A Principal Components Analysis of Common Stochastic Trends in Heterogeneous Panel Data: Some Monte Carlo Evidence
| Date | 01 November 1999 |
| Published date | 01 November 1999 |
| Author | Stephen Hall,Stepana Lazarova,Giovanni Urga |
| DOI | http://doi.org/10.1111/1468-0084.0610s1749 |
A PRINCIPAL COMPONENTS ANALYSIS OF
COMMON STOCHASTIC TRENDS IN
HETEROGENEOUS PANEL DATA: SOME MONTE
CARLO EVIDENCE
Stephen Hall, Stepana Lazarova and Giovanni Urga
I. INTRODUCTION
Over the past few years increasing attention has been paid to the presence
of non-stationarity in panel data sets. This has involved both testing for unit
roots within a panel and assessing cointegration. The main contributions in
this area are Kao and Chiang (1998), McCoskey and Kao (1998a), Pedroni
(1997, 1998) and Phillips and Moon (1999). McCoskey and Kao (1998b)
provide a detailed survey of recent developments and Monte Carlo compari-
son of the tests.
Recent research has pointed out that in dealing with cointegrated panel
data sets it is important to examine not only the relationships between the
dependent variable and regressors but also the structure of the regressors.
Pesaran and Smith (1995) argue that in general in a heterogeneous panel
with individually cointegrated relationships the aggregated relationship does
not cointegrate and that any panel data estimator which imposes homogene-
ity across the panel will give inconsistent estimates of the long-run effects.
However, when certain cointegrating restrictions are placed on the regres-
sors, aggregation bias tends to disappear asymptotically. In a recent paper,
Hall and Urga (1998) have shown that if each regressor in a panel is driven
by a single common stochastic trend and each unit cointegrates then a
standard panel data estimator which imposes homogeneous parameters
across the panel will give rise to consistent estimates of aggregate long-run
effects even if the true model has heterogeneous parameters.
Further, Gonzalo (1993) derives various conditions that must be met if
the aggregation of individual non-stationary series is to be meaningful.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, SPECIAL ISSUE (1999)
0305-9049
749
#Blackwell Publishers Ltd, 1999. Published by Blackwell Publishers, 108 Cowley Road, Oxford
OX4 1JF,UK and 350 Main Street, Malden, MA 02148, USA.
Corresponding author: G. Urga, City University Business School, Department of Investment,
Risk Management and Insurance, Frobisher Crescent, Barbican Centre, London EC2Y 8HB, UK.
E-mail: g.urga@city.ac.uk. We wish to thank the Editor, Anindya Banerjee, for helpful comments.
The usual disclaimer applies. S. Lazarova and G. Urga wish to acknowledge that this work is part
of an ESRC research project (Grant No. R022251032) entitled `An Analysis of the Importance of
Common Stochastic Trends and the Methods of Selecting Them', with Stephen Hall.
Amongst those conditions, an important role is played by the number of
common factors shared by the individual series. Similarly, Ghose (1995)
ascertains the need to know the number of common stochastic trends when
he evaluates the problem of aggregating a subset of non-stationary series in
a cointegrated time series regression. Hence it may often be important to
test a panel for the number of common stochastic trends underlying each
regressor.
All existing tests for cointegration require the underlying variables to be
integrated of order one. This means that every test for the number of
common stochastic trends shared by any set of variables must be preceded
by testing for unit root, augmenting the degree of uncertainty already
involved in the testing procedure. In this paper we propose a new approach
based on principal components which will make it possible to test for the
common stochastic trends regardless of the presence of stationary series in
the data set.
In addition to this, the principal components approach overcomes the
problem of large dimension typical for panel data sets. When the number of
series approaches the number of time observations, it becomes impossible
to use the existing regression methods. The principal components analysis,
instead, can in principle be applied to samples of any dimension. As the
proposed test is asymptotic, we assess the empirical relevance of the
theoretical advantages of the method with a small set of Monte Carlo
simulations.
The organization of the paper is as follows: Section 2 shows the
importance of common stochastic trends in panel data. Section 3 brie¯y
introduces the principal components estimation. In Section 4 we describe a
method for testing the number of common stochastic trends in panel data.
Section 5 reports a series of Monte Carlo experiments and Section 6
concludes.
II. COMMON STOCHASTIC TRENDS IN PANELS
The main motivation of the research on common factors in panels or in
larger time series systems is to determine whether certain suf®cient or
necessary conditions for aggregations are met, as illustrated in recent papers
by Gonzalo (1993), Ghose (1995), and Hall and Urga (1998). The following
example conveniently illustrates the issue. Consider a heterogeneous panel
and let us suppose that yi,tand xi,tare I(1) and that there is a cointegrating
relationship between yi,tand xi,tfor each group, with the parameters
varying randomly across groups, i.e. suppose that
yi,tbixi,tåi,t,i1, ...,N,t1, ...,T, (1)
where åi,tare stationary processes and bi's are assumed to have mean b,
constant covariances ù2
iand ®nite higher-order moments and cross-
moments. The randomness assumption is made only for convenience and
750 BULLETIN
#Blackwell Publishers 1999
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