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

\bigskip \textit{Notes Section 5 }

\textbf{Unit root tests}

Given the distinction between trend-stationary and unit root processes, it
would seem to be very important to be able to determine whether a particular
timeseries which, for instance, generally increases in value is being driven
by some underlying trend, or whether its evolution reflects a unit root in
its data generating process. Those who study macroeconomic phenomena will
want to know whether economic recessions have permanent consequences for the
level of future GDP\ (as they would if GDP exhibits a unit root), or whether
they are merely deviations from a trend rate of growth, temporary downturns
that will be offset by the following recovery. Those who are concerned with
the stock market want to know whether stock prices really do follow a random
walk---i.e. exhibit unit root behavior---rather than some complicated
combination of trends and cycles. If stock prices' behavior reflect a unit
root, then ``technical analysis'' or ``charting'' is no more useful than
astrology. On the other hand, if there are no unit roots in stock prices,
all of the effort applied by stock analysts to studying the behavior of
these series should have a reward.

This concern has given rise to a battery of unit root tests: statistical
procedures that are designed to render a verdict as to whether a given
sample of timeseries data appears to imply the presence of a unit root in
that timeseries, or whether the series may be considered stationary. In
terms of our prior terminology, we are trying to discern whether the series
exhibits $I(1)$ (unit root) or $I(0)$ (stationary) behavior. It turns out
that this is a fairly difficult problem, from a statistical perspective. It
might appear sufficient to merely estimate an equation such as $y_{t}=\phi
_{1}y_{t-1}+\epsilon _{t}$, modified to the form 
\begin{equation}
\Delta y_{t}=\gamma y_{t-1}+\epsilon _{t}  \label{simpledf}
\end{equation}
using the available sample of data, and test the null hypothesis that $
\gamma =(\phi _{1}-1)=1.$ For various reasons, that turns out to be
problematic, in the sense that the distribution of the test statistic is
nonstandard under that null. The $t$ test for $\gamma =0$ does not have a $t$
distribution under the null hypothesis; even as $T\rightarrow \infty ,$ the
distribution of this $t$ statistic will not be $N(0,1).$ Under the
alternative hypothesis, the test statistic is well behaved, but under the
null---the point of interest---it follows the ``Dickey--Fuller''
distribution rather than the Normal or $t.$ The critical points on the D--F
distribution, as established by simulation, are considerably larger than
those of the equivalent $t;$ whereas a value of --1.645 would be on the
borderline of rejection at the 95\% level for a one--tailed $t$ test, the
D-F critical value would be --1.961 for $T=$ 1000.

Of course, the model (\ref{simpledf}) may not be the appropriate special
case of the autoregressive distributed lag model; we may want to allow for
an additional term which would become a constant term in a stable
autoregressive process, or a drift term in a random walk process. Otherwise,
we are specifying a stable autoregressive process with a zero mean under the
alternative hypothesis, which may not be sensible (unless the series has
already been demeaned). With that modification, we would test 
\begin{equation}
\Delta y_{t}=\mu +\gamma y_{t-1}+\epsilon _{t}  \label{constdf}
\end{equation}
which would then allow both a test for a unit root $\left( \gamma =0\right) $
and a joint test for a white noise process (an $F$ test for $\gamma =0$ and $%
\mu =0).$ Note that the critical values for the $t$ test are not the same as
those that would be used in (\ref{simpledf}); for instance, the D--F
critical value for $T=$ 1000 in this test is --2.864. One must also note
that this model would not be appropriate if there was an obvious trend in
the series, since the model under the alternative has no mechanism to
generate such a trend (as the RW--with--drift model does under the null).

The most general form of the standard D--F test allows for both a constant
in the relationship and a deterministic trend: 
\begin{mathletters}
\begin{equation}
\Delta y_{t}=\mu +\gamma y_{t-1}+\beta t+\epsilon _{t}  \label{trenddf}
\end{equation}
Such a model will allow for both a nonzero mean for $y$ (with $\mu \neq 0)$
and trending behavior (with $\beta \neq 0)$ under the alternative
hypothesis, where $\gamma <0.$ The most likely null hypothesis is then that
of a RW--with--drift, so that under $H_{0}$ $\gamma =0$ and $\beta =0$ (no
deterministic trend). This null could be rejected for three reasons: (a)
there could be no unit root, but a deterministic trend; (b) there could be a
unit root, but with a deterministic trend; or (c) there might be no unit
root nor deterministic trend. The most general alternative is (a), for which
an $F$ test is required (since two restrictions on the parameter vector are
implied under the null). The $F$ statistic is calculated in the normal way,
but the distribution is again nonstandard, and tabulated values for the
``D--F $F$ distribution'' must be consulted. More commonly, we consider a $t$
test on $\gamma ;$ once again, the critical values are specific to model (%
\ref{trenddf}). For instance, the D--F critical value for $T=1000$ in this
test is --3.408: larger yet than the critical values in the constant--only
model, which in turn exceed those for the original white noise model.

