Chapter 17 – Random walks, Unit Roots, and Cointegration

\chapter{Random walks, unit roots, and cointegration\label{unitroots.chap}} Most economic time series variables display a significant trend component.
Regression models that are estimated using time-series data will almost always have a high $R^2$ and significant $t$-ratios for some or all of the slope coefficients. These results may be due to the use of an appropriate model specification that accurately represents the data generating process.
It is quite possible, however, that these “statistically significant”
effects may be the result of common trend components existing in the dependent and independent variables. This chapter contains a discussion of the estimation of regression models when the dependent and independent variables exhibit a trend over time.
\section{Stationarity}
A time-series variable is said to be \textbf{stationary} if the probability distribution function for the variable is the same in all time periods. In particular, a time-series variable, $Y_t$, is said to be stationary if: \begin{itemize}
\item the mean of $Y_t$ is constant for all time periods, \item the variance of $Y_t$ is constant for all time periods, and \item the covariance between $Y_t$ and $Y_{t-s}$ depends only on $s$ and is not affected by $t$.
\end{itemize}
Roughly speaking, it can be said that a time-series variable is stationary if the characteristics of the distribution are not affected by the value of the time subscript $t$.
An examination of a graph of most economic time-series variables, however, suggests that the mean of these variables is not constant over time. In particular, most economic time-series variables appear to contain significant trend components. This suggests that most economic time-series variables are nonstationary.
The regression of one nonstationary time-series variable on another will often result in high $t$-statistics and a high value of R$^2$ even when no causal relationship exists between the variables. As evidence of this, recall the example appearing in Chapter \ref{intro.chap} of a regression in which the annual number of deaths was regressed on the number of secondary school teachers. Since the levels of both of these variables have grown substantially over time, a strong (and statistically significant) relationship appears to exist between variables that are unlikely to exhibit any causal relationship.
\subsection{Trend-stationary processes}
A time-series variable, $Y_{t}$, is said to follow a \textbf{% trend-stationary process} if it can be expressed as: \begin{equation}
Y_{t}=\beta _{o}+\beta _{1}t+\epsilon _{t} \label{trend.stat.urc} \end{equation}%
where $t$ = a simple time trend, $\beta _{1}$ is not equal to zero, and $% \epsilon _{t}$ is a white-noise error process.\footnote{% As defined in Chapter \ref{auto.chap}, a white noise error process has a mean of zero, a constant variance, and is uncorrelated across observations.
In mathematical terms, $\epsilon _{t}$ is a white noise error process if: \begin{equation*}
E(\epsilon _{t})=0\text{,}
\end{equation*}%
\begin{equation*}
E(\epsilon _{t}^{2}=\sigma ^{2})\text{,}
\end{equation*}%
and
\begin{equation*}
E(\epsilon _{t}\epsilon _{s})=0\text{ for }t\neq s\text{.} \end{equation*}%
} This series is nonstationary because the mean of the series is not constant. The mean increases over time if $\beta _{1}$ is positive and decreases over time if $\beta _{1}$ is negative. If the dependent variable in a regression is a trend-stationary process, the trend component can be taken into account by simply including a linear trend term as an independent variable in the regression model.
An alternative approach involves \textquotedblleft
detrending\textquotedblright\ the series by estimating the parameters of equation \ref{trend.stat.urc} by OLS and letting the estimated residual serve as the detrended variable. If $Y_{t}$ follows a trend-stationary process, this detrended variable will be stationary (with a mean of zero)..
\subsection{Random walks}
An alternative model of nonstationary time series variables is provided by a \textbf{random walk} process. If $Y_t$ follows a simple random walk process, then the level of $Y_t$ can be expressed as:
\begin{equation}
Y_t=Y_{t-1}+\epsilon _t \label{random.walk.urc}
\end{equation}
where $\epsilon _t$ is a white-noise error process. A random walk model is a special case of a first-order autoregressive process: \begin{equation}
Y_t=\rho Y_{t-1}+\epsilon _t \label{ar1.urc}
\end{equation}
in which the value of $\rho $ equals one. A fundamental difference between a first-order autoregressive process and a random walk process, however, is that the impact of past shocks gradually die out when $\left| \rho \right| $ is less than one but result in a permanent effect when $\left| \rho \right| $ equals one.
An interesting extension of the random walk model is given by: \begin{equation}
Y_t=\beta _o+Y_{t-1}+\epsilon _t \label{random.drift.urc} \end{equation}
The relationship appearing in equation \ref{random.drift.urc} is called a \textbf{random walk with drift}. Subtracting $Y_{t-1}$ from both sides of equation \ref{random.drift.urc}, results in:
\begin{equation}
\Delta Y_t=\beta _o+\epsilon _t \label{dsp.urc}
\end{equation}
As equation \ref{dsp.urc} indicates, the “drift” in this process occurs because the level of $Y_t$, on average, changes by $\beta _o$ units each time period.
Random walk processes are also referred to as \textbf{difference-stationary processes} since the first-difference of such a variable will be stationary (as can be seen by an inspection of equation \ref{dsp.urc}). If the dependent variable in a regression model is a difference-stationary process (a random walk), then an appropriate correction procedure often involves expressing the model in first-difference form before estimating the relationship.\footnote{%
An exception to this principle occurs in the case of cointegrated time series variables. A discussion of this case appears below.} \section{Trend-stationary vs. difference-stationary processes} In a finite sample, the behavior of a random walk with drift will often closely resemble that of a trend-stationary process. Let’s examine why this occurs. Suppose that the $Y_{t}$ is a random walk with drift variable and that the initial level of $Y$ is $Y_{o}$ in period $t=0$. Using equation \ref% {random.drift.urc}, the value of $Y_{t}$ in period $1$ equals: \begin{equation}
Y_{1}=\beta _{o}+Y_{o}+\epsilon _{1} \label{randomdrift2.urc} \end{equation}%
To determine the level of $Y_{t}$ in period 2, the value of $Y_{1}$ from equation \ref{randomdrift2.urc} can be substituted into equation \ref% {random.drift.urc} to form:
\begin{equation*}
Y_{2}=\beta _{o}+\left( \beta _{o}+Y_{o}+\epsilon _{1}\right) +\epsilon _{2} \end{equation*}%
or:
\begin{equation*}
Y_{2}=Y_{o}+2\beta _{o}+\left( \epsilon _{1}+\epsilon _{2}\right) \end{equation*}%
If this process continues for $t$ periods, $Y_{t}$ will equal: \begin{equation}
Y_{t}=Y_{o}+\beta t+\sum_{i=1}^{t}\epsilon _{i} \label{dsp.proc.urc} \end{equation}%
A comparison of equations \ref{dsp.proc.urc} and \ref{trend.stat.urc} indicates the similar nature of a trend-stationary process and a random walk with drift: both models exhibit a linear trend over time. The difference, however, is that the variance of the error process in a random walk with drift increases as the size of the sample rises.
