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Let {(yi* ; xi); 1 ≤ i ≤ n} be an i.i.d sequence of random variables where yi* and xi satisfy the linear relationship
yi* = β0 + β1xi + ∈i
with Cov(xi; ∈i) = 0. Now, suppose there is measurement error in yi* ; that is, suppose yi* is unobserved and instead we observe
yi = yi*+ ui;
where ui is independent of both xi and ∈i.
(a) Will running a regression of yi on xi produce a consistent estimator of the slope parameter β1? Justify your answer.
(b) What impact does measurement error have on the precision of the least squares estimator of the slope parameter? Hint: Compare the conditional variance for the OLS estimator from part (a) to that obtained under no measurement error (i.e. when we observe yi*).
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