Any of the forms of this test presume the existence of white noise errors in
the regression. If that is implausible, the test will lose significant
power. To cope with this issue, any of the `Dickey--Fuller'' tests in
practice are usually employed as the ``augmented Dickey--Fuller'' test, or
ADF\ test, in which a number of lags of the dependent variable are added to
the regression to whiten the errors: 
\end{mathletters}
\begin{equation}
\Delta y_{t}=\mu +\gamma y_{t-1}+\vartheta _{1}\Delta y_{t-1}+\vartheta
_{2}\Delta y_{t-2}+\ldots +\beta t+\epsilon _{t}  \label{adf}
\end{equation}
In this formulation, we consider an $AR(p)$ model as the baseline model,
rather than the $AR(1)$ model of the simple Dickey--Fuller framework. The
choice of appropriate lag length is likely to depend on the frequency of the
data; a general--to--specific strategy (analogous to the Ng--Perron
sequential $t$ procedure) or an information criterion may also be used. We
will discuss the use of a modified AIC\ below.

\textbf{Phillips--Perron tests}

The augmentation of the original D--F\ regression with lags of the dependent
variable is motivated by the need to generate $iid$ errors in that model,
since an OLS\ estimator of the covariance matrix is being employed. An
alternative strategy for allowing for errors that are not $iid$ is that of
Phillips (1987) and Phillips and Perron (1988), known as the
Phillips--Perron (PP) unit root test. The PP test deals with potential
serial correlation in the errors by employing a correction factor that
estimates the long--run variance of the error process with a variant of the
Newey--West formula. Like the ADF\ test, use of the PP test requires
specification of a lag order; in the latter case, the lag order designates
the number of lags to be included in the long--run variance estimate. The PP
test allows for dependence among disturbances of either AR\ or MA\ form, but
have been shown to exhibit serious size distortions in the presence of
negative autocorrelations. In principle, the PP\ tests should be more
powerful than the ADF alternative. The same critical values are used for the
ADF\ and PP tests.

\textbf{The DF--GLS\ test }

Conventional unit root tests are known to lose power dramatically against
stationary alternatives with a low order MA\ process: a characterization
that fits well to a number of macroeconomic time series. Consequently, these
original tests have been largely supplanted in many researchers' toolkits by
improved alternatives. Along the lines of the ADF\ test, a more powerful
variant is the DFGLS test proposed by Elliott, Rothenberg and Stock (ERS,
1996), described in Baum (2000, 2001), and implemented in Stata\footnote{%
This command is not built in to Stata version 7, but can be readily
installed by using the command ``ssc install dfgls''. The command is already
installed in the version of Stata running on fmrisc.bc.edu. It is available
as part of official Stata in version 8.} as command $\mathtt{dfgls}.$ 
\texttt{dfgls} performs the ERS efficient test for an autoregressive unit
root. This test is similar to an (augmented) Dickey-Fuller \textit{t} test,
as performed by \texttt{dfuller}, but has the best overall performance in
terms of small-sample size and power, dominating the ordinary Dickey-Fuller
test. The \texttt{dfgls} test ``has substantially improved power when an
unknown mean or trend is present'' (ERS, p.813).