While trend-stationary and random walk processes exhibit similar characteristics, the appropriate estimation techniques are somewhat different. In each case, the goal is to transform a relationship involving nonstationary time-series variables into one involving stationary time-series variables. Under a trend-stationary process, an appropriate correction procedure involves detrending the affected variables; the correction for a difference-stationary process (a random walk) involves the use of a difference operation.
\section{Caution: Random walks and trend terms}
Econometricians have long recognized that the existence of common trends in dependent and independent variables may lead to spurious regression results.
Until the 1980s, econometricians often dealt with this problem by including a linear time trend as an independent variable to capture any trend effects.
This procedure is appropriate as long as the dependent variable is a trend-stationary random variable. If the dependent variable follows a random walk, however, including time as an independent variable causes some serious problems.
As noted by Nelson and Kang (1984), the regression of a random walk (without drift) on time will produce an $R^{2}$ value of approximately 0.44 even though the mean of the variable is unrelated to time. When a random walk with drift is regressed against time, the value of $R^{2}$ tends toward one as the sample size approaches infinity.
If a random walk process is mistakenly treated as a trend-stationary process, then the variance of the error term increases with time (as an inspection of equation \ref{dsp.proc.urc} indicates). In this case, OLS estimates will be inefficient and the usual $t$-ratios cannot be used for hypothesis testing. Because there are serious consequences of ignoring the presence of a random walk, it would be helpful to be able to test for the existence of such a process. The Dickey-Fuller test is available for this purpose.
\section{Dickey-Fuller test}
As noted above, a random walk process is a special case of a first-order autoregressive process in which $\left| \rho \right| $ equals one. Because of this, a variable that follows a random walk process is said to possess a \textbf{unit root}. Thus, tests for the presence of a random walk are generally referred to as unit root tests.
Consider the model:
\begin{equation}
Y_t=\rho Y_{t-1}+\epsilon _t \label{df.ac}
\end{equation}
\begin{equation*}
\text{where: }\epsilon _t\text{ is a white noise error process} \end{equation*}
As long as $\rho $ is less than one, OLS estimators will result in consistent estimates of the parameter $\rho $. As shown by Dickey and Fuller (1979, 1981), the OLS\ estimate of $\rho $ is downward biased when $\rho $ equals one.
To deal with this problem, Dickey and Fuller proposed the following procedure:
\begin{enumerate}
\item[Step 1:] Estimate the parameters of equation \ref{df.ac} by OLS. The estimated value of $\rho $ is simply the estimated slope coefficient from the OLS estimation procedure.
\item[Step 2:] Construct the Dickey-Fuller statistic defined as: \begin{equation*}
\text{Dickey-Fuller statistic = }\frac{\hat{\rho}-1}{\hat{\sigma}_{\hat{\rho}% }}
\end{equation*}%
(where:\ $\hat{\sigma}_{\hat{\rho}}$ is the estimated standard error of the estimated slope coefficient $\hat{\rho})$. While the Dickey-Fuller statistic may appear to be a $t$-statistic, it is not distributed according to Student’s $t$-distribution since the estimated coefficient is biased when $% \rho $ equals one. Dickey and Fuller have determined the critical values for this estimated statistic under the null hypothesis that $\rho =1$ (these critical values appear in Table \ref{df.tab.1.ac}).
\item[Step 3:] Compare the Dickey-Fuller statistic to the critical values appearing in Table \ref{df.tab.1.ac}. If the estimated Dickey-Fuller statistic is less than the critical value at the preselected significance level, then the null hypothesis can be rejected and it can be assumed that the dependent variable is not a random walk. (Note that this is equivalent to requiring that the absolute value of the Dickey-Fuller statistic is less than the absolute value of the critical value.) If the estimated statistic exceeds the critical value, then it is not possible to reject the hypothesis that $Y_{t}$ is a random walk. In this case, a differenced model should be used when $Y_{t}$ serves as a dependent variable in a regression specification.
%TCIMACRO{%
%\TeXButton{tabular}{\begin {table}
%\begin{center}
%\begin{tabular}{|c|rrrrrrrr|} \hline
