Panel Data and Unobservable Individual Effects

An Important purpose in combining time-series and cross-section data is to control for
individual-specific unobservable effects which ma? be correlated with other explanatorq
variables. Using exogeneitq restrictions and the time-~nvariant characteristic of the latent
variable. we derive ( i ) a test for the presence of thls effect and for the over-identifqing
restrictions we use. (ii) necessarq and sufficient conditions for identification. and ( i i i ) the
asqmptoticallq efficient instrumental variables estimator and conditions under which it
differs from the wlth~n-groups estimator. We calculate efficient estimates of a wage
equation from the Michigan income dqnamics data which indicate substantial differences
from within-groups or Balestra-Nerlove estimates-particularlq, a significantl? higher
estimate of the returns to schooling.
AN IMPORTAKT BEKEFIT from pooling time-series and cross-section data is the
ability to control for individual-specific effects-possibly unobservable-which
may be correlated with other included variables in the specification of an
economic relationship. Analysis of cross-section data alone can neither identif)
nor control for such individual effects. To consider a specific model, let
(1.1) y,, = X,,D + Z,Y+ a, + T,, (I = 1. . . . . N: t = 1 , . . . , T),
where D and y are k and g vectors of coefficients associated with time-varying
and time-invariant observable variables respectively. The disturbance T,, is assumed
uncorrelated with the columns of (X,Z.a) and has zero mean and
constant variance a; conditional on X,, and Z,. The latent individual effect a, is
assumed to be a time-invariant random variable, distributed independently
across individuals, with variance 02.