Consider a regression model yt = x′t ???? + ut t = 1,…, T where ut is identically and independently distributed with mean 0 and variance ????2 and xt (2 × 1) is the vector with elements 1 and t.

(a) Does the matrix T−1

Σ

xtx′t converge to a finite limit?

(b) Is the least squares estimator ????̂ a consistent estimator of ?????

(c) Construct a diagonal matrix KT , depending on T, such that KT (????̂ − ????) has a non-degenerate distribution with finite variance as T → ∞.

(d) Is this limiting distribution normal? (Intelligent guess called for.)

Hint: two well-known identities are

ΣT t=1 t = T(T + 1)

2

,

ΣT t=1 t2 = T(T + 1)(2T + 1)

6

.