Large-N and Large-T Properties of Panel Data Estimators and the Hausman Test

about the relative sizes of N and T . Previous studies of nonstationary panel
data typically assume that T is relatively greater than N . Differently from these
studies, our approximation theories can apply to any panel data set with large
N and large T regardless of their asymptotic ratio.

The main findings of this paper are as follows. Consistent with the pre-
vious studies, we find that the within and GLS estimators (of coefficients on
time-varying regressors) are asymptotically equivalent. Nonetheless, the Haus-
man statistic is asymptotically χ² -distributed under the random effects assump-
tion. This seemingly contradictive result can be explained by our finding that
the differences between the within and GLS estimators converge in probabil-
ity to zeros much faster than the two estimators converge in probability to
the true values of coefficients (at the same speed). For example, for a sim-
ple model with a no-trend time-varying regressor, we find that the within and
GLS estimators (of the coefficient on the only time-varying regressor) are both
NT—-consistent and asymptotically normal. In addition, the two estimators
are NN——equivalent in the sense that the difference between the two estima-
tors is o_p(1∕√NT). However, we also find that the difference is θp(1∕√NT² )
or Op (1∕√NT³) depending on whether data are cross-sectionally heteroskedas-
tic or homoskedastic. This implies that the within and GLS estimators are not
equivalent in the orders of √N7'² or √N—³. Furthermore, it is shown that un-
der the random effects assumption, the differences between the within and GLS
estimators are asymptotically normally distributed. From this, we show that
the Hausman test statistic is asymptotically χ²-distributed. In addition, our
analysis under a series of local alternative assumptions indicates that the Haus-
man test retains power to detect violations of the random effects assumption
even if — →∞.

This paper is organized as follows. Section 2 introduces the panel model of
interest here, and defines the within, GLS estimators and the Hausman test.
For several simple illustrative models, we derive the asymptotic distributions
of the within estimators, the GLS estimators, and the Hausman test statistic.
We show that the convergence rates of the estimators and the Hausman test
statistic are sensitive to the unknown data generating structure. Section 3
defines a conditional α-mixing coefficient. Using this, we propose a conditional
α-mixing process and discuss its properties. In Section 4, we provide our general
asymptotic results. Concluding remarks follow in section 5. All the technical
derivations and proofs are presented in the Appendix.

2 Preliminaries

2.1 Estimation and Specification Test

The model under discussion here is given:

yit = β⁰xit + γ⁰zi + ζ + εit = δ⁰wit + ζ + εit; εit = ui + vit,        (1)

where i = 1,..., N denotes cross-sectional (individual) observations, t = 1,...,

<<   1   2   3   4   5   6   7   >>

More intriguing information