On Maximum Likelihood Estimation of Dynamic Panel Data Models
| DOI | http://doi.org/10.1111/obes.12156 |
| Date | 01 August 2017 |
| Published date | 01 August 2017 |
463
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 79, 4 (2017) 0305–9049
doi: 10.1111/obes.12156
On Maximum Likelihood Estimation of Dynamic
Panel Data Models*
Maurice J.G. Bun,†Martin A. Carree,‡ and Art¯
uras Juodis§
†Amsterdam School of Economics, University of Amsterdam, Amsterdam, The Netherlands
(e-mail: m.j.g.bun@uva.nl)
‡School of Business and Economics, Maastricht University, Maastricht, The Netherlands
(e-mail: m.carree@maastrichtuniversity.nl)
§Faculty of Economics and Business, University of Groningen, Groningen, The Netherlands
(e-mail: a.juodis@rug.nl)
Abstract
We analyse the finite sample properties of maximum likelihood estimators for dynamic
panel data models. In particular, we consider transformed maximum likelihood (TML)
and random effects maximum likelihood (RML) estimation. We show thatTML and RML
estimators are solutions to a cubic first-order condition in the autoregressive parameter.
Furthermore, in finite samples both likelihood estimators might lead to a negative estimate
of the variance of the individual-specific effects. We consider different approaches taking
into account the non-negativity restriction for the variance. We showthat these approaches
may lead to a solution different from the unique global unconstrained maximum. In an
extensive Monte Carlo study we find that this issue is non-negligible for small values of T
and that different approaches might lead to different finite sample properties. Furthermore,
we find that the Likelihood Ratio statistic provides size control in small samples, albeit
with low power due to the flatness of the log-likelihood function. We illustrate these issues
modelling US state level unemployment dynamics.
I. Introduction
Dynamic panel data models have a prominent place in applied research and at the same
time form a challenging field in econometric theory. Many panel data applications have
a relatively small number of time periods T, whereas the cross sectional dimension Nis
JEL Classification numbers: C13, C23.
*This paper greatly benefited from comments made byAnindya Banerjee (Editor) and by two anonymous referees.
The authors thank PeterBoswijk, Geert Dhaene, David Edgerton, Hugo Kruiniger, Peter Schmidt, Joakim Westerlund,
Jeffrey Wooldridge and participants of the Netherlands Econometric Study Group Meeting 2015 in Maastricht, the
21st International Panel Data Conference in Budapest, seminars at University of Amsterdam and Lund University
for helpful comments and suggestions. Financial support from the NWO MaGW Grants ‘Causal inference with
panel data’ and ‘Likelihood-based inference in dynamic panel data models with endogenouscovariates’ is gratefully
acknowledged by the first and third authors respectively.
464 Bulletin
sizeable. It is therefore common to consider the asymptotic behaviour of estimators and
corresponding test statistics with Tfixed and only Ntending to infinity.
A central theme in linear dynamic panel data analysis is the fact that the fixed effects
(FE) estimator is inconsistent for fixed Tand Nlarge.This inconsistency is referred to as the
Nickell (1981) bias, and is an example of the incidental parameters problem. It has therefore
become common practice to estimate the parameters of dynamic panel data models by the
generalized method of moments (GMM), see Arellano and Bond (1991) and Blundell and
Bond (1998). A main reason for using GMM is that it provides asymptotically efficient
inference exploiting a minimal set of statistical assumptions. GMM inference has not been
without its own problems, however.These include small sample biases in both coefficient
and variance estimators, sensitivityto impor tant nuisance parameters and choices regarding
the type and number of moment conditions. A large literature has been devoted to adapting
the GMM approach to limit the impact of these inherent drawbacks, see Bun and Sarafidis
(2015) for a recent overview.
This again has led to an interest in likelihood-based methods that correct for the inciden-
tal parameters problem. Some of these methods are based on modifications of the profile
likelihood, see Lancaster (2002) and Dhaene and Jochmans (2016). Other methods start
from the likelihood function of the first differences, see Hsiao, Pesaran and Tahmiscioglu
(2002). Essentially these methods treat the incidental parameters as fixed in estimation.The
alternative approach is to assume random effects,but in dynamic models it is then necessar y
to be explicit about the non-zero correlation between individual-specific effects and initial
conditions (Anderson and Hsiao, 1982; Alvarez andArellano, 2003; Moral-Benito, 2012).
Random effects type ML estimators therefore typically exploit Chamberlain (1982)-type
projections to model the dependence between individual-specific effects, initial observa-
tions and additional covariates.
