Question: Consider the simple regression model with classical measurement error, y = (0 + (0x* + u, where we have m measures on x*. Write these

Consider the simple regression model with classical measurement error, y = (0 + (0x* + u, where we have m measures on x*. Write these as zh - x* + eh, h - 1, .... m. Assume that x* is uncorrelated with u, e1, ....., em, that the measurement errors are pairwise uncorrelated, and have the same variance, (2e. Let w = (z1 + ... + zm)/m be the average of the measures on x*, so that, for each observation i, wi = (zi1 + ... + zim)/m is the average of the m measures. Let 1 be the OLS estimator from the simple regression yi on 1, wi, i = 1, ...., n, using a random sample of data.
(i) Show that

[The plim of 1 is Cov(w, y)/Var(w).]
(ii) How does the inconsistency in 1 compare with that when only a single measure is available (that is, m = 1)? What happens as m grows? Comment.

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