Estimation of Dynamic Nonlinear Random Effects Models with Unbalanced Panels
| Author | Raquel Carrasco,Pedro Albarran,Jesus M. Carro |
| DOI | http://doi.org/10.1111/obes.12308 |
| Date | 01 December 2019 |
| Published date | 01 December 2019 |
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©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 81, 6 (2019) 0305–9049
doi: 10.1111/obes.12308
Estimation of Dynamic Nonlinear Random Effects
Models with Unbalanced Panels*
Pedro Albarran†Raquel Carrasco‡ and Jesus M. Carro‡
†Fundamentos del An´alisis Econ´omico (FAE), Universidad de Alicante, Alicante, Spain
(e-mail: albarran@ua.es)
‡Department of Economics, Universidad Carlos III de Madrid, Getafe, Spain
(e-mail: rcarras@eco.uc3m.es, jcarro@eco.uc3m.es)
Abstract
This paper presents estimation methods for dynamic nonlinear models with correlated ran-
dom effects (CRE) when having unbalanced panels. Unbalancedness is often encountered
in applied work and ignoring it in dynamic nonlinear models produces inconsistent esti-
mates even if the unbalancedness process is completelyat random. We showthat selecting a
balanced panel from the sample can produce efficiencylosses or even inconsistent estimates
of the average marginal effects. We allow the process that determines the unbalancedness
structure of the data to be correlated with the permanent unobserved heterogeneity. We
discuss how to address the estimation by maximizing the likelihood function for the whole
sample and also propose a Minimum Distance approach, which is computationally simpler
and asymptotically equivalent to the Maximum Likelihood estimation. Our Monte Carlo
experiments and empirical illustration show that the issue is relevant. Our proposed solu-
tions perform better both in terms of bias and RMSE than the approaches that ignore the
unbalancedness or that balance the sample.
I. Introduction
The purpose of this paper is to present and evaluate estimation methods for dynamic non-
linear models with correlated random effects (CRE) when the panel data are unbalanced.1
Unbalanced panels are often encountered in applied work. Forexample, in large households
JEL Classification numbers: C23, C25.
*We are grateful to C´esar Alonso-Borrego, Wiji Arulampalam, Ricardo Mora, Jeff Wooldridge and participants
at the 21st International Panel Data Conference, the IAAE 2015 Annual Conference, the XXXIX SAEe and Econo-
metrics Seminar at Oxford University for helpful comments on this work. All remaining errors are our own.The
authors gratefully acknowledge research funding from the Spanish Ministry of Education, Grants ECO2012-31358,
ECO2015-65204-P, ECO2017-87069-P, MDM 2014-0431 and Comunidad de Madrid, MadEco-CM (S2015/HUM-
3444).
1The CRE approach has been found useful to estimate nonlinear dynamic models in many cases, because it is
not subject to the incidental parameters problem that the fixed-effects (FE) approach suffers and it does not require
a large number of periods. Examples of applications using CRE are Hyslop (1999), Contoyannis, Jones and Rice
(2004), Stewart (2007) and Akee et al. (2010).
Dynamic nonlinear RE in unbalanced panels 1425
panel data sets like the PSID for the U.S. or the GSOEP for Germany, some individuals
drop out (potentially non-randomly) of the sample. At a firm level, Compustat and Datas-
tream International also have an unbalanced structure. In other cases, like in the so-called
‘rotating panels’,the unbalancedness is generated by the sample design (for instance, in the
Monthly Retail Trade Survey for the U.S., or in the Household Budget Continuous Survey
for Spain).
It is well-known how to estimate CRE dynamic nonlinear models with balanced panels.
However, the existing estimation methods cannot be in general directly implemented with
unbalanced panels. Ignoring the unbalancedness produces inconsistent estimates, as wewill
discuss. Obtaining a balanced subsample from the unbalanced panel, so that the existing
CRE methods for balanced panels could then be used, is also problematic. If we balance
the sample by taking a subset of individuals that are observed overthe same periods, we are
making an endogenous selection of the sample unless the unbalancedness is independent
of the individual effects. Another possibility to balance the sample is to take the subset
of periods at which all individuals are observed (see Wooldridge, 2005). But this is in
some cases infeasible because of the lack of a sufficient number of common periods across
individuals and, when feasible, it implies important efficiency losses.
In a dynamic setting under the CRE approach, the so-called ‘initial conditions problem’
arises. Heckman (1981) and Wooldridge (2005) propose solutions to deal with it, but
these are developed only for balanced panels. Furthermore, the initial conditions problem
is exacerbated when the panel is unbalanced because it affects each of the first period
of observation in the data set. This implies that, as we will show, even assuming that
unbalancedness is completely at random is not enough to allow us to ignore it in the
estimation.2
We propose methods to deal with the unbalancedness structure of the data in the esti-
mation of models with lags of the endogenous variable and other explanatory variablesthat
are strictly exogenous. We consider unbalancedness processes that are independent of the
time-varying shocks, but allow them to be correlated with the time-invariant unobserved
heterogeneity. Therefore, we are not restricted to the case of unbalancedness completely
at random. We first discuss how to address the unbalancedness problem by maximizing
the likelihood function for the whole sample. This can be computationally cumbersome
because specific parameters to each subpanel need to be estimated jointly with the com-
mon parameters of the model. We then propose to estimate the model for each subpanel
separately and then to obtain estimates of the common parameters across subpanels by
minimum distance (MD). This method allows us to use the same estimation routines that
we would use if wehad a balanced panel, while keeping the good asymptotic properties of
the maximum likelihood (ML) estimator for the whole sample.
A simulation study shows that these methods perform well compared to other alterna-
tives both in terms of bias and RMSE. As an empirical illustration, we estimate an export
participation equation with dynamic effects using unbalanced data for Spanish manufac-
turing firms. Our results show that the unbalancedness issue is relevant in practice, and
there is evidence of unbalancedness correlated with the unobserved heterogeneity.
2This problem also affects RE models assuming that the time invariant unobserved heterogeneityis independent
of the time-varying covariates.The CRE setting contains RE models as a particular case.
©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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