%\bf{Sample}& \multicolumn{8} {|l|} {\bf {CDF for Dickey-Fuller statistic:} $Y_t=\rho Y_{t-1} + \epsilon_t$ } \\ \cline{2-9} %\bf {Size ($N$)} & \bf 0.01 & \bf 0.025 & \bf 0.05 & \bf 0.10 & \bf 0.90 &\bf 0.95 & \bf 0.975 & \bf 0.99 \\ \hline %25 & -2.66 & -2.26 & -1.95 & -1.60 & 0.92 & 1.33 & 1.70 & 2.16 \\ %50 & -2.62 & -2.25 & -1.95 & -1.61 & 0.91 & 1.31 & 1.66 & 2.08 \\ %100 & -2.60 & -2.24 & -1.95 & -1.61 & 0.90 & 1.29 & 1.64 & 2.03 \\ %250 & -2.58 & -2.23 & -1.95 & -1.62 & 0.89 & 1.29 & 1.63 & 2.01 \\ %500 & -2.58 & -2.23 & -1.95 & -1.62 & 0.89 & 1.28 & 1.62 & 2.00 \\ %$\infty $ & -2.58 & -2.23 & -1.95 & -1.62 & 0.89 & 1.28 & 1.62 & 2.00 \\ \hline %\bf{Sample}& \multicolumn{8} {|l|} {\bf {CDF for Dickey-Fuller statistic:} $Y_t=\beta _o +\rho Y_{t-1} + \epsilon_t$ } \\ \cline{2-9} %\bf {Size ($N$)} & \bf 0.01 & \bf 0.025 & \bf 0.05 & \bf 0.10 & \bf 0.90 &\bf 0.95 & \bf 0.975 & \bf 0.99 \\ \hline %25 & -3.75 & -3.33 & -3.00 & -2.63 & -0.37 & 0.00 & 0.34 & 0.72 \\ %50 & -3.58 & -3.22 & -2.93 & -2.60 & -0.40 & -0.03 & 0.29 & 0.66 \\ %100 & -3.51 & -3.17 & -2.89 & -2.58 & -0.42 & -0.05 & 0.26 & 0.63 \\ %250 & -3.46 & -3.14 & -2.88 & -2.57 & -0.42 & -0.06 & 0.24 & 0.62 \\ %500 & -3.44 & -3.13 & -2.87 & -2.57 & -0.43 & -0.07 & 0.24 & 0.61 \\ %$\infty $ & -3.43 & -3.12 & -2.86 & -2.57 & -0.44 & -0.07 & 0.23 & 0.60 \\ \hline %\bf{Sample}& \multicolumn{8} {|l|} {\bf {CDF for Dickey-Fuller statistic:} $Y_t=\beta _o +\beta_1 t +\rho Y_{t-1} + \epsilon_t$ } \\ \cline{2-9} %\bf {Size ($N$)} & \bf 0.01 & \bf 0.025 & \bf 0.05 & \bf 0.10 & \bf 0.90 &\bf 0.95 & \bf 0.975 & \bf 0.99 \\ \hline %25 & -4.38 & -3.95 & -3.60 & -3.24 & -1.14 & -0.80 & -0.50 & -0.15 \\ %50 & -4.15 & -3.80 & -3.50 & -3.18 & -1.19 & -0.87 & -0.58 & -0.24 \\ %100 & -4.04 & -3.73 & -3.45 & -3.15 & -1.22 & -0.90 & -0.62 & -0.28 \\ %250 & -3.99 & -3.69 & -3.43 & -3.13 & -1.23 & -0.92 & -0.64 & -0.31 \\ %500 & -3.98 & -3.68 & -3.42 & -3.13 & -1.24 & -0.93 & -0.65 & -0.32 \\ %$\infty $ & -3.96 & -3.66 & -3.41 & -3.12 & -1.25 & -0.94 & -0.66 & -0.33 \\ \hline %\end{tabular}
%\caption{Empirical distribution of Dickey-Fuller statistic — random walk model (permission needed- Fuller (1976, p. 373)) \label{df.tab.1.ac}} %\end{center}
%\end{table}}}%
%BeginExpansion
\begin {table}
\begin{center}
\begin{tabular}{|c|rrrrrrrr|} \hline
\bf{Sample}& \multicolumn{8} {|l|} {\bf {CDF for Dickey-Fuller statistic:} $Y_t=\rho Y_{t-1} + \epsilon_t$ } \\ \cline{2-9} \bf {Size ($N$)} & \bf 0.01 & \bf 0.025 & \bf 0.05 & \bf 0.10 & \bf 0.90 &\bf 0.95 & \bf 0.975 & \bf 0.99 \\ \hline 25 & -2.66 & -2.26 & -1.95 & -1.60 & 0.92 & 1.33 & 1.70 & 2.16 \\ 50 & -2.62 & -2.25 & -1.95 & -1.61 & 0.91 & 1.31 & 1.66 & 2.08 \\ 100 & -2.60 & -2.24 & -1.95 & -1.61 & 0.90 & 1.29 & 1.64 & 2.03 \\ 250 & -2.58 & -2.23 & -1.95 & -1.62 & 0.89 & 1.29 & 1.63 & 2.01 \\ 500 & -2.58 & -2.23 & -1.95 & -1.62 & 0.89 & 1.28 & 1.62 & 2.00 \\ $\infty $ & -2.58 & -2.23 & -1.95 & -1.62 & 0.89 & 1.28 & 1.62 & 2.00 \\ \hline \bf{Sample}& \multicolumn{8} {|l|} {\bf {CDF for Dickey-Fuller statistic:} $Y_t=\beta _o +\rho Y_{t-1} + \epsilon_t$ } \\ \cline{2-9} \bf {Size ($N$)} & \bf 0.01 & \bf 0.025 & \bf 0.05 & \bf 0.10 & \bf 0.90 &\bf 0.95 & \bf 0.975 & \bf 0.99 \\ \hline 25 & -3.75 & -3.33 & -3.00 & -2.63 & -0.37 & 0.00 & 0.34 & 0.72 \\ 50 & -3.58 & -3.22 & -2.93 & -2.60 & -0.40 & -0.03 & 0.29 & 0.66 \\ 100 & -3.51 & -3.17 & -2.89 & -2.58 & -0.42 & -0.05 & 0.26 & 0.63 \\ 250 & -3.46 & -3.14 & -2.88 & -2.57 & -0.42 & -0.06 & 0.24 & 0.62 \\ 500 & -3.44 & -3.13 & -2.87 & -2.57 & -0.43 & -0.07 & 0.24 & 0.61 \\ $\infty $ & -3.43 & -3.12 & -2.86 & -2.57 & -0.44 & -0.07 & 0.23 & 0.60 \\ \hline \bf{Sample}& \multicolumn{8} {|l|} {\bf {CDF for Dickey-Fuller statistic:} $Y_t=\beta _o +\beta_1 t +\rho Y_{t-1} + \epsilon_t$ } \\ \cline{2-9} \bf {Size ($N$)} & \bf 0.01 & \bf 0.025 & \bf 0.05 & \bf 0.10 & \bf 0.90 &\bf 0.95 & \bf 0.975 & \bf 0.99 \\ \hline 25 & -4.38 & -3.95 & -3.60 & -3.24 & -1.14 & -0.80 & -0.50 & -0.15 \\ 50 & -4.15 & -3.80 & -3.50 & -3.18 & -1.19 & -0.87 & -0.58 & -0.24 \\ 100 & -4.04 & -3.73 & -3.45 & -3.15 & -1.22 & -0.90 & -0.62 & -0.28 \\ 250 & -3.99 & -3.69 & -3.43 & -3.13 & -1.23 & -0.92 & -0.64 & -0.31 \\ 500 & -3.98 & -3.68 & -3.42 & -3.13 & -1.24 & -0.93 & -0.65 & -0.32 \\ $\infty $ & -3.96 & -3.66 & -3.41 & -3.12 & -1.25 & -0.94 & -0.66 & -0.33 \\ \hline \end{tabular}
\caption{Empirical distribution of Dickey-Fuller statistic — random walk model (permission needed- Fuller (1976, p. 373)) \label{df.tab.1.ac}} \end{center}
\end{table}%
%EndExpansion
\end{enumerate}
Dickey and Fuller have also computed critical values for the Dickey-Fuller statistic in two alternative models:
\begin{equation}
Y_t=\beta _o+\rho Y_{t-1}+\epsilon _t \label{rwd.urc} \end{equation}
and
\begin{equation}
Y_t=\beta _o+\beta _1t+\rho Y_{t-1}+\epsilon _t \label{ltrw.urc} \end{equation}
The model represented by equation \ref{rwd.urc} allows for the possibility of a random walk with drift. A model with a linear trend component as well as a random walk with drift is captured by equation \ref{ltrw.urc}. The critical values for these tests, as computed by Dickey and Fuller (1976) appear in Table \ref{df.tab.1.ac}.
