Testing for cointegration

There are two major approaches to testing for cointegration:

  • The Engle–Granger two-step method
  • The Johansen procedure

The Engle–Granger method involves regressing one series on another, and then applying an ADF unit-root test to the regression residual. If the null hypothesis can be rejected so that we assume the residuals are stationary, then the series are co-integrated. A key benefit of this approach is that the regression coefficient represents the multiplier that renders the combination stationary, that is, mean-reverting. We will return to this aspect when leveraging cointegration for a pairs-trading strategy. On the other hand, this approach is limited to identifying cointegration for pairs of series as opposed to larger groups of series.

The Johansen procedure, in contrast, tests the restrictions imposed by cointegration on a vector autoregression (VAR) model as discussed in the previous section. More specifically, after subtracting the target vector from both sides of the generic VAR(p) preceding equation, we obtain the error correction model (ECM) formulation:

The resulting modified VAR(p) equation has only one vector term in levels, that is, not expressed as difference using the operator, Δ. The nature of cointegration depends on the properties of the coefficient matrix, Π, of this term, in particular on its rank. While this equation appears structurally similar to the ADF test setup, there are now several potential constellations of common trends and orders of integration because there are multiple series involved. For details, see the references listed on GitHub, including with respect to practical challenges regarding the scaling of individual series.

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