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

\bigskip \textit{Notes Section 6 }

\textbf{Cointegration}

Consider the system 
\begin{eqnarray*}
X_{t} &=&X_{t-1}+\epsilon _{t} \\
Y_{t} &=&\alpha +\beta X_{t}+\omega _{t}
\end{eqnarray*}
with $\epsilon _{t}$ and $\omega _{t}$ each $iid,$ standard Normal variates.
There are no dynamics in the DGP, and we assume there is no contemporaneous
correlation between the two error processes. In this case, OLS\ will find
unbiased and consistent estimates of $\alpha $ and $\beta ,$ despite the
fact that the variance of $X$ is not bounded. $X$ and $Y$ are said to be
cointegrated, or to possess a cointegrating relationship. $Y$ will be $I(1),$
since it is composed of an $I(1)$ variable plus a stationary error.

On the other hand, consider two independent random walks, $X$ and $Y,$ and
the regression of $Y$ upon $X.$ In that regression, the true slope
coefficient is zero, since there is no relationship whatsoever between these
two integrated processes. But the regression (run either way) will yield a
nonzero estimate of the slope coefficient, and the significance of that
coefficient will not diminish with sample size. Indeed, the probability of
rejecting the (true) null will increase with sample size. In a simple Monte
Carlo experiment, the rejection rate for the $t$ statistic is 72\% in
samples of size 50, but 80\% for $T=100,$ and 91\% for $T=500,$ whereas the
rejection rate should be 10\%. Furthermore, $R^{2}$ of such a regression
becomes a random variable in a regression of unrelated nonstationary
variables, and the likelihood of finding a sizable $R^{2}$ in that context
is quite sizable.

Demonstrably, OLS\ does not yield consistent estimates of the true slope
parameter in this instance: the case of a \textbf{spurious regression}, as
defined by Granger and Newbold (1974). A theoretical explanation of this
phenomenon was presented by Phillips (1986). The problem of spurious
regressions appears with $I(1)$ variables, so the determination of unit root
processes is essential. Furthermore, the problem will not arise if the
series are cointegrated, so that determining whether cointegration exists is
important as well.

What is cointegration? The notion that two (or more) nonstationary processes
may be following the same stochastic trend, or may share an underlying
common factor. Although $X$ and $Y$ are both $I(1),$ a linear combination of
those nonstationary variables may exist which is itself $I(0).$ The
coefficients in that linear combination form the cointegrating vector, which
will have one element normalized to unity, since the CI\ vector is defined
up to a factor of proportionality. Additionally, the vector may include a
constant, to allow for unequal means of the two variables, so 
\[
Y_{t}=\varphi _{1}+\varphi _{2}X_{t}+\xi _{t} 
\]
and the notion is that $\xi _{t}$ will be a stationary process in the
presence of cointegration.

The concept may be extended to higher orders of integration, and more than
two variables. If $X$ and $Y$ are each $I(2),$ then a linear combination of
them might be $I(1)$, or even $I(0).$ We can speak of series as being
cointegrated $CI(d,b),$ where $d$ is the common order of integration of the
variables, and $b$ is the reduction in the order of integration of the
cointegrating combination. Thus, the case above would be $CI(1,1),$ with two 
$I(1)$ variables forming a combination one order lower, of $I(0).$ A linear
combination of $I(1)$ variables is not a spurious regression if it is
stationary.

\textbf{The Engle--Granger approach}

The original approach to testing for cointegration is that of Engle and
Granger (1987). In this classic paper, they demonstrate that if one
regresses an $I(1)\;$variable upon another $I(1)$ variable (in what is
termed a ``balanced'' regression, in which all variables share the same
order of integration), the residuals from that regression may then be
subjected to a unit root test. The null hypothesis, in this case, is that of 
\textbf{noncointegration}: that is, failing to reject a unit root in the
errors in a Dickey-Fuller style test will yield the conclusion of nonCI,
whereas rejection in favor of stationarity in the error process will be
evidence of a cointegrating relationship among the variables. Stock (1987)
has shown that the OLS\ estimates in this regression have the desirable
property of ``superconsistency''---that is, they are not only consistent
estimates of the underlying parameters of the DGP, but they converge on the
population values more quickly than OLS\ estimates in the context of
stationary regressors. The ADF-type test applied in this instance will not
contain a constant term, since the OLS\ residuals will be mean zero with a
constant included in the CI\ regression. The critical values in this case
are not the same as those of the standard D--F distribution, since the
timeseries being tested is a generated series. They are larger negative
values than those provided for the D--F distribution, and like D--F critical
values, are obtained by simulation. Note also that the regression could be
run with either variable on the left hand side, since it is not a structural
relationship; if a CI\ vector exists, it may be renormalized on the other
variable. If the $R^{2}$ in the CI\ regression is low, however, (less than
0.8), then the inference may differ depending upon normalization.