A variation of the Dickey-Fuller test can be used to test for the presence of a unit root in a model in which the error terms are serially correlated.
Under this alternative procedure, the change in $Y_{t}$ follows an autoregressive process. The basic model is:\footnote{% More generally, the length of the autoregressive process on $\Delta Y_{t}$ can be expanded to allow for more complex forms of autocorrelation. For a more complete discussion of this, see Dickey and Fuller (1981).} \begin{equation}
\Delta Y_{t}=\beta _{o}+\beta _{1}t+\left( \rho -1\right) Y_{t-1}+\theta \Delta Y_{t-1}+\epsilon _{t} \label{aug.df.test.ac} \end{equation}%
If $Y_{t}$ follows a random walk, then $\beta _{1}=0$ and $\rho =1$. A test of this joint hypothesis can be constructed as follows: \begin{enumerate}
\item[Step 1:] Estimate the parameters of the unrestricted model (equation % \ref{aug.df.test.ac}).
\item[Step 2:] Impose the restrictions required by the null hypothesis to form:
\begin{equation*}
\Delta Y_t=\beta _o+\Delta Y_{t-1}+\epsilon _t
\end{equation*}
Estimate the parameters of this restricted model using OLS.
\item[Step 3:] Formulate the usual Wald statistic:
\begin{equation*}
F_{DF}=\frac{\left( \text{RSS}_{R}-\text{RSS}_{UR}\right) /2}{\text{RSS}% _{UR}/(N-(k+1))}
\end{equation*}%
Since the estimated coefficient on $Y_{t-1}$ is biased when $\rho $ equals one, this statistic is not distributed as an $F$ statistic. This statistic is generally referred to as an \textbf{augmented Dickey-Fuller }statistic.
Dickey and Fuller have determined the critical values for this statistic.
Table \ref{df.tab.2.ac} contains these critical values.
\item[Step 4:] Compare the Dickey-Fuller statistic to the critical values for the Dickey-Fuller statistic. If the estimated Dickey-Fuller statistic falls within the rejection region, then the null hypothesis can be rejected and it can be assumed that the dependent variable is not a random walk.
%TCIMACRO{%
%\TeXButton{tabular}{\begin {table}
%\begin{center}
%\begin{tabular}{|c|rrrrrrrr|} \hline
%\bf{Sample}& \multicolumn{8} {|c|} {\bf {CDF for augmented Dickey-Fuller statistic} } \\ \cline{2-9} %\bf {Size ($N$)} & \bf 0.01 & \bf 0.025 & \bf 0.05 & \bf 0.10 & \bf 0.90 &\bf 0.95 & \bf 0.975 & \bf 0.99 \\ \hline %25 & 0.74 & 0.90 & 1.08 & 1.33 & 5.91 & 7.24 & 8.65 & 10.61 \\ %50 & 0.76 & 0.93 & 1.11 & 1.37 & 5.61 & 6.73 & 7.81 & 9.31 \\ %100 & 0.76 & 0.94 & 1.12 & 1.38 & 5.47 & 6.49 & 7.44 & 8.73 \\ %250 & 0.76 & 0.94 & 1.13 & 1.39 & 5.39 & 6.34 & 7.25 & 8.43 \\ %500 & 0.76 & 0.94 & 1.13 & 1.39 & 5.36 & 6.30 & 7.20 & 8.34 \\ %$\infty $ & 0.77 & 0.94 & 1.13 & 1.39 & 5.34 & 6.25 & 7.16 & 8.27 \\ \hline %\end{tabular}
%\caption{Empirical distribution of augmented Dickey-Fuller statistic (permission needed- Dickey and Fuller (1981, p. 373)) \label{df.tab.2.ac}} %\end{center}
%\end{table}} }%
%BeginExpansion
\begin {table}
\begin{center}
\begin{tabular}{|c|rrrrrrrr|} \hline
\bf{Sample}& \multicolumn{8} {|c|} {\bf {CDF for augmented Dickey-Fuller statistic} } \\ \cline{2-9} \bf {Size ($N$)} & \bf 0.01 & \bf 0.025 & \bf 0.05 & \bf 0.10 & \bf 0.90 &\bf 0.95 & \bf 0.975 & \bf 0.99 \\ \hline 25 & 0.74 & 0.90 & 1.08 & 1.33 & 5.91 & 7.24 & 8.65 & 10.61 \\ 50 & 0.76 & 0.93 & 1.11 & 1.37 & 5.61 & 6.73 & 7.81 & 9.31 \\ 100 & 0.76 & 0.94 & 1.12 & 1.38 & 5.47 & 6.49 & 7.44 & 8.73 \\ 250 & 0.76 & 0.94 & 1.13 & 1.39 & 5.39 & 6.34 & 7.25 & 8.43 \\ 500 & 0.76 & 0.94 & 1.13 & 1.39 & 5.36 & 6.30 & 7.20 & 8.34 \\ $\infty $ & 0.77 & 0.94 & 1.13 & 1.39 & 5.34 & 6.25 & 7.16 & 8.27 \\ \hline \end{tabular}
\caption{Empirical distribution of augmented Dickey-Fuller statistic (permission needed- Dickey and Fuller (1981, p. 373)) \label{df.tab.2.ac}} \end{center}
\end{table}
%EndExpansion
%TCIMACRO{%
%\TeXButton{unit roots box}{\exbox{Are most economic time-series variables random walks?}{ %Nelson and Plosser (1982) examined the behavior of 14 basic macroeconomic time-series %variables (including variables such as real and nominal GNP, the unemployment rate, %nominal and real wages, interest rates, and stock prices). The results of Dickey-Fuller tests %applied to these series suggest that these variables (with the exception of the unemployment %rate) follow a random
%walk with drift process, rather than a trend-stationary process.
%
%Critics of the Nelson and Plosser results, however, note that while one cannot reject the %hypothesis that $\rho$ = 1, a similar test would also fail to reject a hypothesis stating that %$\rho$ = 0.95. As long as the value of $\rho$ is less than one, a stationary (or trend-stationary) %process occurs. While a unit root tests involves a failure to reject the hypothesis that %$\rho$ equals one, this does not necessarily mean that the true value of $\rho$ equals one.
