man test would have low power to detect any violation of the random effects
assumption when T is large? This paper is concerned to answer these questions.
The analysis of panel data with both large N and T becomes increasingly
important in the literature. While a variety of estimation and model specifica-
tion testing techniques have been introduced and proposed in the panel data
literature, most of these methods are limited to the analysis of data with large
N and small T . An obvious reason for this limited approach is that most of
the available panel data have only a short history. However, panel data with a
large number of time-series observations have been increasingly more available
in recent years in many economic fields such as international finance, finance, in-
dustrial organization, and economic growth. Furthermore, popular panel data,
such as the Panel Study of Income Dynamics (PSID) and the National Longitu-
dinal Surveys (NLS), contain increasingly more time-series observations as they
are updated regularly over the years. Consistent with this trend, some recent
studies have examined the large-N and large-T properties of the within and
GLS estimators for error-component models.2 For example, Phillips and Moon
(1999) and Kao (1999) establish the asymptotic normality of the within estima-
tor for the cases in which regressors follow unit root processes. Extending these
studies, Choi (1998) considers a general random effects model which contains
both unit-root and covariance-stationary regressors. For this model, he derives
the asymptotic distributions of both the within and GLS estimators. These
papers however do not consider the asymptotic properties of the Hausman test.
The general model we consider in this paper is different from the models
considered by these studies in two ways. First, our model contains the time-
varying regressors that are serially dependent and heteroskedastic over time with
or without time trends. These variables could be cross-sectionally heteroskedas-
tic or homogeneous. Second, our model contains time invariant regressors that
are correlated or uncorrelated with other time-varying regressors. If a time-
varying regressor is correlated with time-invariant regressors, the time series
of it is non-ergodic because the influence of the time invariant random regres-
sors is persistent in all time periods.3 However, under the assumption that such
time-varying regressors satisfy mixing properties conditionally on time invariant
random regressors, we can derive the limiting distributions of various forms of
sample averages of panel data when both N, T →∞. These intermediate results
are used to establish the asymptotic distributions of the panel data estimators
and the Hausman test.
Most of the asymptotic results derived in the paper hold as N, T →∞
without any particular sequence. In addition, we do not make any assumption
2 Some other studies have considered different panel data models with large N and large
T . For example, Levin and Lin (1992, 1993), Quah (1994), Pesaran, Shin and Im (1997),
and Higgin and Zakrajsek (1999) develop unit-root tests for data with large N and large T.
Alvarez and Arellano (1998) and Hahn and Kuersteiner (2000) examine the large-N and large-
T properties of generalized method of moments (GMM) and within estimators for stationary
dynamic panel data models.
3The time series of the panel data considered in Phillips and Moon (1999) is also non-
ergodic. However, in their paper the non-ergodicity arises due to stochastic trends generated
by unit root processes.