let yi xi 1 le i le n be an iid sequence of

Question

Let {(y_i* ; x_i); 1 ≤ i ≤ n} be an i.i.d sequence of random variables where y_i* and x_i satisfy the linear relationship

y_i* = β₀ + β₁x_i + ∈_i

with Cov(x_i; ∈_i) = 0. Now, suppose there is measurement error in y_i* ; that is, suppose y_i* is unobserved and instead we observe

y_i = y_i*+ u_i;

where ui is independent of both x_i 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 y_i*).