Introductory Econometrics 4th Edition by Jeffrey Wooldridge

تمرين 7

Consider a model at the employee level,
where the unobserved variable f i is a "firm effect" to each employee at a given firm i. The error term v i,e is specific to employee e at firm i. The composite error is u i,e = f 1 + v i,e , such as in equation.
(i) Assume that Var(f 1 ) = 2 f , Var(v ie ) = 2 v , and f i and v i,e are uncorrelated. Show that Var(u i,e ) = 2 f + 2 v ; call this 2.
(ii) Now suppose that for e g, v i,e and v i,g are uncorrelated. Show that Cov(u ie ,u ig ) = 2 f.
(iii) Let be the average of the composite errors within a firm. Show
(iv) Discuss the relevance of part (iii) for WLS estimation using data averaged at the firm level, where the weight used for observation i is the usual firm size.