Introductory Econometrics 4th Edition by Jeffrey Wooldridge

Exercise 10

Consider the simple regression model with classical measurement error, y = 0 + 1x* + u, where we have m measures on x*. Write these as z h = x* + e h , h = 1,..., m. Assume that
x* is un-correlated with u, e1,.... , em, that the measurement errors are pairwise uncorrelated,
and have the same variance, a2. Let w = (z1 +... + zm)/m be the average of the measures on x*, so that, for each observation i, w. = (z., +... + z.)/m is the average of the m measures. Let 31 be the OLS estimator from the simple regression y. on 1, w. , i = 1,..., n, using a random sample of data.
(i) Show that
[Hint: The plim of 3-1 is Cov(w,-y)/Var(w).]
(ii) How does the inconsistency in 31 compare with that when only a single measure is available (that is, m = 1) What happens as m grows Comment.