Standard statistical methods applied to general time series might not work.

Cointegration extends regression methods to cover more cases.

I'll talk about implementation of solution methods.

I(0) time series are formally expressed

A series x_t is called I(1) if

The I(1) vector x_t is cointegrated if there exists a vector β such that

The I(1) series x_t has cointegration rank 1 if

Require each series to be I(1).

^{α, β full column rank}

Find α, β of rank r (not unique!).

Johansson developed a maximum likelihood estimate of α and β.

Not that Johansson!

Actually, it was Søren Johansen, the Danish economist.

We need that

Johansen shows that

We desire the eigenvalues and a basis for the span of the columns of β.

Write

Then there exist U, V w/orthonormal columns and nonsingular Z s.t.:

Generalized singular values

Doornik and O'Brien (2001) use the SVD. Their example



shows a complete loss of accuracy for δ < 10^-6 when solved directly but works fine with the SVD approaches.