Midterm Exam 1999

1. Let Y be n × 1, X be n × k (rank k), and Z = XB, where B is k × k with rank h. Let (β^ˆ, e^ˆ) denote the OLS coefficients and residuals from regression of y on X. Similarly, let (β^˜, e^˜) denote these from OLS regression of y ou Z.  Find the relationship between β^ˆ and β^˜, and the relationship between e^ˆ and e^˜.

2. Let Y be n × 1, X be n × k (rank k). Suppose that T (Y | E) = Xβ. Define the ridge regression estimator

β^ˆ = (X' X + λI_k)^-1 (X'Y)

where λ > 0 is a fixed constant.  Find E(β^{^}|X). Is β^{^} biased for β?

3. Of the random variables (Y^*, Y, X) only the pair (Y, X) are observed.  (In this case, we say that Y^* is a latent variable.) Suppose E(Y^*| X)= Xβ and Y = Y* + u, where u is a measurement error satisfying E(u|Y^*, X) = 0. Let β^ˆ denote the OLS coefficient from the regression of Y on X.

(a) Find E(Y|X).

(b) Is β^ˆ consistent for β as n →∞?

(c) Find the asymptotic distribution of √n(β^ˆ - β). as n →∞.

4.  You run an OLS regression of the form y^ˆ = βˆx₁ + β^ˆ₂x₂, where y=executive salaries on x₁ = sales and x₂ = profits, across a sample of 102 firms.  The results are

(All variables are expressed as deviations about their means. The numbers in parenthesis are standard errors. V^ˆ is the estimated covariance matrix for β^ˆ)

(a) Someone suggests that the high collinearity between sales and profits has prevented precise estimation of the parameters. Does this seem reasonable, based on the evidence presented?

(b) Someone else suggests a method to eliminate this problem. First, regress profits on sales, and obtain the residuals x^*₂. Second, regress y on x1 and x^*₂ to estimate the salary function. Denote the results of the second step by y^˜ = β^˜₁x₁ + β^˜₂x₂. Find an expression for x^*₂.

(c) Calculate β^˜₁ and β^˜₂.

(d) Calculate their conventional standard errors.

(e) Evaluate this proposal as a devise to eliminate (or reduce) collinearity.