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\textit{EC821:\ Time Series Econometrics, Spring 2003}

\bigskip \textit{Notes Section 9}

\textbf{Panel Unit Root Tests}

A\ variety of procedures for the analysis of unit roots in a panel context
have been developed. The emphasis in this development is the attempt to
combine information from the time--series dimension with that obtained from
the cross--sectional dimension, in the hope that inference about the
existence of unit roots and cointegration can be made more straightforward
and precise by taking account of the latter. Given that many interesting
relations involve relatively short time--series dimensions, and the
well--known low power of conventional unit root tests when applied to a
single time series, there may be considerable potential for tests that can
be employed in an environment where the time series may be of limited
length, but very similar data may be available across a cross--section of
countries, regions, firms, or industries. With the increasing availability
of quite rich panel data sets in a number of contexts, tests that can be
applied to these data would seem very attractive.

However, a variety of issues arise when panel data are employed in testing
for unit roots. Some of the tests proposed require a balanced panel (no
missing data for any $i$ nor $t)$ whereas others allow for an unbalanced
panel setting. In a panel context---that is, with a set of time series---one
may form the null hypothesis as a generalization of the standard
Dickey--Fuller test, in that all series in the panel are assumed to exhibit
nonstationary behavior. This null might be rejected if a fraction of the
series in the panel appear to be stationary. Conversely, one could form the
null hypothesis of the sort employed in the KPSS\ test or the
Leybourne--McCabe test, in which one presumes that all of the series in the
panel are $I(0)$ processes, rejecting when there is sufficient evidence of
nonstationarity. In any case, the consideration of a set of timeseries lead
to a ``box score'' concept, wherein one makes an inference on the set of
series depending on the predominance of the evidence.

An important theoretical consideration in the development of this literature
is the issue of the asymptotic behavior of the panel's two dimensions, $N$
and $T.$ Various assumptions may be made about the rates at which those
parameters tend to infinity. For instance, one may fix $N$ and let $T$ tend
to infinity, and subsequently let $N$ tend to infinity. Alternatively, one
may allow the two indices to pass to infinity at a controlled rate, such as $%
T=T(N).$ A third possibility, as expressed in Phillips and Moon's work, is
to allow both indices to tend to infinity simultaneously.

\textbf{The MADF\ Test}

Sarno and Taylor developed a multivariate analogue to the ADF\ test (1998,
1998) as an extension of a test developed by Abuaf and Jorion several years
earlier. In this test, a single autoregressive parameter is estimated over a
panel, by applying Zellner's SUR estimator to $N$ equations, corresponding
to the $N$ units of the panel. Since SUR\ can only be employed where $T>>N,$
the test may only be used where this condition is satisfied. Thus, it is not
a suitable test for small-$T,$ large-$N$ panels, such as those often
employed in a cross--country context. Each equation is specified as a $%
k^{th}-$order autoregression, and the test involves testing the hypothesis
that the sum of the coefficients on the autoregressive polynomial is unity.
The null hypothesis states that this condition is satisfied over the $N$
equations. Thus, this null will be violated if even one of the series in the
panel is stationary. A\ rejection should thus not be taken to indicate that
each of the series in the panel is stationary, but rather an indication that
the condition that all series are $I(1)$ does not receive empirical support.
Critical values are nonstandard, and have been generated by simulation of a
response surface. The Sarno and Taylor paper also present Johansen's
likelihood ratio test, which has the null that at least one of the series in
the panel is a nonstationary process. The MADF\ test (Baum, 2001) is
available in Stata (version 7 or later) as routine \texttt{madfuller (findit
madfuller}).

\textbf{The Levin--Lin--Chu Test}

One of the first unit root tests to be developed for panel data is that of
Levin and Lin, as originally circulated in working paper form in 1992 and
1993. Their work was finally published, with Chu as a coauthor, in 2002.
Their test is based on analysis of the equation: 
\begin{eqnarray*}
\Delta y_{i,t} &=&\alpha _{i}+\delta _{i}t+\theta _{t}+\rho
_{i}y_{i,t-1}+\varsigma _{i,t}, \\
i &=&1,2,..N,t=1,2,...T.
\end{eqnarray*}

This model allows for two--way fixed effects ($\alpha \,$and $\theta )$ and
unit--specific time trends. The unit--specific fixed effects are an
important source of heterogeneity, since the coefficient of the lagged
dependent variable is restricted to be homogeneous across all units of the
panel. The test involves the null hypothesis $H_{0}:\rho _{i}=0$ for all $i$
against the alternative $H_{A}:\rho _{i}=\rho <0$ for all $i,$ with
auxiliary assumptions under the null also being required about the
coefficients relating to the deterministic components. Like most of the unit
root tests in the literature, LLC\ assume that the individual processes are
cross--sectionally independent. Given this assumption, they derive
conditions (and correction factors) under which the pooled OLS\ estimate of $%
\rho $ will have a standard normal distribution under the null hypothesis.
Their work focuses on the asymptotic distributions of this pooled panel
estimate of $\rho $ under different assumptions on the existence of fixed
effects and homogeneous time trends.

