T denotes time, wit =(xit,zi)0, and δ = (β0, γ0)0. In model (1), xit is a k × 1
vector of time-varying regressors, zi is a g × 1 vector of time-invariant regres-
sors, ζ is an overall intercept term, and the error εit contains a time-invariant
individual effect ui and random noise vit. We consider the case of both large
numbers of individual and time series observations, so asymptotic properties of
the estimators and statistics for model (1) apply as N, T →∞. The orders
of convergence rates of some estimators depend on whether or not the model
contains an overall intercept term. This problem will be addressed later.
We assume that data are distributed independently (but not necessarily
identically) across different i, and that the vit are independently and identically
distributed (i.i.d.) with var(vit) = σv2. We further assume that ui, xi1, ..., xiT
and zi are strictly exogenous with respect to vit ; that is,
E(vit | ui,xi1,...,xiT)=0,
for any i and t. This assumption rules out the cases in which the set of regres-
sors includes lagged dependent variables or predetermined regressors. Detailed
assumptions about the regressors xi1, ..., xiT,zi will be introduced later.
For convenience, we adopt the following notational rule: For any p × 1 vector
ait, we denote aii = 1 ∑t ait'; eit = ait - ai' a = N1 ∑i âi'; ⅛ = ai - a∙ Thus,
for example^ for wit = (χit, z0)0, we have wi = (x,i,z'0)'; wit = (X0u, 01×g)0;
w = (X0, z); wîi = ((Xi — x)0, (Zi — z)0)0.
When the regressors are correlated with the individual effect, both of the
OLS and GLS estimators of δ are biased and inconsistent. This problem has
been traditionally addressed by the use of the within estimator (OLS on data
transformed into deviations from individual means):
βbw = (Pi,t xeitxe0it)-1 Pi,t xeityei0t.
Under our assumptions, the variance-covariance matrix of the within estimator
is given:
V ar(βbw) = σ2ε(Pi,t xeitxe0it)-1. (2)
Although the within method provides a consistent estimate of β, a serious
defect is its inability to identify γ , the impact of time-invariant regressors. A
popular treatment of this problem is the random effects (RE) assumption under
which the ui are random and uncorrelated with the regressors:
E(ui | xi1, ..., xiT,zi)=0. (3)
Under this assumption, all of the parameters in model (1) can be consistently
estimated. For example, a simple but consistent estimator is the between esti-
mator (OLS on data transformed into individual means):
0
bδb = (βbb, γb0b)0 = (Pi weiwei0)-1 Pi weiyei.
However, as Balestra and Nerlove (1966) suggest, under the RE assumption,
an efficient estimator is the GLS estimator of the following form:
bδg = [Pi,t(weit + θT wei)(weit + θT wei)0]-1 Pi,t(weit + θT wei)(yeit + θT yei)
=[Pi,tweitwei0t+Tθ2TPiweiwei0]-1[Pi,tweityei +Tθ2TPiweiyei],