How would one operate with more than two variables?\ One may still form a
cointegrating vector among three or more $I(1)$ series, and estimate the CI\
regression. As in the two--variable case, the estimates of the CI\
coefficients will be superconsistent. However, with more than two variables,
the weakness of the E-G approach emerges: the test can determine whether a
CI\ vector exists that yields stationary errors, but it will generate only
one of the possible multiple CI\ vectors that could exist in this setting.
When there are three variables, for instance, they could all be driven by
the same common factor (stochastic trend), or their behavior could reflect
two common factors--coinciding with the existence of two CI\ vectors. The
E--G approach is not capable of finding more than one CI\ vector; other
approaches, such as Johansen and Juselius' ML\ approach, can do so. (Of
course, it could be that there are zero common factors underlying these
three variables' dynamics, in which case the E--G approach will correctly
reflect the absence of CI\ among them).

\textbf{Cointegration and the error correction model}

A simple error correction model (ECM)\ may be written as
\[
Y_{t}-Y_{t-1}=\theta _{1}\Delta X_{t}+\theta
_{2}(Y_{t-1}-Y_{t-1}^{*})+\epsilon _{t}
\]
In this formulation, also known as a partial adjustment scheme, there is
adjustment of $Y$ toward a target $Y^{*}$ which depends on the lagged
disequilibrium. Imagine that there is a constant ratio, in equilibrium,
between consumption and income (in logs) $c$ and $y,$ so that $c_{t}=k+y_{t}.
$ Then a measure of disequilibrium may be written as $\zeta
_{t}=c_{t}-(k+y_{t}).$ An error correction scheme might be written as
\[
\Delta c_{t}=\theta _{1}\Delta y_{t}+\theta _{2}\zeta _{t-1}+\epsilon _{t}
\]
where consumers react to last period's disequilibrium by revising their
consumption. Substituting in, we have
\[
\Delta c_{t}=\theta _{0}+\theta _{1}\Delta y_{t}+\theta _{2}\left(
c_{t}-y_{t}\right) _{-1}+\epsilon _{t}
\]
where the parenthesized quantity is the error correction term. Consumption
will change if income changes, or if there was a disequilibrium in the
relationship last period. (Note that the proportionality factor $k$ has been
subsumed in the constant term of this relationship; we could, instead, leave
it in the error correction term as a coefficient on $y$). The coefficient   $%
\theta _{2}$ has limits $-1\leq \theta _{2}<0,$ since for stability one
should not overadjust to the disequilibrium (which would correspond to
closing the entire gap this period) nor to fail to adjust at all (a
coefficient of zero), let alone a positive coefficient (which would drive $c$
away from its equilibrium relationship). The ECM\ contains both the
short--run mechanism by which consumption will adjust to current changes in
income, as well as the long--run adjustment to equilibrium. The relative
importance of short-run and long-run fluctuations in consumption will be
governed by the relative magnitudes of the $\theta _{1}$ and $\theta _{2}$
coefficients. Incorporating the equilibrium coefficient in the relationship,
we have
\begin{equation}
\Delta c_{t}=\theta _{0}+\theta _{1}\Delta y_{t}+\theta _{2}\left(
c_{t}-\varphi _{2}y_{t}\right) _{-1}+\epsilon _{t}  \label{ecm}
\end{equation}
corresponding to the ``levels'' or static relationship
\begin{equation}
c_{t}=\varphi _{1}+\varphi _{2}y_{t}+\xi _{t}  \label{levels}
\end{equation}
If these variables are both $I(1),$ this relationship may be
(super)consistently estimated by OLS, and the residuals tested for
noncointegration. If the null of nonCI\ is rejected, then the relationship (%
\ref{ecm})\ may be estimated, replacing the unknown $\zeta _{t-1}$ with the
lagged residuals from (\ref{levels}), by OLS. 