The LLC\ test may be viewed as a pooled Dickey--Fuller (or ADF)\ test,
potentially with differing lag lengths across the units of the panel. Unlike
the MADF\ test, it is applicable to small--$T,$ large--$N$ panels. The LLC\
test (Bornhorst and Baum, 2001) is available in Stata (version 7 or later)
as routine \texttt{levinlin (findit levinlin}).

\textbf{The Im--Pesaran--Shin Test}

The Im--Pesaran--Shin (IPS, 1997)\ test extends the LLC\ framework to allow
for heterogeneity in the value of $\rho _{i}$ under the alternative
hypothesis. Given the same equation: 
\begin{eqnarray*}
\Delta y_{i,t} &=&\alpha _{i}+\delta _{i}t+\theta _{t}+\rho
_{i}y_{i,t-1}+\varsigma _{i,t}, \\
i &=&1,2,..N,t=1,2,...T.
\end{eqnarray*}
The null and alternative hypotheses are defined as: 
\begin{eqnarray*}
H_{0} &:&\rho _{i}=0~\forall i \\
H_{A} &:&\rho _{i}<0,i=1,2,...,N_{1};\rho _{i}=0,i=N_{1}+1,N_{1}+2,...N
\end{eqnarray*}
Thus under the null hypothesis, all series in the panel are nonstationary
processes; under the alternative, a fraction of the series in the panel are
assumed to be stationary. This is in contrast to the LLC\ test, which
presumes that all series are stationary under the altenative hypothesis. The
errors $\varsigma _{i,t}$ are assumed to be serially autocorrelated, with
different serial correlation properties and differing variances across
units. IPS\ propose the use of a group--mean Lagrange multiplier statistic
to test the null hypothesis. The ADF\ regressions (perhaps of differing lag
lengths) are computed for each unit, and a standardized statistic computed
as the average of the LM\ tests for each equation. Adjustment factors
(available in their paper) are used to derive a test statistic that is
distributed standard Normal under the null hypothesis. IPS\ also propose the
use of a group--mean $t-$bar statistic, where the $t$ statistics from each
ADF\ test are averaged across the panel; again, adjustment factors are
needed to translate the distribution of $t-$bar into a standard Normal
variate under the null hypothesis. IPS\ demonstrate that their test has
better finite sample performance than that of LLC.

The IPS\ test (Bornhorst and Baum, 2001) is available in Stata (version 7 or
later) as routine \texttt{ipshin (findit ipshin}).

\textbf{The Hadri LM\ Test}

The Lagrange multiplier test of Hadri (2000) differs from the other tests in
that its null hypothesis is that all series in the panel are stationary.
Just as the null of the KPSS\ test differs from that of Dickey--Fuller style
tests in assuming stationarity rather than nonstationarity, Hadri's test
generalizes this notion to the panel context. The test statistic is
distributed as standard Normal under the null hypothesis. As in the
univariate KPSS\ test, the series may be stationary around a deterministic
level (specific to the unit--i.e. a fixed effect) or around a unit--specific
deterministic trend. The error process may be assumed to be homoskedastic
across the panel, or heteroskedastic across units. Serial dependence in the
disturbances may also be taken into account using a Newey--West estimator of
the long--run variance. The residual--based test is based on the squared
partial sum process of residuals from a demeaning (detrending) model of
level (trend) stationarity.

The Hadri LM\ test (Baum, 2001) is available in Stata (version 7 or later)
as routine \texttt{hadrilm (findit hadrilm}).

\textbf{The Nyblom--Harvey Test of Common Stochastic Trends}

Nyblom and Harvey (2000) have developed a number of tests for common
stochastic trends. They test the validity of a specific value of the rank of
the covariance matrix of the disturbances driving the multivariate random
walk, which is equal to the number of common trends in the set of series. As
they show, this test is very simple, since it does not require the
specification of a model (in contrast to, say, the Johansen approach).