In this study, we consider the transformed maximum likelihood approach (TML) as
in Hsiao et al. (2002) and the random effects maximum likelihood estimator (RML) as
in Alvarez and Arellano (2003).1There is a close connection between TML and RML
in the sense that TML can be expressed as a restricted version of RML. Under suitable
regularity conditions, ML estimators are consistent and asymptoticallynor mallydistributed
(Amemiya, 1985; Kruiniger, 2013). Monte Carlo evidence in Hsiao et al. (2002), Alvarez
and Arellano (2003) and Hayakawa and Pesaran (2015), suggest that these likelihood-
based approaches can serve as viable alternatives to the usual GMM estimators. Just like
for GMM, however, the application of ML estimators is not without its own problems.
We address two important issues when implementing ML for dynamic panel data
models. First, we show that in the simple setup without time-series heteroscedasticity,
both the TML and RML estimators give rise to a cubic first-order condition in the au-
toregressive parameter. We therefore have either one or three real solutions to the first-
order conditions. As a result, even asymptotically the log-likelihood function can be
bimodal. This result is different from Kruiniger (2008) and Han and Phillips (2013), who
find a quartic equation assuming covariance stationarity.
Second, because both TML and RML can be seen as correlated random effect ML
estimators, we address the issue of negative variance estimates as mentioned in Maddala
1The former estimator is also referred to as fixed effects ML in Kruiniger (2013).
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
On ML estimation of dynamic panel data models 465
(1971) and Alvarez and Arellano (2003). An important consequence of bimodality is that
unconstrained maximization of the log-likelihood may lead to ML estimates which do
not satisfy the implicit restriction of non-negative variances. Enforcing this non-negativity
constraint may furthermore lead to a boundary solution.
We further investigate the impact of multiple roots and boundary conditions in finite
samples in a Monte Carlo study. We consider the finite sample bias and RMSE of co-
efficient estimators as well as size and power of corresponding t- and LR statistics. We
find that, despite the robustness of TML and RML to initial conditions (i.e. they remain
consistent as long as initial conditions have finite variances), the finite sample properties
of both estimators for small values of Tdepend heavily on the initial condition. A partial
explanation is that the behaviour of the initial condition has a direct effecton the bimodality
of the log-likelihood function. We find that when there are three solutions to the first-order
condition, the left solution alwayssatisfies the non-negativity restriction, while the right so-
lution violates it. Estimators taking into account non-negativity constraints perform much
better than unconstrained counterparts. Furthermore, we find that inference based on the
LR statistic is size correct, while t- statistics show large size distortions. Using the data set
in Bun and Carree (2005) we show how these theoretical results can influence empirical
estimates of US state level unemployment dynamics.
Throughout the analysis, we limit ourselves to an asymptotic analysis in which Tis
fixed and N→∞. When Tis large the influence of initial conditions becomes negligible,
hence our main results become less relevant. For results with Tlarge, see e.g. Bai (2013).
Furthermore, we do not analyse the unit root case (or explosive root). Distribution theory
becomes rather different in this case, see Ahn and Thomas (2006) and Kruiniger (2013).
The plan of this study is as follows. In section II, weintroduce the Maximum Likelihood
estimators for the panel AR(1) model including the cubic first-order condition for the
autoregressive parameter, with extensions to dynamic models with additional covariates
in subsection ‘Extension to exogenous regressors’. Section III deals with the possibility of
multiple solutions and proposes bounded estimation as a solution. Section IV reports the
results from the Monte Carlo study, while section Vshows the empirical results. Section VI
concludes. Proofs of all propositions are provided in the supplementary online appendix.
II. ML estimation for the panel AR(1) model
We consider the following simple AR(1) specification without exogenous regressors:2
yi,t=i+yi,t−1+"i,t,E["i,t|yi,0,i]=0, (1)
for i=1,…, N,t=1,…, T. We assume that the idiosyncratic errors i,tare i.i.d. (0, 2) and
that initial conditions yi,0 are observed.3Stacking the observations over time, we can write
the AR(1) model for each individual as follows:
yi=yi−+Ti+εi,εi=("i,1,…,"i,T),(2)
2Time-specific effects can be accommodated by taking the variablesin deviations from the cross-sectional mean.
3The estimators considered in this paper remain consistent even if in the population some variance components
are heteroscedastic over i. In this case both estimators presented should be treated as pseudo maximum likelihood
estimators, see also Hayakawa and Pesaran(2015).
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
To continue reading
Request your trial