Maximum Likelihood Estimation in Panels with Incidental Trends
| Date | 01 November 1999 |
| Author | Peter C. B. Phillip,Hyungsik R. Moon |
| DOI | http://doi.org/10.1111/1468-0084.0610s1711 |
| Published date | 01 November 1999 |
MAXIMUM LIKELIHOOD ESTIMATION IN PANELS
WITH INCIDENTAL TRENDS
Hyungsik R. Moon and Peter C. B. Phillips
I. INTRODUCTION
In recent non-stationary time series applications, it has been extremely
common to model time series with roots near unity using the device of an
autoregressive root that is local to unity. Some early studies of near unit root
non-stationary time series include developments of local alternatives to unit
root speci®cations (Bobkoski, 1983; Phillips, 1987), derivations of power
functions and power envelopes of unit root tests (e.g., Cavanagh, 1985;
Phillips, 1987; Johansen, 1991), and the construction of con®dence intervals
for the long-run autoregressive coef®cient (Stock, 1991). More recent
research on near unit root non-stationary time series investigates the
ef®cient extraction of deterministic trends (Phillips and Lee, 1996; Canjels
and Watson, 1997), and the construction of point optimal invariant tests for
a unit root (Elliott, Rotherberg and Stock, 1996) and cointegrating rank
(Xiao and Phillips, 1999). For further examples, readers can refer to recent
surveys on unit root processes (e.g., Stock, 1994; Phillips and Xiao, 1998).
Like other parameters in econometric models, localizing parameters in
near integrated processes are not usually observable. But, implementation
of some methods in the aforementioned studies requires knowledge of the
localizing parameter or a consistent estimate of it. For example, it is well
known that ef®ciency gains in the estimation of deterministic trends can be
obtained by quasi-differencing the data using the unknown localizing
parameter (e.g. Phillips and Lee, 1996; Canjels and Watson, 1997). How-
ever, if we implement this procedure using inconsistent estimates of the
localizing parameter, then the limit distribution of the resulting trend coef-
®cient estimator is highly non-standard, which introduces new dif®culties,
for example, in constructing con®dence intervals for the trend coef®cient.
Largely because of this problem, Cavanagh, Elliott and Stock (1995) and
Canjels and Watson (1997) suggested the use of Bonferroni-type con®dence
intervals, which are often very conservative.
Finding a consistent estimate of the localizing parameter is not straight-
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, SPECIAL ISSUE (1999)
0305-9049
711
#Blackwell Publishers Ltd, 1999. Published by Blackwell Publishers, 108 Cowley Road, Oxford
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Moon thanks the Academic Research Committee of UCSB for research support and Phillips
thanks the NSF for research support under Grant No. SBR97-30295.
forward. Obvious procedures like the use of least squares are well known to
be inconsistent (Phillips, 1987); and, even in the simplest framework,
consistent estimation inevitably involves the introduction of additional
information. In view of its potential applications in both estimation and
inference, the problem of consistent estimation of the localizing parameter
in local to unity models poses an interesting problem with important
implications. Two recent studies that consider the subject are Moon and
Phillips (1998) and Phillips, Moon and Xiao (1998).
The main purpose of this paper is to investigate the asymptotic properties
of the Gaussian maximum likelihood estimators (MLE) of the localizing
parameter in local to unity dynamic panel regression models. The model we
consider here allows for the panel to be generated with deterministic and
stochastic trends, and a common localizing parameter is assumed to apply
across individuals. Commonality of the localizing parameter is restrictive,
but is no more restrictive than the conventional assumption of common AR
parameters in stationary dynamic panels (e.g., Nickell, 1981). Two different
models are considered: a homogeneous trend model in which the determi-
nistic trends are homogeneous across the individuals in the panel; and a
heterogeneous trend model where the deterministic trends may vary across
individuals, much like ®xed individual effects. In the homogeneous trend
model we show that the Gaussian MLE of the common localizing parameter
is
N
p-consistent and has a limiting normal distribution that is the same as
that in the case where the trends are known. In the heterogeneous trends
model it is shown that the Gaussian MLE of the localizing parameter is
inconsistent.
The inconsistency of the MLE of the localizing parameter in the hetero-
geneous trend model is an instance of the so-called incidental parameter
problem originally explored by Neyman and Scott (1948). In this model, the
heterogenous trend coef®cients correspond to incidental parameters whose
number goes to in®nity as the cross-section dimension N!1. Such
problems frequently appear in panel data models with ®xed effects, a well-
known example being the dynamic panel regression model with ®xed
effects. In this case, the MLE of the lagged dependent variable coef®cient
that is common over individuals is inconsistent if N!1while the sample
size dimension, T, is ®xed (Nickell, 1981). In most panel data situations this
incidental parameter problem disappears when Tpasses to in®nity also
(e.g., Alvarez and Arellano, 1998; Hahn, 1998). A particularly interesting
aspect of the incidental parameter problem discovered in this paper is that
the inconsistency of the MLE of the localizing parameter does not disappear
even when both Nand Ttend to in®nity.
The paper is organized as follows. Section 2 lays out the model and
assumptions, and shows that when the deterministic components are known,
the Gaussian MLE of the localizing parameter is consistent. Section 3
studies asymptotic properties of the Gaussian MLE of the panel regression
model with unknown deterministic trends. Section 4 reports some Monte
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#Blackwell Publishers 1999
Carlo simulations that investigate the magnitude of the inconsistency.
Section 5 concludes and offers some suggestions for dealing with the
inconsistency. Proofs and technical derivations are collected in the Appen-
dix in the last section.
Our notation is mostly standard. We use `!p' and `)' to denote
convergence in probability and convergence in distribution, respectively.
The notation (N,T!1) implies that Nand Ttend to in®nity together,
while (N,T!1)seq means that the indices pass to in®nity sequentially
(®rst Tand then N). Standard Brownian motion is denoted by W(r).
II. NEAR INTEGRATED PANELS: PRELIMINARY THEORY
We start by introducing a panel regression model where data zi,tare gener-
ated by deterministic trends Gi(t) and near integrated stochastic trends yi,t
as follows:
zi,tGi(t)yi,t,t1, ...,T;i1, ...,N, (1)
yi,tayi,tÿ1åi,t,aexp c
T
1c
T
:
The parameter cin (1) is a local to unity parameter that is common to all
individuals in the panel. The main purpose of this paper is to investigate
asymptotic properties of the MLE of the localizing parameter c.
To provide some intuition, we ®rst consider the simple case where
yi,tzi,tÿGi(t) is observable, abstracting from the problem of ®tting the
deterministic component in (1). Assume that the errors åi,tare i.i.d.
N(0, ó2), and, for simplicity in this section, that ó2is known and that the
initial observations yi,0 0 for all i. Under these assumptions the standar-
dized log-likelihood function of the panel data yN,T(y1,1,... ,yN,T)9is
LN,T(yN,T;c)ÿ 1
2ó2NX
N
i1X
T
t1
Äyi,tÿc
Tyi,tÿ1
2
constant:(2)
Let c0denote the true localizing parameter, and assume that c0is an
element of the interior of a convex set of R. De®ne åi,t(c0)
Äyi,tÿ(c0=T)yi,tÿ1. Then, the MLE of cis obtained by maximizing the
standardized log-likelihood
MLE IN PANELS WITH INCIDENTAL TRENDS 713
#Blackwell Publishers 1999
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