\texttt{dfgls} applies a generalized least squares (GLS) detrending
(demeaning) step to the \texttt{varname}:

$y_{t}^{d}=y_{t}-\hat{\beta}^{^{\prime }}z_{t}$

For detrending, $z_{t}=\left( 1,t\right) ^{^{\prime }}$ and $\hat{\beta}%
_{0}, $ $\hat{\beta}_{1}$ are calculated by regressing $\left[ y_{1},\left(
1-\bar{\alpha}L\right) y_{2},...,\left( 1-\bar{\alpha}L\right) y_{T}\right] $
onto $\left[ z_{1},\left( 1-\bar{\alpha}L\right) z_{2},...,\left( 1-\bar{%
\alpha}L\right) z_{T}\right] $ where $\bar{\alpha}=1+\bar{c}/T$ with $\bar{c}
=-13.5$, and $L$ is the lag operator. For demeaning, $z_{t}=\left( 1\right)
^{\prime }$ and the same regression is run with $\bar{c}=-7.0.$ The values
of $\bar{c}$ are chosen so that ``the test achieves the power envelope
against stationary alternatives (is asymptotically MPI\ \textit{(most
powerful invariant))} at 50 percent power'' (Stock, 1994, p.2769; emphasis
added). The augmented Dickey-Fuller regression is then computed using the $%
y_{t}^{d}$ series:

$\Delta y_{t}^{d}=\alpha +\gamma t+\rho y_{t-1}^{d}+\sum_{i=1}^{m}\delta
_{i}\Delta y_{t-i}^{d}+\epsilon _{t}$

where $m$=\texttt{maxlag}.The \texttt{notrend} option suppresses the time
trend in this regression.

Approximate critical values for the GLS detrended test are taken from ERS,
Table 1 (p.825). Approximate critical values for the GLS demeaned test are
identical to those applicable to the no--constant, no--trend Dickey--Fuller
test, and are computed using the \texttt{dfuller} code.

The \texttt{dfgls} routine includes a very powerful lag selection criterion,
the ``modified AIC'' (MAIC) criterion proposed by Ng and Perron (2001). They
have established that use of this MAIC criterion may provide ``huge size
improvements'' (2001, abstract) in the \texttt{dfgls} test. The criterion,
indicating the appropriate lag order, is printed on \texttt{dfgls}' output,
and may be used to select the test statistic from which inference is to be
drawn.

It should be noted that all of the lag length criteria employed by \texttt{%
dfgls} (the sequential \textit{t} test of Ng and Perron (1995), the SC, and
the MAIC) are calculated, for various lags, by holding the sample size fixed
at that defined for the longest lag. These criteria cannot be meaningfully
compared over lag lengths if the underlying sample is altered to use all
available observations. That said, if the optimal lag length (by whatever
criterion) is found to be much less than that picked by the Schwert
criterion, it would be advisable to rerun the test with the \texttt{maxlag}
option specifying that optimal lag length, especially when using samples of
modest size.

\textbf{The KPSS\ test}

An alternative test is that proposed by Kwiatkowski et al. (1992), the
so-called KPSS\ test, which has a null hypothesis of stationarity (that is, $%
H_{0}:y\sim I(0)$). It is also described in Baum (2000) and implemented in
Stata\footnote{%
This command is not built in to Stata, but can be readily installed in any
version of Stata with access to the Web by using the ``ssc install kpss''
command. The command is already installed in the version of Stata running on
fmrisc.bc.edu.} as command $\mathtt{kpss}.$ \texttt{kpss} performs the
Kwiatkowski, Phillips, Schmidt, Shin (KPSS, 1992) test for stationarity of a
time series. The test may be conducted under the null of either trend
stationarity (the default) or level stationarity. Inference from this test
is complementary to that derived from those based on the Dickey--Fuller
distribution (such as \texttt{dfgls}, \texttt{dfuller} and \texttt{pperron}%
). The KPSS test is often used in conjunction with those tests to
investigate the possibility that a series is fractionally integrated (that
is, neither \textit{I(1)} nor \textit{I(0)}): see Lee and Schmidt (1996).

The series is detrended (demeaned) by regressing $y$ on $z_{t}=\left(
1,t\right) ^{^{\prime }}$ $\left( z_{t}=\left( 1\right) ^{\prime }\right) ,$
yielding residuals $e_{t}.$ Let the partial sum series of $e_{t}$ be $s_{t}.$
Then the zero-order KPSS\ statistic $k_{0}=T^{-2}\sum_{t=1}^{T}s_{t}^{2}$ / $%
T^{-1}\sum_{t=1}^{T}e_{t}^{2}.$ For \texttt{maxlag}$>0,$ the denominator is
computed as the Newey-West estimate of the long run variance of the series;
see \texttt{[R] newey.}

