Elsevier

Journal of Econometrics

Volume 87, Issue 1, November 1998, Pages 115-143
Journal of Econometrics

Initial conditions and moment restrictions in dynamic panel data models

https://doi.org/10.1016/S0304-4076(98)00009-8Get rights and content

Abstract

Estimation of the dynamic error components model is considered using two alternative linear estimators that are designed to improve the properties of the standard first-differenced GMM estimator. Both estimators require restrictions on the initial conditions process. Asymptotic efficiency comparisons and Monte Carlo simulations for the simple AR(1) model demonstrate the dramatic improvement in performance of the proposed estimators compared to the usual first-differenced GMM estimator, and compared to non-linear GMM. The importance of these results is illustrated in an application to the estimation of a labour demand model using company panel data.

Introduction

In dynamic panel data models where the autoregressive parameter is moderately large and the number of time series observations is moderately small, the widely used linear generalised method of moments (GMM) estimator obtained after first differencing has been found to have large finite sample bias and poor precision in simulation studies (see Alonso-Borrego and Arellano, 1996). Lagged levels of the series provide weak instruments for first differences in this case. Here we consider two alternative estimators that impose further restrictions on the initial conditions process, designed to improve the properties of the standard first-differenced GMM estimator.

The first type of restriction justifies the use of an extended linear GMM estimator that uses lagged differences of yit as instruments for equations in levels, in addition to lagged levels of yit as instruments for equations in first differences (see Arellano and Bover, 1995). Monte Carlo simulations and asymptotic variance calculations show that this extended GMM estimator offers dramatic efficiency gains in the situations where the basic first-differenced GMM estimator performs poorly. This estimator is also shown to encompass the GMM estimator based on the non-linear moment conditions available in the dynamic error components model (see Ahn and Schmidt, 1995), and we find substantial asymptotic efficiency gains relative to this non-linear GMM estimator.

The second type of restriction validates the use of the error components GLS estimator on an extended model that conditions on the observed initial values. This provides a consistent estimator under homoskedasticity which, under normality, is asymptotically equivalent to conditional maximum likelihood (see Blundell and Smith, 1991). A Monte Carlo analysis also suggests that this estimator has good finite sample properties. However, the conditional GLS estimator requires homoskedasticity, and only extends to a model with regressors if the regressors are strictly exogenous. This is not the case for the GMM estimators.

Both types of restrictions are satisfied under stationarity but both are also valid under weaker assumptions. The gain in precision that results from exploiting these initial condition restrictions in these two alternative estimators is shown to increase for higher values of the autoregressive parameter and as the number of time series observations gets smaller. Our Monte Carlo analysis finds both a large downward bias and very low precision for the standard first-differenced estimator in these cases. The initial condition information not only greatly improves the precision but also greatly reduces the finite sample bias.

The main contributions of the paper are: to characterise the weak instruments problem for the first-differenced GMM estimator in terms of the concentration parameter (cf. Staiger and Stock, 1997); to demonstrate that the levels restrictions suggested by Arellano and Bover (1995) remain informative in the cases where the first-differenced instruments become weak; to relate these restrictions explicitly to the initial conditions process; to evaluate the asymptotic efficiency gains that result from exploiting these restrictions compared to both the differenced GMM estimator and the non-linear GMM estimator of Ahn and Schmidt (1995); to relate the conditions needed for consistency of the Blundell and Smith (1991) conditional GLS estimator to the initial conditions process; and finally to evaluate the performance of these estimators using both Monte Carlo simulations and an application to company panel data.

