Question: Let xt and yt be random variables, having mean zero, and related by E(yt |xt) = ????xt w.p.1. Given a random sample of the pairs

Let xt and yt be random variables, having mean zero, and related by E(yt

|xt) = ????xt w.p.1.

Given a random sample of the pairs (yt, xt), for t = 1,…, T, consider the ordinary least squares estimator of ????

????̂ =

Σ

Σtxtyt tx2t

.

(a) Is ????̂ unbiased for ?????Why? (or,Why not?)

(b) If E(u2t

|xt) = ????2 w.p.1, where ut = yt − ????xt, what is the variance of ????̂?

(c) If ut

|xt is normally distributed where ut = yt − ????xt, what can be said about the distribution of ????̂?

(d) Assuming ut

|xt as defined in part

(c) is normally distributed, how would you perform a test of the null hypothesis ???? = 0?

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