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 (βˆ, eˆ) 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 βˆ and β˜, and the relationship between eˆ 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
βˆ = (X' X + λIk)-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 βˆ denote the OLS coefficient from the regression of Y on X.
(a) Find E(Y|X).
(b) Is βˆ consistent for β as n →∞?
(c) Find the asymptotic distribution of √n(βˆ - β). as n →∞.
4. You run an OLS regression of the form yˆ = βˆx1 + βˆ2x2, where y=executive salaries on x1 = sales and x2 = profits, across a sample of 102 firms. The results are
![485_Figure.png](https://secure.tutorsglobe.com/CMSImages/485_Figure.png)
(All variables are expressed as deviations about their means. The numbers in parenthesis are standard errors. Vˆ is the estimated covariance matrix for βˆ)
(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.