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

Lastly, the following theorem finds the asymptotic distribution of the Haus-
man test statistic under the random effect assumption and the local alternatives:

Theorem 9 Suppose that Assumptions 1-8 and 12 hold. Corresponding to the
size of (χi,z0)⁰, partition Ξ and λ, respectively, as follows:

Ξxx Ξxz            λx

Ξ =   _Ξ0_{xz Ξzz}   ; λ =   λ_z   .

Then, as (N,T →∞),

(a) under Assumption 9 (random effects),

HMNT ⇒ χ²_k ;

(b) under Assumption 11 (local alternatives to random effects),

HMNT ⇒ χ²_k (η),

where η = λ⁰x (Ξ_xx — Ξ_xzΞ^-1Ξ⁰_xz)λ_x/σ2_l is the noncentral parameter.

Theorem 9 shows that under the random effects assumption, the Hausman
statistic is asymptotically χ²-distributed with degrees of freedom equal to k
(the number of the time-varying regressors). Furthermore, Theorem 9 (ii) shows
that the Hausman statistic has significant local power to detect any correlation
between the time-varying regressors x_it and the effect u_i . This is so, because
the noncentral parameter η equals zero if, and only if, λ_x = 0_k×1 . In contrast,
the noncentrality parameter η does not depend on λ_z , indicating that the Haus-
man test has no power to detect nonzero correlations between time invariant
regressors zi and the individual effect ui in the direction of our local alternative
hypotheses (Assumption 11). This result holds even if T is finite and fixed. As
discussed earlier, this is due to the fact that the conditional mean of the effect u_i
is a linear function of regressors under our local alternative hypotheses. When
the conditional mean is not linear, the Hausman test could have a power to
detect nonzero correlations of the effect ui with the time invariant regressors.
However, the power of the Hausman test to such correlations is generally lim-
ited. This is so because the Hausman test can detect such correlations only if
they can cause a large bias in the between estimator of β, the coefficient vector
on time-varying regressors (see Ahn and Low, 1996).

5 Conclusion

This paper has considered the asymptotic properties of the popular panel data
estimators and the Hausman test. We find that the convergence rates of the
estimators and the test statistic are sensitive to data generating process. In
particular, the convergence rates of the between estimator crucially depend on
whether the data are cross-sectionally heteroskedastic or homoskedastic. De-
spite the different convergence rates, however, the estimators are consistent and

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