Approximate critical values for the KPSS test are taken from KPSS (1992).
The \texttt{kpss} routine includes two options recommended by the work of
Hobijn et al. (1998). An automatic bandwidth selection routine has been
added, rendering it unnecessary to evaluate a range of test statistics for
various lags. An option to weight the empirical autocovariance function by
the Quadratic Spectral kernel, rather than the Bartlett kernel employed by
KPSS, has also been introduced. These options may be used separately or in
conjunction. It is in conjunction that Hobijn et al. found the greatest
improvement in the test: ``Our Monte Carlo simulations show that the best
small sample results of the test in case the process exhibits a high degree
of persistence are obtained using both the automatic bandwidth selection
procedure and the Quadratic Spectral kernel.'' (1998, p.14) The \texttt{qs}
option specifies that the autocovariance function is to be weighted by the
Quadratic Spectral kernel, rather than the Bartlett kernel. Andrews (1991)
and Newey and West (1994) ``indicate that it yields more accurate estimates
of $\sigma _{\epsilon }^{2}$ than other kernels in finite samples.'' (Hobijn
et al., 1998, p.6) The \texttt{auto} option specifies that the automatic
bandwidth selection procedure proposed by Newey and West (1994) as described
by Hobijn et al. (1998, p.7) is used to determine \texttt{maxlag}, in two
stages. First, the ``a priori nonstochastic bandwidth parameter'' $n_{T}$ is
chosen as a function of the sample size and the specified kernel. The
autocovariance function of the estimated residuals is calculated, and used
to generate $\gamma $ as a function of sums of autocorrelations. The \texttt{%
maxlag} to be used in computing the long-run variance, $\hat{m}_{T}$, is
then calculated as $\min \left[ T,int\left[ \hat{\gamma}T^{\theta }\right]
\right] $ where $\theta $=1/3 for the Bartlett kernel and 1/5 for the
Quadratic Spectral kernel.

\textbf{The Leybourne--McCabe test}

Like the KPSS test, the test proposed by Leybourne and McCabe (LMc, 1994,
1999) has a null of stationarity, and a unit root alternative hypothesis.
The difference lies in the specification. The LMc test, like the ADF\ test,
is parametric; it posits a null of $ARIMA(p,0,0)$ (that is, $AR(p)$ with
deterministic trend) with an alternative of $ARIMA(p,1,1).$ The LMc test is
more cumbersome, as it requires estimation of this nonlinear model under the
null hypothesis, but it has been shown to be more powerful than the KPSS\
test. Just as the PP test is a semiparametric alternative to the ADF---that
is, the PP\ test uses a Newey--West long run estimate to deal with
dependence in the error process, rather than an explicit $AR(p)$
specification---the KPSS\ test may be considered as a semiparametric
alternative to the LMc test. The tests differ ``in how they take account of
autocorrelation in $y_{t}$ under H0'' (1994, p.160): the LMc test accounts
for autocorrelation in a parametric manner by including lagged terms in the
first difference in the series, as does the augmented Dickey-Fuller test. In
contrast, the KPSS test modifies the test nonparametrically, ``in a manner
similar to that in which the Phillips-Perron test is a nonparametric
adjustment of the simple DF test.'' (op.cit.) The authors find that ``once $%
y_{t}$ does not resemble white noise, the size of [the KPSS test] is likely
to be quite badly approximated by its asymptotic distribution, even when the
lag length $l$ is relatively high.'' (1994, p.161) The critical values for
the LMc test are identical to those used for KPSS.

\textbf{Combining inference from I(1) and I(0) tests}

The two families of unit root tests may be used in conjunction to establish
the nature of the data generating process for a given timeseries, and in
particular to signal the presence of fractional integration in the series.
If inference from the DFGLS\ test rejects its null hypothesis of unit root
behavior, or nonstationarity, while the KPSS test also rejects its null,
then we might conclude that both $I(1)$ and $I(0)$ are rejected by the data.
That sets the stage for an alternative explanation of the timeseries'
behavior: that of fractional integration, or long-range dependence, in which
the series may be characterized as $I(d),$ $0<d<1,$ neither $I(0)$ nor $%
I(1). $

