Asymptotic inference for dynamic panel estimators of infinite order autoregressive processes

Abstract

In this paper we consider the estimation of a dynamic panel autoregressive (AR) process of possibly infinite order in the presence of individual effects. We employ double asymptotics under which both the cross-sectional sample size and the length of time series tend to infinity and utilize the sieve AR approximation with its lag order increasing with the sample size. We establish the consistency and asymptotic normality of the fixed effects estimator and propose a bias-corrected fixed effects estimator based on a theoretical asymptotic bias term. Monte Carlo simulations demonstrate the usefulness of bias correction. As an illustration, the proposed methods are applied to dynamic panel estimation of the law of one price deviations among US cities.

Introduction

In recent decades, an increasing number of panel datasets with longer time series have become available for economic analysis. In this paper, we investigate the possible benefits of using such panel data in estimating a general dynamic structure described by an infinite order panel autoregressive (AR) model. To this end, we follow recent studies in dynamic panel analyses by using an asymptotic approximation with not only a cross-sectional dimension $N$ but also a time series dimension $T$ that tends to infinity. For example, using this type of asymptotic framework, Hahn and Kuersteiner (2002), Alvarez and Arellano (2003) and Hayakawa (2009) among others, have investigated the properties of various estimators for finite order panel AR models. We consider a more general dynamic linear model that is less subject to problems caused by possible model misspecification.¹ Our approach is to approximate a panel AR model of infinite order by letting the AR order $p$ increase with $T$. Such an idea of the AR sieve approximation in estimating a general linear model has long been used in the time series analysis literature. To the best of our knowledge, however, it has yet to be used in the inference of dynamic panel models. It is our intention to fill the gap between these two bodies of literature.

There are a number of important empirical issues to which our method can be applied. In macroeconomic analysis, the long-run cumulative effect of productivity or demand shocks on the economy is often of interest and time series data have been used to measure the persistence of shocks. Once the general linear model is expressed as an AR model of infinite order, the sum of the AR coefficients (SAR) can be used as a formal measure of the persistence. The AR sieve estimator of the SAR, however, is known to converge at a rate of $\sqrt{T∕p}$ which is slower than the order of $\sqrt{T}$. By incorporating cross-sectional information, our dynamic panel procedure can offer increased precision of the persistence estimator with its convergence rate $\sqrt{NT∕p}$ which can be faster than $\sqrt{T}$. As an empirical illustration, we estimate the SAR of the law of one price (LOP) deviations among US cities based on the micro price panel data of individual goods used in Crucini et al. (2015). Another useful application of our approach is the literature on dynamic panel vector AR (VAR) models. Allowing for heterogeneity among households and firms by using micro data has become an important issue in recent VAR analyses. For example, Franco and Philippon (2007) and Head et al. (2014), among others, estimate structural panel VAR models with a moderately large number of time series observations $T$. The use of such VAR models without prespecifying the lag length can be justified by our results for the multivariate case.

We begin our analysis with the fixed effects (FE) estimator. Under some regularity conditions, we show the consistency and asymptotic normality of the FE estimator which are comparable to those of the ordinary least squares (OLS) estimator in a time series setting, including the ones obtained by Lewis and Reinsel (1985). The presence of the individual fixed effects in the dynamic panel setting, however, makes the analysis more complicated than in the time series setting, creating an asymptotic bias of order $\sqrt{p}∕T$. If an intercept term is included in the analysis of Lewis and Reinsel (1985), there will also be a bias term of the same order. However, it converges to zero at a rate faster than the rate of convergence of the OLS estimator ($\sqrt{T}$) because $\sqrt{T}\times \sqrt{p}∕T=\sqrt{p∕T}\to 0$ under $p∕T\to 0$ which is implied by the rate conditions used to prove the consistency and asymptotic normality of the OLS estimator. Therefore, no bias term shows up in the asymptotic distribution when $N=1$. In panel data settings, the order of the bias of the FE estimator is still $\sqrt{p}∕T$, but the FE estimator converges at a faster rate of $\sqrt{NT}$. As a consequence, the asymptotic distribution is contaminated by a bias of order $\sqrt{pN∕T}$ $(=\sqrt{NT}\times \sqrt{p}∕T)$.

Because of the incidental parameters problem of Neyman and Scott (1948), the FE estimator in dynamic panel data models is known to be biased when $N∕T$ is not very small (see, e.g., Nickell (1981), Kiviet (1995), Hahn and Kuersteiner (2002)). One of the important implications of the paper is that the bias can be even more problematic in the estimation of panel AR models of infinite order because the bias increases with the lag order $p$ used in the AR sieve approximation, so that using a sieve AR approach to mitigate the effects of lag order misspecification can adversely contribute to a larger bias.

To eliminate the increased magnitude of the first order bias, we propose a bias-corrected FE (BCFE) estimator based on the consistent estimator of the theoretical bias term. A Monte Carlo simulation suggests that our proposed BCFE estimator works well in reducing the bias of the FE estimator which is not negligible with the sample sizes typically available in practice.² Based on the theoretical results for the asymptotic normality, we can consider asymptotically valid standard errors and an asymptotically valid automatic lag selection procedure in an AR sieve approximation, both of which are useful in conducting inference.

The remainder of this paper is organized as follows. Our model is described in Section 2. The FE estimator and the BCFE estimators are introduced and their asymptotic properties are investigated in Section 3. The finite-sample performance of the estimators is examined in Section 4, and our approach is applied to the real data in Section 5. Concluding remarks are made in Section 6. All mathematical proofs are collected in the Appendix and the supplemental material available on the authors’ web sites.

We use the following notation: for a sequence of vector $a_{it}$, we let $a_t={(a_{1t},\dots ,a_{Nt})}^{\prime }$. The same convention applies to a sequence of a vector denoted by $a_{it}\left(p\right)$ so that $a_t\left(p\right)={(a_{1t}\left(p\right),\dots ,a_{Nt}\left(p\right))}^{\prime }$. A constant $C$ represents an arbitrary constant.