The test may be considered as a generalization of the Nyblom and Makelainen
(1983) and KPSS\ univariate tests for stationarity of a series. Those tests
consider the null hypothesis that the series is stationary, or stationary
around a deterministic trend, against the alternative that a random walk
component is present. The Nyblom--Harvey test considers the same structure
in the context of multiple time series. The ``random walk with noise''
model, for a univariate time series, can be written as:
\begin{eqnarray*}
y_{t} &=&\mu _{t}+\epsilon _{t} \\
\mu _{t} &=&\mu _{t-1}+\eta _{t}
\end{eqnarray*}
where each error process is  $i.i.d.$ normal with variances $\sigma
_{\epsilon }^{2}$ and $\sigma _{\eta }^{2},$ respectively. If $\sigma _{\eta
}^{2}=0,$ the random walk becomes a constant level, and $y$ is a (level)
stationary series.

The multivariate analog, for a vector of time series processes, may be
written in the same notation; now the vector $\epsilon _{t}$ has covariance
matrix $\Sigma _{\epsilon },$ while the vector $\eta _{t}$ has covarance
matrix $\Sigma _{\eta }.$ We assume that  $\Sigma _{\epsilon }$ is a
positive definite $n$ $\times $ $n$ matrix. The null hypothesis is then $%
H_{0}:\Sigma _{\eta }=0,$ which a test for whether there is any
nonstationarity in the system. Under the condition that there are $N$
independent random walks in the set of timeseries, $\Sigma _{\eta }$ is of
full rank ($N).$ The test involves roots of the matrix equation:
\[
\left| \Sigma _{\eta }-q_{i}\Sigma _{\epsilon }\right| =0,i=1,N
\]
where $Q=diag(q_{1}...q_{N),}$ $P\Sigma _{\eta }P^{\prime }=Q,$ and $P$ is
defined from the factorization $P\Sigma _{\epsilon }P^{\prime }-I.$ The null
hypothesis involves the test that all of these roots are zero; without any
beliefs about relative magnitudes under the alternative, assume  $%
q_{1}=q_{2}=...=q_{N}=q>0,$ corresponding to the ``homogeneous'' model $%
\Sigma _{\eta }=q\Sigma _{\epsilon }.$ The test has critical values, for the
constant and constant--trend case, tabulated by the authors, which depend
only on the number of series and the hypothesized number of common trends.

One form of the Nyblom--Harvey\ test (Baum and Bornhorst, 2001) is available
in Stata (version 7 or later) as routine \texttt{nharvey (findit nharvey}).
This program considers the special case that the rank of the covariance
matrix, $k,$ equals zero:\ that is, there are no common trends among the
variables. In that context, it may be considered a test for cointegration,
since in the authors' words, ``common trends imply cointegration, and vice
versa.'' Thus, a failure to reject the null hypothesis of zero common trends
is also an indication that the variables do not form a cointegrated
combination. The test could also be extended to consider the null hypothesis
that $k$ takes on a certain value, less than $N$ (the number of series),
versus the alternative that it takes on a greater value. Another useful
reference in this field is Nyblom and Harvey (2001).

{\small References}

\begin{thebibliography}{9}
\bibitem{banerjee}  {\small Banerjee, A., 1999. Panel data unit root tests
and cointegration: An overview. Oxford Bulletin of Economics and Statistics,
Special Issue, 607--629.}

\bibitem{hadrilm}  {\small Hadri, K., 2000. Testing for stationarity in
heterogeneous panel data. Econometrics Journal, 3, 148--161.}

\bibitem{ipshin}  {\small Im, K., Pesaran, M., and Y. Shin, 1997. Testing
for unit roots in heterogeneous panels. Mimeo, Department of Applied
Economics, University of Cambridge.}

\bibitem{llc2002}  {\small Levin, A., Lin. C.-F. and C-S. Chu, 2002. Unit
root tests in panel data: Asymptotic and finite sample properties. Journal
of Econometrics 108, 1--24.}

\bibitem{nharvey}  {\small Nyblom, J. and A. Harvey. Tests of common
stochastic trends. Econometric Theory, 16, 2000, 176-199.}

\bibitem{nharvey01} {\small Nyblom, J. and A. Harvey. Testing against smooth
stochastic trends. Journal of Applied Econometrics, 16, 415--429.}

\bibitem{nmake}  {\small Nyblom, J. and T. Makelainen, 1983. Comparison of
tests for the presence of random walk components in a simple linear model.
Journal of the American Statistical Association, 78, 856--864.}

\bibitem{sartay98}  {\small Sarno, L. and M. Taylor, 1998. Real exchange
rates under the current float: Unequivocal evidence of mean reversion.
Economics Letters 60, 131--137.}

\bibitem{taysar98}  {\small Taylor, M. and L. Sarno, 1998. The behavior of
real exchange rates during the post--Bretton Woods period. Journal of
International Economics, 46, 281--312.}
\end{thebibliography}

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