\textbf{Seasonal unit root tests}

The implicit assumption in applying unit root tests to data which have been
deseasonalized is that the adjustment method does not affect inference on
the stationarity of the time series. However, several authors have called
that assumption into question in the case where the SA data have been
generated by a moving average filter. The most popular SA\ technique is the
US\ Census Bureau's X-11 seasonal adjustment program, which passes the data
through a sequence of moving average filters. Monte Carlo simulations show
that the power of standard unit root tests applied to SA data generated in
this manner is reduced, so that the null of nonstationarity is not rejected
frequently enough. Although one could deal with this issue by testing NSA\
data, they are not always available. One can modify the standard $AR$ model
that gives rise to the unit root test to 
\begin{eqnarray*}
y_{t} &=&\phi _{1s}y_{t-s}+\epsilon _{t} \\
(1-\phi _{1s}L^{s})y_{t} &=&\epsilon _{t}
\end{eqnarray*}
where $s=4$ for quarterly data, $s=12$ for monthly data, etc. If $\phi
_{1s}=1,$ then there is a unit root at the $s^{th}$ seasonal frequency, and
the $s^{th}$ difference of $y_{t}$ will remove it. Although this model could
be directly tested with the D--F methodology, it is likely to be too simple
for most applications, since it restricts the dynamics of $y$ to depend only
on seasonal differences. A natural extension of this model would be 
\begin{equation}
\phi (L)\phi _{s}(L^{s})y_{t}=\epsilon _{t}  \label{hegyeqn}
\end{equation}
where $\phi (L)$ is a standard autoregressive polynomial, and $\phi
_{s}(L^{s})=(1-\phi _{1s}L^{s}-...-\phi _{rs}L^{rs})$ is the seasonal
polynomial. One could have unit roots in either, both or neither of the
polynomials, and if they exist in both of the polynomials, both an ordinary
difference and a seasonal difference would have to be applied to render the
resulting series stationary. The test for a unit root in this context is
that of Hylleberg et al. (HEGY, 1990). For quarterly data, the test may be
implemented within Stata via the routine \texttt{hegy4} of Baum and Sperling
(\texttt{findit hegy4}).

In this representation, the unit root at the quarterly frequency can be
written as 
\begin{eqnarray*}
(1-L^{4}) &=&(1-L)(1+L+L^{2}+L^{3}) \\
&=&(1-L)(1+L)(1-iL)(1+iL)
\end{eqnarray*}
and the composite polynomial in (\ref{hegyeqn}) can be expanded about its
roots with a remainder term: 
\begin{eqnarray*}
\phi (L)\phi _{4}(L^{4}) &=&-\gamma _{1}L(1+L+L^{2}+L^{3}) \\
&&-\gamma _{2}(-L)(1-L+L^{2}-L^{3}) \\
&&-(\gamma _{3}L+\gamma _{4})(-L)(1-L^{2})+\phi ^{*}(L)(1-L^{4}) \\
\phi ^{*}(L)\Delta ^{4}y_{t} &=&\gamma _{1}y_{1,t-1}+\gamma
_{2}y_{2,t-1}+\gamma _{3}y_{3,t-2}+\gamma _{4}y_{3,t-1}+\epsilon _{t} \\
y_{1t} &=&(1+L+L^{2}+L^{3})y_{t} \\
y_{2t} &=&-(1-L+L^{2}-L^{3})y_{t} \\
y_{3t} &=&-(1-L^{2})y_{t}
\end{eqnarray*}
and this regression may be run as the equivalent of the D--F\ regression. A
test of $\gamma _{1}=0$ corresponds to a test of the standard unit root
hypothesis against the alternative of stationarity. A test of $\gamma _{2}=0$
allows for the semi--annual root of $-1$ versus the stationary alternative.
Seasonal unit roots at the quarterly frequency correspond to $\gamma
_{3}=\gamma _{4}=0.$ Thus there will ne no seasonal unit roots in the series
if $\gamma _{2}\neq 0$ and either $\gamma _{3}\neq 0$ or $\gamma _{4}\neq 0,$
corresponding to a rejection of the null that $\gamma _{2}\neq 0$ and the
joint null that $\gamma _{3}=\gamma _{4}=0.$

The Stata routine \texttt{hegy4} performs the test for seasonal unit roots
by estimating the four roots of the timeseries representation and presents
estimates of these roots as $\pi _{1}...\pi _{4}.$ A joint test for $\pi
_{3}=\pi _{4}=0$ is also presented. Critical values are those appropriate
for T=100, taken from HEGY Table 1. Joint tests for $\pi _{2}=\pi _{3}=\pi
_{4}=0$ and $\pi _{1}=\pi _{2}=\pi _{3}=\pi _{4}=0,$ with critical values,
are those presented by Ghysels et al. (1994). Critical values for the case
of muliplicative seasonality (see below) are from tables 1a-c in Smith and
Taylor (1998). Critical values are linearly interpolated for sample sizes in
the ranges (48,100) and (100,200).