If $c_{t}$ and $y_{t}$ are both $I(1)$ and cointegrated $CI(1,1),$ and
neither has a trend in the mean, then by the Granger Representation Theorem
(Granger and Weiss, 1983) there will always exist an error correction
representation of the form
\begin{eqnarray}
\Delta c_{t} &=&lagged(\Delta c_{t},\Delta y_{t})+\theta _{2c}\xi
_{t-1}+\upsilon _{1t}  \label{ecform} \\
\Delta y_{t} &=&lagged(\Delta c_{t},\Delta y_{t})+\theta _{2y}\xi
_{t-1}+\upsilon _{2t}  \nonumber
\end{eqnarray}
where $\xi _{t}=(c_{t}-\varphi _{1}+\varphi _{2}y_{t}),$ $\upsilon
_{1t}=\omega (L)\varepsilon _{1t}$ and $\upsilon _{2t}=\omega (L)\varepsilon
_{2t},$ with the $\varepsilon $ sequences white noise. It must be so that $%
\left| \theta _{2c}\right| +\left| \theta _{2y}\right| \neq 0;$ that is, the
lagged disequilibrium term must appear in at least one of the equations. In
general, it will appear in both. Note that this model is essentially a VAR\
augmented by the error correction term. The transformation of the dynamic
model into this form illustrates that a regular VAR\ in the differences of
these variables would be misspecified, in that it would omit the error
correction term. That term is required to fully specify the dynamics of the
model. In its absence, the reversion of these variables to a long--run
equilibrium is not modeled.

These equations are ``balanced'' in that if the levels variables are $I(1),$
their differences are $I(0).$ If the variables are $CI(1,1),$ then the error
correction term will be $I(0),$ and all terms in (\ref{ecform}) are $I(0).$
If these two variables are $I(1)$ and $CI(1,1),$ then knowledge of the one
variable helps to forecast the other---at least in one direction. Having
established cointegration as a long-run property of the data, it is natural
to think of an ECM\ as an appropriate way of capturing the dynamic
adjustments of these variables to the long run. 

\textbf{Multiple cointegrating relationships}

Consider a set of time series variables. If they are each (trend--)stationary, a VAR 
may be employed to estimate their joint evolution and interdependence. If one or more
of the variables in the VAR are nonstationary, it would be inappropriate to estimate
the VAR in levels. However, it might also be inappropriate to estimate a VAR in differences,
even in the case where all variables in the VAR possess unit roots. If there are 
cointegrating relationships among the level variables, the proper representation
will be the error correction model (ECM), and the VAR in differences may be
seen to be a misspecified version of that model (excluding, as it does, the 
error correction term). The VAR in differences is also uninformative about the long--run
behavior of the series, since it only expresses their short--run paths of adjustment,
without any link to the long--run equilibrium relationships among the variables. The
error correction model explicitly provides that link, capturing the short--run 
adjustment toward the long--run equilibria.

When there are more than two variables in the set, there may be multiple 
cointegrating relationships among them. In a two--variable system, the variables either
form a cointegrating combination or they do not. In a three--variable system, there
may be zero, one, or two cointegrating vectors. If zero, then these are three independent 
random walks. If there are two CI vectors, the Engle--Granger procedure will locate
one of them, but it is incapable of identifying the multiplicity, or of estimating 
a second relationship. Likewise, for higher--order systems of order $k$, there may
be up to $k-1$ CI vectors defining long--run equilbria among the variables.

The most common methodology employed to evaluate multiple cointegrating relationships
is that of Johansen (1988) and Johansen and Juselius (1990), which is based on the estimation of a $p^{th}$--order 
VAR in the $k$ variables. The VAR in the $k$--vector $y$ is:

\begin{equation}
y_t = \Pi_1 y_{t-1} + \Pi_2 y_{t-2} + \dots + \Pi_p y_{t-p} + \Psi D_t + \epsilon_t
\end{equation}

where $D_t$ is a $d$--vector of deterministic terms, such as a constant, trend and
seasonal dummies. The VAR may be reparameterized into an ECM:

\begin{equation}
\Delta y_t = \Pi y_{t-1} + \Gamma_1 \Delta y_{t-1} + \Gamma_2 \Delta y_{t-2} + \dots 
+ \Gamma_{p-1} \Delta y_{t-(p-1)} + \Psi D_t + \epsilon_t
\end{equation}

No assumption is made about the rank of $\Pi$. In the decomposition $\Pi = \alpha \beta'$, 
$\alpha$ and $\beta$ are $k$ x $k$ matrices. We seek to determine whether any columns
of $\beta$ (that is, rows of $\beta'$) are statistically indistinguishable from zero
vectors. The existence of $r$ cointegrating vectors reduces the rank of $\Pi$ by $k-r$: that is,
if there are $r$ cointegrating relationships among the variables, then there will be
$r$ nonzero eigenvalues in the dynamic system, and $k-r$ zero eigenvalues.
The decomposition will then relate  $\Pi = \alpha \beta'$ where $\alpha$ and $\beta$ are
both $k$ x $r$ matrices. If the CI rank is full, that is, $r=k$, then the VAR is stationary
in the levels. If the rank is zero, then there is no implied long run, and the VAR
may be safely reformulated in first differences. 

