Let be the OLS estimate from the regression of y on X. Let A be a

Let β̂ be the OLS estimate from the regression of y on X. Let A be a (k + 1) × (k + 1) nonsingular matrix and define z_t ; x_tA, t = 1, . . . , n. Therefore, z_t is 1 × (k + 1) and is a nonsingular linear combination of x_t. Let Z be the n × (k + 1) matrix with rows z_t. Let β̂ denote the OLS estimate from a regression of y on Z.

(i) Show that β̂ = A^–1 β̂.

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

(iii) Show that the estimated variance matrix for β̂ is ŝ²A^–1 (X’X)^–1A^–1’, where ŝ² is the usual variance estimate from regressing y on X.

(iv) Let the β̂_j be the OLS estimates from regressing y_t on 1, x_t1, . . . , x_tk, and let the β̂_j be the OLS estimates from the regression of y_t on 1, a₁x_t1, . . . , α_kx_tk, where α_i ≠ 0, j = 1, . . . , k. Use the results from part (i) to find the relationship between the β̂_j and the β̂_j.

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

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