1. Consider the linear regression model
y = Xβ + u,
where y is an n x 1 vector of observations on the dependent variable, X is an n x k matrix of observations on k non-stochastic explanatory variables, β = (β1,...................,βk) is a k x 1 vector of unknown coefficients, u is an n x 1 vector of unobservable random disturbances such that E(u) = 0 and E(uu') = σ2In, σ2 is an unknown positive constant, and In is the it n x n identity matrix. Let β' = (X'/X)-1X'y be the ordinary least squares (OLS) estimator of β.
(a) What assumption about X guarantees the existence of β'? Give an example where this assumption fails.
(b) Define the OLS residuals as u^ = y XB^. Show that X'u^ = 0.
(c) Show that β^ is an unbiased estimator of β and obtain its variance-covariance matrix. How can the standard errors of the elements of β^ be estimated?
(d) Let β = Cy be an unbiased estimator of β, where C is a non-stochastic k x it matrix. Is there a matrix C for which le is a more efficient estimator than β? Explain.
(e) How do your answers to (c) and (d) change if E(uu') = σ2Ω, where Ω is a symmetric and positive definite it n x n matrix (with Ω ≠ In)? Explain in detail.
2. Consider the linear regression model
Yt = β0 + β1X1t + β2X2t + ut t =1,2,...,n,
where Xit and X2t are non-stochastic stochastic explanatory variables and ut is a random disturbance such that E(ut) = 0 for all t.
(a) Suppose that E(utus) = 0 for all t ≠ s and E(ut2) = σ2 exp(X1t + X2t), where σ2 is an unknown positive constant. What are the statistical properties of the OLS estimator of β = ( β0, β1, β2)' in this model? Explain how to obtain the weighted least squares (WLS) estimate of p. What are the statistical properties of WLS estimates?
(b) Suppose that ut = Φut-4 + εt, where Φ is a known parameter and εt are random variables such that E(εt) = 0, E(εt2) = σε2 > 0, and E(εtεs) = 0 for all t ≠ s. Explain how to obtain an efficient estimate of β.
(c) Suppose that the ut's are homoskedastic and serially uncorrelated, and that X1t and X2t are stochastic with E(X1tut) = 0 for all t and E(X2tut)≠ 0. Assuming that three valid instruments (W1t, W2t, W3t) are available, explain how to obtain the instrumental-variables (IV) estimate of β. Would you recommend the use of OLS or IV in this case?
3. The EViews workfile and Excel worksheet contain monthly U.S. data, from February 1959 to February 1996, on the three-month laeasury bill rate (R), the growth rate of the index of industrial production (GIP), the growth rate of nominal money supply (GM), and the growth rate of the producer price index (GP). A model for the Treasury bill rate is:
Rt = βo+ β1GIPt + β2GMt + β3GPt + ut, t = 1,..........., 445,
where ut is an unobservable random disturbance.
(a) Fit the model to the data using OLS.
(b) Comment on the goodness of fit of the estimated model.
(c) Test the hypothesis Ηo: β1 = 0 against H1 : β1 < 0 at the 5% significance level.
(d) Test the hypothesis that β1, β2, and β3 are jointly equal to zero (at the 5% significance level).
(e) Test the hypothesis Ho : β2 = - β1 against H1 β1 ≠ - β1 (at the 5% significance level).
(f) Under what assumptions about ut are the tests in (c), (d) and (e) valid? Test whether these assumptions are satisfied in your model.
(g) If the assumptions you state in (f) are not satisfied, discuss how to construct valid tests of the hypotheses in (c), (d) and (e), and carry them out.
(h) Test the hypothesis that the coefficients of the model underwent a structural change in 1979:10.