The Johansen methodology provides inference on the number of
nonzero eigenvalues, or CI relationships, by setting up an eigenvalue problem derived
from the levels and differences of the $k$ variables. The eigenvalues are ordered, from
largest to smallest. The space spanned by the $r$ largest eigenvalues is the $r$--dimensional
cointegrating space. If $r=1$, $\beta$ is $k$ x 1, and is the eigenvector corresponding
to the largest eigenvalue. If $r=2$, $\beta$ is $k$ x 2; the first column is as before,
and the second column is the eigenvector corresponding to the second largest eigenvalue.


Two statistics are defined in Johansen's work to determine the CI rank:
first, the trace statistic,

\begin{equation}
trace = -T \sum_{i=r+1}^{k} ln(1-\lambda_i)
\end{equation}

which allows for the test of $H(r)$: the rank of $\Pi$ is r, against the alternative 
that the rank of $\Pi$ is $k$. A large value of the trace statistic is evidence against
$H(r)$: that is, with $r=1$, a value of the trace statistic greater than the appropriate
critical value allows us to reject $r=1$ in favor of $r>1$. The test may then be 
repeated for $r=2$, and so on. 

Alternatively, the $\lambda_{max}$ statistic may be used:

\begin{equation}
\lambda_{max} = -T ln(1-\lambda_{r+1})
\end{equation}

This test allows for the comparison of a CI rank of $r$ against the alternative of 
a CI rank of $r+1$. This test also may then be repeated for larger values of $r$ until
one fails to reject the null hypothesis.

The distribution of both statistics is nonstandard, and depends on nuisance parameters
in $D_t$. Critical values have been tabulated by Johansen and Osterwald--Lenum, and
are reproduced in the textbook. Research
by Reimers and Cheung and Lai (1993) have identified small--sample biases in the tabulated
values of these test statistics, and they recommend applying a small--sample adjustment.

Extensions of the Johansen methodology include tests of various restrictions on the CI
vectors: either zero (exclusion) restrictions, indicating that certain variables should
not appear in certain of the equilibrium relationships, or restrictions on parameters' 
values, such as those forthcoming from theory (e.g. purchasing power parity not only
specifies a long run relationship, but indicates that the coefficients in the CI 
combination should be (1, 1, -1)). 


{\small References}

\begin{thebibliography}{9}
\bibitem{cl93}  {\small Cheung, Y.-W., and K. S. Lai. 1993. Finite--sample
sizes of Johansen's likelihoood ratio tests for cointegration. Oxford
Bulletin of Economics and Statistics, 55, 313--328.}

\bibitem{eg87}  {\small Engle, R., and C.W.J. Granger. 1987. Cointegration
and error correction: Representation, estimation and testing. Econometrica, 55, 251--276.}

\bibitem{gnewbold}  {\small Granger, C.W.J., and P. Newbold. 1974. Spurious regressions
in econometrics. Journal of Econometrics, 2, 111--120.}

\bibitem{gweiss}  {\small Granger, C.W.J., and A. Weiss. 1983. Time--series analysis
of error correction models, in S. Karlin, T. Amemiya and L. Goodman, eds., Studies in 
Econometrics, Time Series and Multivariate Statistics. Academic Press.}

\bibitem{joh88}  {\small Johansen, S. 1988. Statistical analysis of cointegration
vectors. Journal of Economic Dynamics and Control, 12, 231--254.}
 
\bibitem{jj90}  {\small Johansen, S. and K. Juselius. 1990. Maximum likelihood 
estimation and inference on cointegration with applications to the demand for money. 
Oxford Bulletin of Economics and Statistics, 522, 169--210.} 

\bibitem{tsstat}  {\small Phillips, P.C.B. 1986. Understanding spurious regressions
in econometrics. Jorunal of Econometrics, 33, 311--340.}

\bibitem{stock87}  {\small Stock, James. 1987. Asymptotic properties of least squares
estimators of co--integrating vectors. Econometrica, 55, 1035--1056.}
\end{thebibliography}

\end{document}