%
%The work of Nelson and Plosser, hower, convinced most econometricians of the importance of %testing for stationary error processes when working with time-series models.This study %helped to encourage the current widespread use of the DIckey-Fuller test.
%}}}%
%BeginExpansion
\exbox{Are most economic time-series variables random walks?}{ Nelson and Plosser (1982) examined the behavior of 14 basic macroeconomic time-series variables (including variables such as real and nominal GNP, the unemployment rate, nominal and real wages, interest rates, and stock prices). The results of Dickey-Fuller tests applied to these series suggest that these variables (with the exception of the unemployment rate) follow a random
walk with drift process, rather than a trend-stationary process.
Critics of the Nelson and Plosser results, however, note that while one cannot reject the hypothesis that $\rho$ = 1, a similar test would also fail to reject a hypothesis stating that $\rho$ = 0.95. As long as the value of $\rho$ is less than one, a stationary (or trend-stationary) process occurs. While a unit root tests involves a failure to reject the hypothesis that $\rho$ equals one, this does not necessarily mean that the true value of $\rho$ equals one.
The work of Nelson and Plosser, hower, convinced most econometricians of the importance of testing for stationary error processes when working with time-series models.This study helped to encourage the current widespread use of the DIckey-Fuller test.
}%
%EndExpansion
\end{enumerate}
If either of the above Dickey-Fuller tests suggests the presence of a unit root in a dependent variable, then it is generally desirable to estimate the regression equation in differenced form.\footnote{%
To be sure that first-differencing has eliminated all unit roots in the series, it is appropriate to apply a Dickey-Fuller test to the differenced variable. If the hypothesis of a random walk cannot be rejected for the differenced variable, a second difference operation is appropriate (this involves computing the change in the first-differenced variable).
\par
In practice, first or second differencing will nearly always be sufficient to convert an economic time-series variable to a stationarity process.
\par
As is discussed in Chapter \ref{ARIMA.chap}, a time-series variable is said to be integrated of order $k$ if $k$th-order differencing is required to induce stationarity.}
\subsection{Example: Consumption expenditures and disposable personal income \label{df.example.urc}}
Various versions of the consumption function were estimated in previous chapters. Under each specification, the level of consumption expenditures was assumed to be a function of the level of personal disposable income.
Let’s examine whether these two time-series variables possess unit roots.
For most aggregate economic time-series variables, it is standard practice to apply a log transformation to the variable before applying a Dickey-Fuller test for a unit root. The argument behind this is that most economic time-series tend to exhibit an exponential growth trend over time.
The log transformation converts this exponential trend into a linear trend.
This transformation has been applied to the consumption and personal disposable income series used in the discussion that follows.
To test for the presence of a unit root in the consumption expenditure series, the following equation was estimated using 226 quarterly observations on the natural log of real consumption expenditures:\footnote{% The data used to estimate this model appears in Table \ref{cons3.dat} in Appendix \ref{data.appendix} (and in the file \textquotedblleft cons3.dat\textquotedblright ). The alternative unit root tests for a simple random walk and a random walk with drift are left to the reader as an exercise. The consumption variable is the natural log of consumption expenditures expressed in billions of 1996 chained dollars.} \begin{equation*}
\hat{C}_{t}=\underset{(0.10404)}{0.18269}-\underset{(0.0001323)}{0.00021493}% t+\underset{(0.015143)}{0.97476}C_{t-1}
\end{equation*}%
\begin{equation*}
\text{(standard errors in parentheses)}
\end{equation*}%
The Dickey-Fuller statistic is computed as:
\begin{equation}
\text{Dickey-Fuller statistic = }\frac{0.97476-1}{0.0151427}=-1.667 \label{df.1.urc}
\end{equation}%
The null and alternative hypotheses for the Dickey-Fuller test are:% \begin{equation*}
\text{H}_{o}\text{: }\rho =1
\end{equation*}%
\begin{equation*}
\text{H}_{1}\text{:\ }\rho \neq 1
\end{equation*}%
Since this is a two-tailed hypothesis test, critical values of the Dickey-Fuller statistic are chosen so that the probability of an outcome in the rejection region under the null hypothesis is equal to the pre-selected significance level, $\alpha .$At a 5\% significance level, Table \ref% {df.tab.1.ac} indicates that the critical values of the Dickey-Fuller statistic are approximately -3.69 and -0.64.\footnote{% Since $N=226$ in this case, the critical values for the Dickey-Fuller test cannot be read directly from Table~\ref{df.tab.2.ac}.This sample size falls between the sample sizes of 100 and 250 listed in the table. In such cases, there are two commonly accepted practices:
\par
\begin{itemize}
\item use linear interpolation to estimate the critical values, or \par
\item use the critical value corresponding to the smaller of the sample sizes (in this case, the selected critical values would be -3.73 and -0.62).
By using the smaller sample size to determine the critical value, the actual significance level of the test is less than or equal to the pre-selected significance level.
\end{itemize}
\par
In this case, the linear interpolation method is used to determine the critical values. sample size is much closer to 250 than to 100, and linear interpolation results in significance levels that round to the values corresponding to a sample size of 250.} The null hypothesis (at a 5\% significance level) if the Dickey-Fuller statistic is less than -3.69 or greater than -0.64. Since the estimated Dickey-Fuller statistic does not fall within the rejection region, the null hypothesis cannot be rejected.
Thus, the null hypothesis of a unit root cannot be rejected at a 5\% significance level. This suggests that the behavior of consumption expenditures cannot be distinguished from a random walk with drift process.
A test for a random walk in the natural log of the disposable personal income series can be based on the following estimated equation:\footnote{% The personal disposable income variable is measured as the natural log of disposable income (in billions of 1997 chained dollars).} \begin{equation*}
\widehat{\text{\textit{YD}}}_{t}=\underset{(0.0.090625)}{0.13216}+\underset{% (0.00011325)}{0.00013786}t+\underset{(0.012981)}{0.98253}\text{\textit{YD}}% _{t-1}
\end{equation*}%
\begin{equation*}
\text{(standard errors in parentheses)}
\end{equation*}%
The Dickey-Fuller statistic is computed as:
\begin{equation}
\text{Dickey-Fuller statistic = }\frac{0.98253-1}{0.012981}=-1.346 \label{df.2.urc}
\end{equation}%
As noted above, the null hypothesis should be rejected at a 5\% significance level only if the Dickey-Fuller statistic is less than -3.69 or greater than -0.64. As in the case of the consumption variable, the hypothesis of a unit root cannot be rejected at a 5\% significance level for the \textit{YD}$_{t}$ variable.