L:ike the standard D--F test, it may be necessary to augment the HEGY\ test
with additional lags of the dependent variable. The option \texttt{%
Lags(numlist)} specifies the lag orders to be used in augmenting the model
with lags of the fourth difference of the timeseries. Its default is zero.
If sequential lags are specified starting with 1, HEGY4 automatically
conducts a sequential t-test to determine the optimal lag length and optimal
lags to be included in the auxiliary regression. It may also be desirable to
include deterministic terms such as a constant, trend and in this case
seasonal dummy variables in the model. In the \texttt{hegy4} routine, the
option \texttt{det} may take on values \texttt{none, const, seas, trend,
strend} or \texttt{mult}, specifying the process to be tested. The default,
as suggested by HEGY and Ghysels et al. (1994), is \texttt{seas}, indicating
that a set of 3 seasonal dummies plus constant are to be included in the
regression. \texttt{none} specifies that no deterministic variables are to
be included; \texttt{const} specifies only a constant. \texttt{trend}
specifies that a trend is to be included along with a constant term. \texttt{
strend} specifies that a trend is to be included along with seasonal dummies
and a constant term. \texttt{mult} specifies that seasonal intercepts (the
case of multiplicative seasonality, recommended by Smith and Taylor (1998))
are to be included along with seasonal dummies and a constant term.

The HEGY\ model may also be defined for monthly data (although the algebra
in that case is rather menacing); a Stata routine for that purpose is under
development.

\textbf{Testing for unit roots with structural breaks}

A well known problem in the unit root literature is the potential for a
series which exhibits structural shifts to fail to reject the unit root
null. In the simplest case, a series which undergoes a mean shift is not
covariance stationary, but could be made so if regressed on a dummy that
identified the shift period (zero before, one after). Early work along these
lines was that of Perron (1989) and Perron and Vogelsang (1992).  If there
is a known break in a sample of $T$ observations at point $T_{b},$ we may
consider three extensions of the random--walk--with--drift model, where $y$
is considered to be in logs, and no further dynamics are present:
\begin{eqnarray*}
y_{t} &=&\mu +\delta _{1}DVTB_{t}+y_{t-1}+\epsilon _{t} \\
y_{t} &=&\mu +\delta _{2}DVU_{t}+y_{t-1}+\epsilon _{t} \\
y_{t} &=&\mu +\delta _{1}DVTB_{t}+\delta _{2}DVU_{t}+y_{t-1}+\epsilon _{t}
\end{eqnarray*}
where $DVTB_{t}=1$ in the period $T_{b}+1,$ and $DVU_{t}=1$ for $t>T_{b}.$
The first model considers a level shift (jump) at time $T_{b}+1;$ the second
model considers a change in the growth rate of the series effective at time $%
T_{b}+1;$ and the third model considers the possibility that both occur. The 
$DVTB_{t}$ dummy is an ``impulse'' dummy, picking out the single period of
the shift, while $DVU_{t}$ is a shift dummy, which changes the underlying
slope of the stochastic trend. The alternative hypotheses to each of these
models are, respectively:
\begin{eqnarray*}
y_{t} &=&\mu +\beta t+\delta _{2}DVU_{t}+\epsilon _{t} \\
y_{t} &=&\mu +\beta t+\delta _{3}DVT_{t}^{*}+\epsilon _{t} \\
y_{t} &=&\mu +\beta t+\delta _{2}DVU_{t}+\delta _{3}DVT_{t}^{{}}+\epsilon
_{t}
\end{eqnarray*}
where $DVT_{t}^{*}\ $and $DVT_{t}^{{}}$ both equal zero if $t\leq T_{b}.$  $%
DVT_{t}^{*}=$ $t-T_{b}$ if $t>T_{b},$ and $DVT_{t}^{{}}=t$ if $t>T_{b}.$
Since there is drift in the first and third models under the null
hypothesis, the alternative includes a deterministic trend. In the first
model, the null is a unit root with level change; under the alternative, we
have a trend stationary series with a change in the intercept to $\mu
+\delta _{2}.$ In the second model, the null hypothesis is a unit root with
a change in the drift, and the alternative is a trend stationary series with
a change in the slope to  $\beta +\delta _{3}.$ In the third model, the null
is a unit root series with change in both level and drift, and the
alternative is a trend stationary series with changes in the intercept and
slope. In practice, further dynamics may be necessary to whiten the
residuals. 

