What assumption about x guarantees the existence of beta

Question

1. Consider the linear regression model

y = X_β + 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,...................,β_k) 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') = σ²I_n, σ² is an unknown positive constant, and I_n 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 ≠ I_n)? Explain in detail.

2. Consider the linear regression model

Y_t = β₀ + β1X_1t + β₂X_2t + u_t   t =1,2,...,n,

where Xit and X_2t are non-stochastic stochastic explanatory variables and u_t is a random disturbance such that E(u_t) = 0 for all t.

(a)  Suppose that E(u_tu_s) = 0 for all t ≠ s and E(u_t²) = σ² exp(X_1t + X_2t), 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 u_t = Φu_t-4 + ε_t, where Φ is a known parameter and ε_t are random variables such that E(ε_t) = 0, E(ε_t²) = σ_ε² > 0, and E(ε_tε_s) = 0 for all t ≠ s. Explain how to obtain an efficient estimate of β.

(c)  Suppose that the u_t's are homoskedastic and serially uncorrelated, and that X_1t and X_2t are stochastic with E(X_1tu_t) = 0 for all t and E(X_2tu_t)≠ 0. Assuming that three valid instruments (W_1t, W_2t, W_3t) 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 = β_o+ β₁GIP_t + β₂GM_t + β₃GP_t + u_t, t = 1,..........., 445,

where u_t 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 Η_o: β₁ = 0 against H₁ : β₁ < 0 at the 5% significance level.

(d) Test the hypothesis that β_1, β_{2, and} β₃ are jointly equal to zero (at the 5% significance level).

(e) Test the hypothesis H_o : β₂ = - β₁ against H₁ β₁ ≠ - β₁ (at the 5% significance level).

(f)  Under what assumptions about u_t 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.