14.12 Consider the fixed-effects panel data model Yjt = aj + ujt for j = 1,c, k and t = 1,c, T. Assume that ujt is i.i.d. across entities j and over time t with E1ujt2 = 0 and var1ujt2 = s2 u.

a. The OLS estimator of aj is the value of aj that makes the sum of squared residuals gk j = 1gTt

= 11Yjt - aj2 2 as small as possible. Show that the OLS estimator is an j = Yj =

1 T

gTt

= 1 Yjt.

b. Show that i. an j is an unbiased estimator of aj.

ii. var1an j2 = s2 u >T.

iii. cov1an i, an j2 = 0 for i  j.

c. You are interested in predicting an out-of-sample value for entity j—that is, for Yj,T+1—and use an j as the predictor. Show that MSPE = s2 u + s2 u>T.

d. You are interested in predicting an out-of-sample value for a randomly selected entity—that is, for Yj,T+1, where j is selected at random. You again use an j as the predictor. Show the MSPE = s2 u + s2 u >T.

e. The total number of in-sample observations is n = kT. Show that in both

(c) and

(d) MSPE = s2 u11 + k>n2.