Perron suggests the use of two procedures for modeling this process,
depending on whether adjustment following the break is assumed to be
instantaneous or gradual. The former is known as the $AO\;$(additive
outlier) case, where there is a single effect at the breakpoint.
Alternatively, the $IO\;$(innovational outlier) model may be applied, which
allows for a gradual adjustment of the series following the break. 

The original derivation of these tests was performed conditional on a known
breakpoint. More realistically, we may not know when (or even if) such a
breakpoint exists. If the tests are performed conditional on an a priori
breakpoint, they may not have maximal power. More recent derivations of unit
root tests in the presence of structural change have focused on unknown
breakpoints, and in some cases on multiple structural breaks and the methods
needed to consistently detect them. For instance, Perron (1997) presents a
procedure for locating the single breakpoint with highest likelihood by
considering all possible breakpoints in the interior of the sample,
selecting that point which maximizes the absolute value of the $t-$statistic
for the structural change term. That article presents asymptotic critical
values for the unit root test statistic in the presence of a single
breakpoint. 

A paper using this methodology to examine the impact of structural breaks on
unit root testing is Baum et al. (1999), which considers the possibility
that common findings of nonstationarity in real exchange rates may be an
artifact of structural breaks in the series. In the end, they conclude that
unit roots are present even when structural breaks (and the potential for
fractional integration) are accounted for. The unit-root test statistics
forthcoming from the $AO$\ and $IO$\ models will account for one-time level
shifts which might otherwise be identified as departures from stationarity.
However, the behavior of real exchange rate series over our sample period
may not be adequately characterized by a single shift; as Lothian (1998)\
has noted, US dollar-based real exchange rates appear to have exhibited two
shifts in mean over the 1980-1987\ period, approximately reverting to their
pre-1980\ level after 1987. In these circumstances, allowing for a single
level shift will not suffice. The Perron-Vogelsang methodology has been
extended to double mean shifts by Clemente et al. (1988), who demonstrate
that a two-dimensional grid search for breakpoints ($T_{b1}$ and $T_{b2})$
may be used for either the $AO$\ or $IO$\ models, and provide critical
values for the tests. In this context, the $AO$ model involves the
estimation of:

\begin{equation}
y_{t}=\mu +\delta _{1}DU_{1t}+\delta _{2}DU_{2t}+\tilde{y}_{t}  \label{nineA}
\end{equation}
and subsequently searching for the minimal $t-$ratio for the hypothesis $
\alpha =1$ in the model:
\begin{eqnarray*}
\tilde{y}_{t} &=&\sum_{i=0}^{k}\omega _{i}DT_{b1,t-i}\sum_{i=0}^{k}\omega
_{i}DT_{b2,t-i}+\alpha \tilde{y}_{t-1}+ \\
&&\sum_{i=1}^{k}\theta _{i}\Delta \tilde{y}_{t-i}+e_{t},\;
\end{eqnarray*}
for $t=k+2,...,T.$

For the $IO$\ model, the modified equation to be estimated becomes:

\begin{eqnarray}
y_{t} &=&\mu +\delta _{1}DU_{1t}+\delta _{2}DU_{2t}+\vartheta
_{1}DT_{b1,t}+\vartheta _{2}DT_{b2,t}  \label{elevenA} \\
&&+\alpha y_{t-1}+\sum_{i=1}^{k}\theta _{i}\Delta y_{t-i}+e_{t},
\end{eqnarray}
for $\;t=k+2,...,T,$ with a search for the minimal $t-$ratio for the
hypothesis $\alpha =1.\footnote{%
These tests customarily are applied to a trimmed sample; we trimmed 5\% of
the sample from each end when searching for the breakpoints.}$

Code to estimate unit root tests, allowing for one or two structural breaks
in either an $AO\;$or $IO$ context, is available for Stata as routines 
\texttt{clemao1, clemao2, clemio1}, and \texttt{clemio2}. As it has not yet
been documented, it is not in the $SSC$ archive, but it is available from
the instructor on request.

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\end{document}
