Consider the simple regression model with classical measurement error, y = β0 + β 1x* + u,...

Consider the simple regression model with classical measurement error, y = β0 + β 1x* + u, where we have m measures on x*. Write these as zh = x* + eh, 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.