\section{Cointegration}
As noted above, the regression of a variable with a unit root on another variable possessing a unit root will often result in spurious regression results. Since many economic time series variables appear to possess unit roots, this problem is fairly serious. One solution to this problem involves a regression using a first-differenced version of the original regression equation.
While differencing can be used as a way of eliminating unit roots, this involves the loss of some information about the underlying series.
Fortunately, however, it is often possible to estimate the relationship that exists between levels of two time-series variables if the variables are tied together through some form of a long-run equilibrium relationship. For example, the long-run relationship between consumption spending and disposable income may be relatively stable even though the consumption and disposable income series themselves may each possess a unit root.
Suppose that two time-series variables $X_{t}$ and $Y_{t\text{ }}$are each found to possess a unit root. These series are said to be \textbf{% cointegrated} if a linear combination of $X_{t}$ and $Y_{t}$ does not possess a unit root. This occurs when a stable long-run equilibrium relationship exists between these variables. Roughly speaking, two series are said to be cointegrated if they share the same unit root.\footnote{% A time-series variable is said to be integrated of order 1 (I(1)) if a first difference of the variable is stationary. A process is said to be integrated of order $d$ if a $d$th difference operation is required to induces stationarity. In the discussion above, the random walk processes are I(1) variables. The term \textquotedblleft cointegrated\textquotedblright\ is derived from this concept of integrated variables. If $X_{t}$ and $Y_{t}$ are cointegrated, they share the same unit root and the residual in a regression of $Y_{t}$ on $X_{t}$ is stationary.}
When two time-series variables are cointegrated, they tend to move together over long periods of time (although they may diverge in the short run). In many cases, economic theory predicts that long-run equilibrium relationships will exist among particular variables. Examples of time-series that are likely to be cointegrated include:
\begin{itemize}
\item short-run and long-run interest rates,
\item consumption expenditures and disposable personal income, \item nominal wages and the price level,
\item wage rates in different industries (or geographical regions), \item the price of a given commodity in different geographical locations, \item prices of close substitutes in a given market, and \item real income and productivity.
\end{itemize}
\section{Tests for the existence of cointegration}
Suppose that an econometrician wishes to analyze the relationship existing between two time-series variables $X_t$ and $Y_t$. If a Dickey-Fuller test applied to the $X_t$ and $Y_t$ series separately suggests that each series possesses a unit root, then a test for the presence of cointegration is desirable. If two random walk processes are cointegrated, the residual in a regression of one variable on the other should be stationary. A simple test based on this principle was constructed by Engle and Granger (1987). This \textbf{Engle-Granger test} is conducted in the following manner: \begin{enumerate}
\item[Step 1:] Estimate the parameters of the \textbf{cointegrating regression} defined as:
\begin{equation*}
Y_t=\gamma _o+\gamma _1X_t+u_t
\end{equation*}
and save the residuals ($\hat{u}_t$). If $Y_t$ and $X_t$ are cointegrated, the residuals should be a stationary process.
\item[Step 2:] Use an OLS\ estimation procedure to estimate the model given by:
\begin{equation}
\hat{u}_{t}=\rho \hat{u}_{t-1}+\epsilon _{t} \label{egtest.urc} \end{equation}%
When estimating this equation, be sure to suppress the estimation of a constant term if you regression software defaults to including a constant.
Use the estimated value of $\hat{\rho}$ to construct a Dickey-Fuller statistic:
\begin{equation*}
\frac{\hat{\rho}-1}{\hat{\sigma}_{\hat{\rho}}}
\end{equation*}%
As noted by Engle and Granger, this statistic does not follow the same distribution as in the random walk process discussed by Dickey and Fuller.
Through a simulation process, Engle and Granger have derived critical values of this statistic for a sample of 100 observations. Their results indicate that the critical values are equal to -1.6177, -1.9439, and -2.5899 at the 10\%, 5\% and 1\% significance levels. If the value of this statistic is less than the critical value at the preselected significance level, then the Engle-Granger test suggests that a unit root is not present and the variables are cointegrated. If the value of this statistic exceeds the critical value, then the Engle-Granger test indicates that the variables are not cointegrated.
\end{enumerate}
Notice that the Engle-Granger test essentially involves the application of a Dickey-Fuller test to the residuals in the cointegrating regression. If the error terms in the regression appearing in equation \ref{egtest.urc} are serially correlated, then an augmented Engle-Granger test may be applied that is analogous to the augmented Dickey-Fuller test described above.% \footnote{%
A detailed discussion may be found in Engle and Granger (1987). Critical values for the augmented Engle-Granger statistic may be found in Davidson and MacKinnon (1993), Table 20.2, p.~722. Several modern econometrics packages also provide critical values for this statistic.} An alternative and somewhat simpler test for the existence of cointegration is provided by the \textbf{cointegrating regression Durbin-Watson test}. The following procedure is used to conduct this test:
\begin{enumerate}
\item[Step 1:] Use an OLS estimation procedure to estimate the parameters of the cointegrating regression:
\begin{equation*}
Y_t=\gamma _o+\gamma _1X_t+u_t
\end{equation*}
\item[Step 2:] Use the estimated Durbin-Watson statistic from the regression performed in Step 1 to test the hypothesis that $\rho $ equals 1. Since the usual Durbin-Watson test is based on a test of the hypothesis that states that $\rho $ equals 0, the usual Durbin-Watson table cannot be used for this test. If the true value of $\rho $ equals 1, then the estimated Durbin-Watson statistic should be close to 0. Using a simulation procedure, Sargan and Bhargava (1983) have found that the appropriate critical values for this statistic are 0.322, 0.386, and 0.511 at the 10\%, 5\%, and 1\% significant levels when a sample of 100 observations is used. If the estimated Durbin-Watson statistic exceeds the appropriate critical value, then this test suggests that $X_t$ and $Y_t$ are cointegrated.
