a. Just substitute b_i = β + d_i into (1) and rearrange:

where v_it ≡ x_itd_i + u_it.

b. The time-demeaned equation is simply

where

.

c. The fixed effects estimator is simply the pooled ordinary least squares estimator applied to the time-demeaned equation (5). So we can write the fixed effects estimator as

By the rank condition (3) and the weak law of large numbers (WLLN),

, where

. Consistency follows, again by the WLLN, if

Sufficient is

. But

, t = 1,…,T, under (2), which implies

. Further, (4) is equivalent to

; the law of iterated expectations then implies that

.

Condition (4) only restricts the mean dependence between b_i, and the time-demeaned variables,

; it places no direct restriction on the dependence between b_i, and _i. Often, it is reasonable to think that although the time averages of some covariates would be correlated with unobserved heterogeneity—in particular, with b_i—deviations about the time average might not be. So the fixed effects estimator, although not robust to arbitrary dependence between b_i and (x_i1,x_i2,…,x_iT), does have some useful robustness properties in the general random coefficient panel data model.

d. Zero correlation does not suffice because we need b_i to be uncorrelated with

.