Assume that the model y = Xβ + u satisfies the Gauss-Markov assumptions, let G be a (k + 1...

Assume that the model y = Xβ + u satisfies the Gauss-Markov assumptions, let G be a (k + 1) × (k + 1) nonsingular, nonrandom matrix, and define δ =Gβ, so that δ is also a (k + 1) × 1 vector. Let be the (k + 1) × 1 vector of OLS estimators and define =G as the OLS estimator of δ.

(i) Show that E(X) =δ.

(ii) Find Var(X) in terms of σ2, X, and G.

(iii) Use Problem

Problem Let be the OLS estimate from the regression of y on X. Let A be a (k + 1) × (k + 1) nonsingular matrix and define zt = xtA, t = 1,….,n. Therefore, zt is 1 × (k + 1) and is a nonsingular linear combination of xt. Let Z be the n × (k + 1) matrix with rows zt. Let denote the OLS estimate from a regression of y on Z.

(i) Show that =A-1.

(ii) L et be the fitted values from the original regression and let . y t be the fitted values from regressing y on Z. Show that =, for all t =- 1, 2,…., n. How do the residuals from the two regressions compare?

(iii) Show that the estimated variance matrix for is A1(XʹX)1A1ʹ, where is the usual variance estimate from regressing y on X.

(iv) L et the be the OLS estimates from regressing yt on 1, xt1,...,xtk, and let the be the OLS estimates from the regression of yt on 1, a1xt1,…,ak xtk, where aj ≠ 0, j =1,…,k. Use the results from part (i) to find the relationship between the and the

(v) Assuming the setup of part (iv), use part (iii) to show that se() = se()/aj.

(vi) Assuming the setup of part (iv), show that the absolute values of the t statistics for and are identical.

to verify that and the appropriate estimate of Var( X) are obtained from the regression of y on XG1.

(iv) Now, let c be a (k +1) × 1 vector with at least one nonzero entry. For concreteness, assume that ck ≠ 0. Define θ=cʹβ, so that θ is a scalar. Define δj=βj, j =1, ...,k 1 and δk = 0. Show how to define a (k+ 1) × (k + 1) nonsingular matrix G so that δ =Gβ.

(v) Show that for the choice of G in part (iv),

Use this expression for G1 and part (iii) to conclude that and its standard error are obtained as the coefficient on xtk /ck in the regression of yt on [1 (c0/ck)xtk], [xt1 (c1/ck)xtk],..., [xt,k1 (ck1/ck)xtk], xtk/ck, t = 1,...,n. This regression is exactly the one obtained by writing βk in terms of θ and β0, β1,...,βk1, plugging the result into the original model, and rearranging. Therefore, we can

formally justify the trick we use throughout the text for obtaining the standard error of a linear combination of parameters.