Econometrics II

where the unobserved variable f i is a ’firm effect’ to each employee at a given firm i. Theerror term v i,e is specific to employee e at firm i. The composite error is u i,e = f i + v i,e ,such as in equation (8.28).(a) Assume that V ar(f i ) = σ 2 f , V ar(v i,e) = σ 2 v, and f i and v i,e are uncorrelated. Showthat V ar(u i,e ) = σ 2 f + σ2 v; call this σ 2 .(b) Now suppose that for e ≠ g, v i,e and v i,g are uncorrelated. Show that Cov(u i,e , u i,g ) =σ 2 f .(c) Let ū i = m −1ithat V ar(ū i ) = σ 2 f + σ2 v/m i .∑ mii=1 u i,e be the average of the composite errors within a firm. Show(d) Discuss the relevance of part (c) for WLS estimation using data averaged at the firmlevel, where the weight used for observation i is the usual firm size.5. Wooldridge C8.22