Consider the linear model y = X???? + u (T × 1), where X is T × k and ???? is k × 1, plim T−1X′X = MXX < ∞ having full rank, but plim T−1X′u ≠ ????.

(a) Show that the ordinary least squares estimator is inconsistent.

(b) Suppose there exists a matrix W (T × k) such that plim T−1W′u = ????, while MWX = plim T−1W′X is a matrix with full rank k. Show howthismatrix can be used to derive a consistent estimator of ????.

(c) Further suppose that T−1∕2W′u d−
→N(0, ????2MWW), where d−
→ denotes convergence in distribution and MWW = plim T−1W′W, which by assumption is a finite matrix of full rank. Use this information to derive the asymptotic distribution of the estimator you obtained in part (b).

(d) Comment on any generalizations and applications of this procedure that you know of.