This is just a quick note on deriving an impulse response function for a VECM system. Basically we want to get the system into a form where we can take the partial derivatives at various lags. Starting with a simplified VECM:
We determine the partial derivative of ∂vj / ∂vk (i.e. the impact of a change in the kth variable on the ith) after n time periods (t+n) to be:
where Si is a selection vector with 1 at the ith position and 0 elsewhere.
Normally the cholesky decomposition is used to orthogonalize the covariance (U U’ = Σ), however other decompositions can be used, providing different measures of response such as the Bernanke-Sims approach.