\end{enumerate}
\subsection{Example: consumption and disposable personal income} Figure~\ref{fig17_1} contains a time-series plot of the natural logs of consumption expenditures and disposable personal income for the period from 1947 to 2003. An inspection of this graph suggests that these two time-series variables tend to move together over time. This suggests the possible presence of a cointegrating relationship between these two variables.
\begin{center}
\FRAME{ftbpFU}{4.8386in}{3.7602in}{0pt}{\Qcb{Time path of consumption and disposable personal income variables (in natural log form)}}{\Qlb{fig17_1}}{% fig17_1.png}{\special{language “Scientific Word”;type “GRAPHIC”;maintain-aspect-ratio TRUE;display “USEDEF”;valid_file “F”;width 4.8386in;height 3.7602in;depth 0pt;original-width 8.8885in;original-height 6.6668in;cropleft “0”;croptop “1.0343”;cropright “1”;cropbottom “0”;filename ‘GRAPHS/fig17_1.png’;file-properties “XNPEU”;}}
\end{center}
To test for the presence of a cointegrating relationship between consumption and personal disposable income, the following cointegrating equation is estimated:\footnote{%
Both variables are expressed in natural logs.}
\begin{equation}
\hat{C}_{t}=\underset{(0.0225)}{-0.11555}+\underset{(0.0028129)}{1.0011}% \text{\textit{YD}}_{t} \label{coint.reg.urc}
\end{equation}%
\begin{equation*}
\text{R}^{2}=0.998\hspace{1in}\text{DW = 0.2006}
\end{equation*}%
\begin{equation*}
\text{(standard errors in parentheses)}
\end{equation*}%
To compute the Engle-Granger statistic, the following equation was estimated:
\begin{equation*}
\hat{u}_{t}=\underset{(0.02936)}{0.9019}\hat{u}_{t-1} \end{equation*}%
The Engle-Granger statistic for this regression is given by: \begin{equation*}
\text{Engle-Granger statistic = }\frac{0.9019-1}{0.02936}=-3.34 \end{equation*}%
Since the absolute value of this statistic exceeds the absolute value of the critical value for an Engle-Granger statistic at a 1\% significance level, the hypothesis of a unit root in the error process can be rejected. This suggests that the two variables are cointegrated.
The cointegrating regression Durbin-Watson test, however, does not reject the hypothesis that $\rho =1$ even at a 0.1 significance level. To see this, note that the value of the Durbin-Watson statistic (= 0.2006) in equation % \ref{coint.reg.urc} is less than the critical values at the 0.1, 0.05 and 0.01 significance levels.
\section{Error correction model}
If two time-series variables are cointegrated, econometricians may specify an \textbf{error correction model} to capture the relationship that exists between the two series.\footnote{%
The Granger Representation Theorem (discussed in Engle and Granger (1987)) states that the relationship between any two cointegrated time-series variables may be represented by an error correction model.} Under an error correction model, it is assumed that:
\begin{itemize}
\item a long-run stable equilibrium relationship exists between two variables,\footnote{%
Hamilton (1994) and Maddala and Kim (1998) provide a good (although mathematically sophisticated) discussion of models involving more than two variables.}
\item a disequilibrium situation may exist in the short run, and \item a fixed proportion of the disequilibrium in period $t-1$ is “corrected” in period $t$.
\end{itemize}
A common implementation of the error correction model involves the specification:\footnote{%
A more complete, though mathematically more advanced, discussion of error correction models may be found in Hamilton (1994) or Maddaka and Kim (1998).} \begin{equation}
\Delta Y_{t}=\beta _{o}+\beta _{1}\Delta X_{t}+\beta _{2}\left( Y_{t-1}-\gamma _{1}X_{t-1}\right) +u_{t} \label{ecm.urc} \end{equation}%
In this equation, the coefficient $\beta _{1}$ captures the long-run relationship that exists between $X_{t}$ and $Y_{t}$. If $X_{t}$ and $Y_{t}$ are measured in log form, the coefficient $\beta _{1}$ measures the long-run elasticity of $Y_{t}$ with respect to $X_{t}$. The $\left( Y_{t-1}-\gamma _{1}X_{t-1}\right) $ term captures the short-run adjustment to a disequilibrium situation. Thus, a nonzero value of $\beta _{2}$ provides evidence of an error correction process. A desirable feature of error correction models is that they can account for the existence of long-run equilibrium relationships while also allowing for the existence of a short-run disequilibrium adjustment process.
Consistent estimates of the parameters of equation \ref{ecm.urc} cannot be generated through an OLS procedure applied directly to this equation. Engle and Granger (1987) proposed the following two-stage estimation procedure for this model:
\begin{enumerate}
\item[Step 1:] Use OLS to estimate the parameters of the cointegrating regression:
\begin{equation*}
Y_t=\gamma _o+\gamma _1X_t+u_t
\end{equation*}
and store the fitted residuals, $\hat{u}_t$.
\item[Step 2:] Use OLS to estimate the parameters of the following equation: \begin{equation*}
\Delta Y_t=\beta _o+\beta _1\Delta X_t+\beta _2\hat{u}_{t-1}+v_t \end{equation*}
This involves estimating the error correction model given in equation \ref% {ecm.urc} after replacing the ($Y_t-\gamma _1X_{t-1}$) term with the fitted residuals $\hat{u}_{t-1}$. This procedure provides consistent estimates of $% \beta _1$ and $\beta _2$.
\end{enumerate}
\subsection{Example: Consumption function error correction model} Since the tests described above provides some evidence of cointegration between the consumption and disposable personal income series, an error correction model can be specified that relates these variables. The use of the Engle-Granger two-stage procedure described above results in the following estimated version of this equation:\footnote{% The first-stage estimates used to generate the values of $\hat{u}_{t}$ are the same as those reported above in equation \ref{coint.reg.urc}.} \begin{equation}
\widehat{\Delta C}_{t}=\underset{(8.81)}{0.00595}+\underset{(6.32)}{0.317}% \Delta \text{\textit{YD}}_{t}+\underset{(3.29)}{0.0715}\hat{u}_{t-1} \label{c.erc.eq.urc}
\end{equation}%
\begin{equation*}
(t\text{-statistics in parentheses})
\end{equation*}%
\begin{equation*}
\text{R}^{2}=0.173\hspace{1in}\text{DW statistic = 2.23} \end{equation*}%
Notice that the estimated long-run elasticity of consumption with respect to personal disposable income is substantially less under the error correction model than it is when a consumption function is estimated in levels form.
While a regression in levels form provides substantially higher $t$% -statistics and R$^{2}$, these results are at least partly due to the spurious regression results that occur when one random walk variable is regressed upon another random walk variable. The significant $t$-ratios in equation \ref{c.erc.eq.urc} indicate, however, that the relationship between consumption expenditures and personal disposable income is not entirely a spurious phenomenon. The positive and statistically significant coefficient of 0.317 on $\Delta $\textit{YD}$_{t}$ provides a measure of the long-run equilibrium relationship that exists between $C_{t}$ and $Yd_{t}$. It indicates that an increase in disposable personal income results in an increase in the equilibrium level of consumption spending.
