7 Consider the simple regression model with classical measurement error, y 5 b0 1 b1x* 1 u, where we have m measures on x*. Write these as zh 5 x* 1 eh, h 5 1, …, m. Assume that x* is uncorrelated with u, e1, ...., em, that the measurement errors are pairwise uncorrelated, and have the same variance, s2

e. Let w 5 (z1 1 … 1 zm)/m be the average of the measures on x*, so that, for each observation i, wi 5 (zi1 1 … 1 zim)/m is the average of the m measures. Let - b1 be the OLS estimator from the simple regression yi on 1, wi, i 5 1, …, n, using a random sample of data. (i) Show that plim( - b1) 5 b1 s2 _____________ x* [s2 x* 1 (s2 em)] . [Hint: The plim of - b1 is Cov(w, y)/Var(w).] (ii) How does the inconsistency in - b1 compare with that when only a single measure is available (that is, m 5 1)? What happens as m grows? Comment.