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.