\section{Summary}
In this chapter, the effect of nonstationarity in regression models has been examined. If the dependent and independent variables in a regression equation possess a unit root, a spurious regression relationship may appear in a regression involving the levels of the series. Tests proposed by Dickey and Fuller, however, make it possible to test for the presence of a unit root in these variables. If unit roots are found, a first-differenced version of the model may eliminate the problem of spurious regression results.
In some cases, however, both the dependent and independent variables may possess a unit root, but tend to move together over long periods of time. In this case, a cointegration relationship may exist between these variables.
If a test suggests the presence of cointegration, an error correction model may be specified that captures both the long-run equilibrium relationship between the series and the short-run adjustment process.
\section{Key Concepts}
stationarity
trend stationary process
random walk
random walk with drift
unit root
Dickey-Fuller test
augmented Dickey-Fuller test
cointegration
Engle-Granger test
cointegrating regression
augmented Engle-Granger test
cointegrating regression Durbin-Watson test
error correction model
\newpage\
\section{Exercises and problems}
\begin{enumerate}
\item Show that the trend-stationary process represented in equation \ref% {trend.stat.urc} is nonstationary. (Hint: Determine $E(Y_t$).) \item
\begin{enumerate}
\item Explain the difference between a trend-stationary process and a random walk with drift.
\item In which of these models is a first-difference operation appropriate?
\item Why is it often difficult to determine which of these two models provides the most appropriate description of a nonstationary economic time-series variable?
\end{enumerate}
\item Use the \textquotedblleft cons3.dat\textquotedblright\ data file described in Table \ref{cons3.dat} in Appendix \ref{data.appendix} to verify the Dickey-Fuller statistics appearing in equations \ref{df.1.urc} and \ref% {df.2.urc}.
\item An alternative method of conducting the Dickey-Fuller test involves estimating one of the following models:
\begin{equation*}
\Delta Y_t=\delta Y_{t-1}+\epsilon _t
\end{equation*}
\begin{equation*}
\Delta Y_t=\beta _o+\delta Y_{t-1}+\epsilon _t
\end{equation*}
and
\begin{equation*}
\Delta Y_t=\beta _o+\beta _1t+\delta Y_{t-1}+\epsilon _t \end{equation*}
\begin{enumerate}
\item Show that the estimation of these three equations is equivalent to the estimation of equations \ref{df.ac}, \ref{rwd.urc}, and \ref{ltrw.urc}, respectively. (Hint: subtract $Y_{t-1}$ from each side of equations \ref% {df.ac}, \ref{rwd.urc}, and \ref{ltrw.urc} and simplify.) \item What is the relationship between the value of $\delta $ and the value of $\rho $ under these alternative models?
\item Is a test of $\rho $ = 1 equivalent to a test of $\delta =0$?
\item How would the Dickey-Fuller statistic be computed under these alternative models?
\end{enumerate}
\item Use the \textquotedblleft cons3.dat\textquotedblright\ data described in Table \ref{cons3.dat} in Appendix \ref{data.appendix} to test for the presence of a unit root in the consumption and disposable personal income using the specifications:
\begin{equation*}
Y_{t}=\rho Y_{t-1}+\epsilon _{t}
\end{equation*}%
\begin{equation*}
Y_{t}=\beta _{o}+\rho Y_{t-1}+\epsilon _{t}
\end{equation*}%
(Apply a log transformation to each variable before applying this test.) Does the outcome of these tests differ from the tests appearing in Section % \ref{df.example.urc}?
\item Apply an augmented Dickey-Fuller test to the consumption and personal disposable income series appearing in \textquotedblleft cons3.dat\textquotedblright\ (this data is described in Table \ref{cons3.dat} in Appendix \ref{data.appendix}). Use a log transformation on each series before applying this test.
\item
\begin{enumerate}
\item At time $t$, what is the variance of the error term in the random walk with drift model appearing in equation in equation \ref{dsp.proc.urc}?
\item Express this variance as a function of $t$. What happens to this variance as $t$ increases?
\end{enumerate}
\item What is the interpretation of the coefficient $\beta _o$ in the error correction model appearing in equation \ref{ecm.urc}.
\item Use the \textquotedblleft money.dat\textquotedblright\ data file (described in Table \ref{money.dat} in Appendix \ref{data.appendix}) to: \begin{enumerate}
\item transform the GDP and M2 series into log form.
\item test for the presence of a unit root in the money supply and GDP time series variables using a Dickey-Fuller test at the 5\% significance level.
\item test for the existence of costationarity between the two series using an Engle-Granger test at a 5\% significance level.
\item test for the existence of costationarity using a cointegrating regression Durbin-Watson test at a 5\% significance level.
\item estimate the error correction model given by:
\begin{equation*}
\Delta \text{Y}_t=\beta _o+\beta _1\Delta \text{M2}_t+\beta _2(\text{Y}% _{t-1}-\gamma _1\text{M2}_{t-1})+v_t
\end{equation*}
Interpret these results.

\end{enumerate}
\item Why might economic theory support the use of an error-correction model in the following cases? In each situation explain why economic theory would suggest that the variables move together over long periods of time, but might behave differently in the short term.
\begin{enumerate}
\item the relationship between wage rates in different geographical locations (for workers in a given occupation).
\item the relationship between short-term and long-term interest rates.
\end{enumerate}
\item Use the data in the file \textquotedblleft int.dat\textquotedblright\ (this data is described in Table \ref{int.dat} in Appendix \ref% {data.appendix}) to:
\begin{enumerate}
\item test for the presence of a unit root in the yield on 3-month Treasury bills and in the yield on 30-year Treasury bonds using a Dickey-Fuller test at the 5\% significance level.
\item test for the existence of costationarity between the two series using an Engle-Granger test at a 5\% significance level.
\item test for the existence of costationarity using a cointegrating regression Durbin-Watson test at a 5\% significance level.
\item estimate the error correction model given by: \begin{equation*}
\Delta \text{INT3M}_{t}=\beta _{o}+\beta _{1}\Delta \text{INT30Y}_{t}+\beta _{2}(\text{INT3M}_{t-1}-\gamma _{1}\text{INT30Y}_{t-1})+v_{t} \end{equation*}
\item Interpret these results.
\end{enumerate}
\end{enumerate}
\newpage