Re: CointegrationMLE

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s_11 is singular when at least one element of the error term epsilon_{1, t} in auxiliary regression 1 (i.e. delta(y_t) on the lagged differences)  is zero. This happens if at least one component of delta(y_t) is nearly perfectly related to some components of delta(y_{t-i}) (1 <= i <= p-1), resulting in a perfect fit for some components in auxiliary regression 1. This indicates a linear causal relationship amongst components of {y_t}, so it may not be appropriate to use cointegration. Replacing delta(y_t) by y_t in the argument above, this also applies to s_22.

We can check the linear dependence between the components of delta(y_t) (and/or y_t) and the lagged differences first before Johansen’s test is applied. However, this is not absolutely necessary, since an “almost” singular matrix (s_11 or s_22) does indicate a clear causal relationship or linear dependency.