Abstract
A dynamic panel data model is considered that contains possibly stochastic individual components and a common stochastic time trend that allows for stationary and nonstationary long memory and general parametric short memory. We propose four different ways of coping with the individual effects so as to estimate the parameters. Like models with autoregressive dynamics, ours nests I(1) behaviour, but unlike the nonstandard asymptotics in the autoregressive case, estimates of the fractional parameter can be asymptotically normal. For three of the estimates, establishing this property is made difficult due to bias caused by the individual effects, or by the consequences of eliminating them, which appears in the central limit theorem except under stringent conditions on the growth of the cross-sectional size N relative to the time series length T, though in case of two estimates these can be relaxed by bias correction, where the biases depend only on the parameters describing autocorrelation. For the fourth estimate, there is no bias problem, and no restrictions on N. Implications for hypothesis testing and interval estimation are discussed, with central limit theorems for feasibly bias-corrected estimates included. A Monte Carlo study of finite-sample performance is included.
Highlights
Important features of many econometric models for panel data are unobserved individual fixed effects and temporal dynamics that possibly allow for nonstationarity
There may be scope for relaxing our restrictions on D, though these restrictions appear to play a role in ensuring that the approximation errors stemming from the presence of the individual effects αi, or from the measures we take to eliminate them, are small enough to enable our estimates to be consistent and asymptotically normally distributed
The bias decays to zero for δ > 0, but more or less slowly, and its presence explains the need for asymptotic theory with T → ∞, in order to achieve consistent estimation of θ0
Summary
Important features of many econometric models for panel data are unobserved individual fixed effects and temporal dynamics that possibly allow for nonstationarity. A consequence established in that literature is that large sample inference based on an approximate Gaussian pseudo likelihood can be expected to entail standard limit distribution theory; in particular, Lagrange multiplier tests on θ0 (for example of the I(1) hypothesis δ0 = 1) are asymptotically χ 2 distributed with classical local power properties, and estimates of θ0 are asymptotically normally distributed with the usual parametric rate (see Robinson (1991, 1994), Beran (1995), Velasco and Robinson (2000), Hualde and Robinson (2011)) This is the case whether δ0 lies in the stationary region (0, 1/2) or the nonstationary one [1/2, ∞) (or, the negative dependent region (−∞, 0)). Our proofs use technical lemmas, stated and proved in Appendix C; we draw attention here to Lemma 3, which is a technical tool that is central to the consistency proofs, and Lemma 4, which is of some independent interest
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