Cointegration

Many time series appear to be $I(1)$ and have useful long-run equilibrium relationships. We can difference to avoid the problem of spurious regression, but this loses the useful relationships!

Sometimes we can carry out meaningful regressions involving $I(1)$ variables

Definition: Two $I(1)$ time series $\{Y_t\}$ and $\{X_t\}$ are cointegrated if there exists a linear combination $Y_t-\lambda X_t=(1,-\lambda )(Y_t,X_t{)}^{\prime }$ such that $\{Y_t-\lambda X_t\}\sim I(0)$

$(1,-\lambda {)}^{\prime }$ is called the cointegrating vector and the two variables share a common stochastic trend

Example:
Let $Y_t=a{R}_t+Z_t$ and $X_t=b{R}_t+W_t$ where $a,b\ne 0$, $\{Z_t\}$ and $\{W_t\}$ are weakly stationary and mutually independent, and $\{R_t\}$ is a random walk, $R_t=R_{t-1}+u_t$, $\{u_t\}\sim \text{WN}(0,{\sigma }_u^2)$

The cointegrating vector is $(1,-a/b{)}^{\prime }$,
$(1,-a/b)(Y_t,X_t{)}^{\prime }=Y_t-a{X}_t/b=a{R}_t+Z_t-a{R}_t-a{W}_t/b$
$=Z_t-a{W}_t/b$

Definition: A vector ${\mathbf{X}}_t=(X_{1,t},\dots ,X_{m,t})$ is cointegrated of order $(r,k)$, denoted ${\mathbf{X}}_{\mathbf{t}}\sim \text{CI}(r,k)$, if each element of ${\mathbf{X}}_{\mathbf{t}}$ is $I(r)$ and there exists a linear combination ${\mathbf{c}}^{\prime }{\mathbf{X}}_t\sim I(r-k)$ for $k\ge 1$

How do we test if two variables are cointegrated?

We can reformulate cointegration to make this easier
Let $Y_t=\lambda X_t+Z_t$ where $\{X_t\}$ and $\{Z_t\}$ are mutually independent
If $\{X_t\}\sim I(1)$ and $\{Z_t\}\sim I(0)$ then $(Y_t,X_t)\sim \text{CI}(1,1)$

Now testing for cointegration is equivalent to testing $\{Z_t\}$ is $I(0)$

First we perform a regression and obtain parameter estimates $\hat{\delta }$ and $\hat{\lambda }$ for $Y_t=\delta +\lambda X_t+Z_t$

The residuals are ${\hat{Z}}_t=Y_t-\hat{\delta }-\hat{\lambda }{X}_t\approx Z_t$
So we test if $\{{\hat{Z}}_t\}\sim I(0)$

Definition: The Dickey-Fuller test for no-cointegration between $\{Y_t\}$ and $\{X_t\}$ makes use of the $t$-ratio test statistic $\text{DFNC}=(\hat{\varphi }-1)/\text{SE}(\hat{\varphi })$ where $\hat{\varphi }$ is the estimated parameter from the $\text{AR}(1)$ model ${\hat{Z}}_t=\varphi {\hat{Z}}_{t-1}+{ϵ}_t$ and $\{{\hat{Z}}_t\}$ are residuals from the static regression $Y_t=\delta +\lambda X_t+Z_t$

The null hypothesis is $H_0:\varphi =1$ (no-cointegration) against $H_1:|\varphi |<1$ (cointegration)

Critical values are taken from a special table, which accounts for kinds of trends, sample size, and number of time series being tested

• Instead of single numbers, they fit a more complex function to accommodate different values of $T$, ${\text{DFNC}}_{\text{crit}}={\beta }_{\infty }+{\beta }_1/T+{\beta }_2/T^2+{\beta }_3/T^3$

In practice,

1. Start with performing ADF unit root tests to determine the integration order of $\{Y_t\}$ and $\{X_t\}$ (if either is stationary then we should stop testing)
2. Perform regression and obtain ${\hat{Z}}_t$ residuals
3. Perform unit root test on residuals and check against the special critical values

Error Correct Models

The ECM helps us model relationships between cointegrated non-stationary time series

$\Delta Y_t=\gamma [Y_{t-1}-\delta -\lambda X_{t-1}]+{ϵ}_t$
If we fix for long-term equilibriums, we get $\bar{Y}=\delta +\lambda \bar{X}$

$Y_{t-1}-\delta -\lambda X_{t-1}$ is called the error correction term, since it corrects $Y_t$ towards the long-run equilibrium (if $\gamma <0$)

Granger’s Representation Theorem: If $(Y_t,X_t)\sim \text{CI}(1,1)$ then $\{Y_t\}$ admits the error-correction representation $\Delta Y_t=\gamma [Y_{t-1}-\delta -\lambda X_{t-1}]+{ϵ}_t$

In fact, the theorem is a bit more general and allows for for lags of the differences of our time series,
$\Delta Y_t=\gamma [Y_{t-1}-\delta -\lambda X_{t-1}]+{\sum }_{i=1}^p{\varphi }_i\Delta Y_{t-i}+{\sum }_{j=1}^q{\beta }_j\Delta X_{t-j}+{ϵ}_t$ (in practice, ${ϵ}_t$ can be zero-mean, weakly stationary, which is more general than white-noise)

Written in another way,
$\Delta Y_t=\gamma Z_{t-1}+{\sum }_{i=1}^p{\varphi }_i\Delta Y_{t-i}+{\sum }_{j=1}^q{\beta }_j\Delta X_{t-j}+{ϵ}_t$
where $Z_{t-1}=Y_{t-1}-\delta -\lambda X_{t-1}$

So our steps to estimate these parameters are,

1. Regress $X_t$ on $Y_t$ to obtain ${\hat{Z}}_t$
2. Perform a second regression for the final model parameters

This works because our estimator $(\hat{\delta },\hat{\lambda }{)}^{\prime }$ is super-consistent and is therefore unusually accurate and suitable for a second regression

Engle and Granger showed that ${\hat{\varphi }}_i$ and ${\hat{\beta }}_j$ have the same asymptotic distribution as if $\{Z_t\}$ is known