Cointegration

Summary

Cointegration, definition

We say that two processes $y_1,…,y_T$ and $x_1,….,x_T$ are cointegrated if

• Both processes are $I(1)$ , that is, they are nonstationary while $Δy_t$ and $Δx_t$ are stationary processes.
• There exists a parameter $β$ such that $y_t-βx_t$ is stationary. $β$ is called the cointegration parameter .

Estimating the cointegration parameter

If $y_1,…,y_T$ and $x_1,….,x_T$ are cointegrated then $β$ can be consistently estimated from an OLS regression of $y_t$ on $x_t$ .

Testing for a cointegrating relationship

• Argue that $y_t$ and $x_t$ are $I(1)$ – for example using a unit root test.
• Run a regression of $y_t$ on $x_t$ (with an intercept) and save the residuals
• Test if the residuals are stationary. The critical values from the standard unit root test are not valid. You need appropriate critical values for a residual-based unit root test. These are available in most econometrics software.