The layout of the paper is as follows. In Section 2we briefly review the standard moment conditions for the autoregressive error components model, in the framework of Anderson and Hsiao (1981), Holtz-Eakin et al. (1988), Arellano and Bond (1991), and Ahn and Schmidt (1995). In Section 3we evaluate the problem of weak instruments in the first-differenced instrumental variable estimator. Section 4goes on to consider restrictions on the initial condition process that render lagged values of Δyit valid as instruments for the levels equations and discusses the extended GMM estimator which is available when these restrictions are satisfied. In Section 5we consider the conditional GLS estimator in which initial conditions are explicitly added to the model. Section 6presents the results of Monte Carlo simulations which highlight the potential importance of exploiting the extra moment restrictions relating to the properties of the initial condition process for the efficiency of the AR coefficient estimators. Section 7discusses the extensions to models with strictly exogenous and predetermined regressors. An application to a panel data model of labour demand is presented which illustrates the usefulness of the extended GMM estimator in practice. Section 8concludes.

Section snippets

The model

We consider an autoregressive panel data model of the formyit=αyi,t−11′xit2′xit−1i+vitfor i=1,…,N and t=2,…,T, where uitηi+vit is the usual ‘fixed effects’ decomposition of the error term; N is large, T is fixed and |α|<1.1 This has the corresponding ‘common factor’ restricted (β2=−αβ1) formyit1′xit+fiitwithζit=αζi,t−1+vitandηi=(1−α)fi.

In our application to

The standard moment conditions

In the absence of any further restrictions on the process generating the initial conditions, the autoregressive error components model (2.3)–(2.6) implies the following m=0.5(T−1)(T−2) orthogonality conditions which are linear in the α parameterE(yi,t−sΔvit)=0fort=3,…,Tands⩾2where Δvit=vitvi,t−1. These depend only on the assumed absence of serial correlation in the time-varying disturbances vit, together with the restriction in Eq. (2.6).

The moment restrictions in Eq. (3.1)can be expressed

Non-linear moment conditions and restrictions on the initial conditions process

In this section we consider an additional, but in many cases relatively mild, restriction on the initial conditions process which allows the use of additional linear moment conditions for the levels equations in the GMM framework. This allows the use of lagged differences of yit as instruments in the levels equations. These additional moment conditions are likely to be important in practice when α is close to unity or when ση2/σv2 is high, since we have seen that lagged levels will be weak

The conditional GLS estimator

In the autoregressive error components model with homoskedasticity across both individuals and time,10 restrictions on the initial conditions process can be used to derive a consistent conditional GLS (CGLS) estimator by including yi1 in each of the T−1 levels equations and then applying the standard error components GLS estimator. Under joint normality, this is the conditional ML estimator (CMLE) proposed by Blundell

Monte Carlo results

In this section we report the results of a Monte Carlo study which investigates the potential gains from exploiting the moment conditions (4.3) and (4.4) in finite samples, as well as the potential gains from using the conditional GLS estimator.

The model

In this section we illustrate the benefits of exploiting the additional linear moment restrictions (4.3) and (4.4) in an application using real data. We consider a simple dynamic labour demand equation of the formnit=αni,t−10wit1wi,t−10kit1ki,t−1+(ηi+vit),where nit is the log of employment in firm i in year t, wit is the log of the wage rate and kit is the log of the capital stock. This log-linear model can be derived as the static conditional labour demand equation for a competitive

Summary and conclusions

In this paper we have discussed the importance of exploiting initial condition information in generating efficient estimators for dynamic panel data models where the number of time-series observations is small. We have focused on the individual effects autoregressive model yit=αyi,t−1+ηi+vit although our results extend naturally to dynamic models with regressors.

We considered two estimators that can improve the precision of the standard first-differenced GMM estimator for this model. One

Acknowledgements

An earlier version of this paper was read to the Econometric Society Meetings in San Francisco January 1996. We are grateful to participants at that meeting, Seung Ahn, Manuel Arellano, Andrew Chesher, Zvi Griliches Jinyong Hahn, Bronwyn Hall, Bo Honore, Joel Horowitz Ekaterini Kyriazidou, Jacques Mairesse, Whitney Newey, Neil Shephard, Richard Smith, Richard Spady, Alain Trognon, Frank Windmeijer, three anonymous referees and seminar participants at Berkeley, INSEE, Manchester, MIT and

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