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:

Convert this into a form expressing in terms of X instead of ΔX:

We change variable to simplify the form:

Via Pesaran and Shin (1996) we transform this into the following recursive expression:

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.

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