Spectral estimation of the multivariate impulse response

One of the applications of cross-spectral estimation of stationary time series, developed some five decades ago, is the estimation of the lagged response of an output series causally dependent on an input series, in the absence of feedback from the output to the input. The direct application of cross-spectral analysis for this purpose is no longer appropriate in the presence of feedback or more general inter-dependence of the series. In that case, vector autoregressive modeling has been used, particularly in the econometric context, to estimate the response of one series to a shock or impulse in the innovations of another series. To achieve the same end, cross-spectral analysis requires the application of spectral factorization, and in this article, we demonstrate this methodology, explaining how it may be used to construct impulse response function estimates and their statistical properties. Our presentation includes an information criterion for choosing the smoothing bandwidth to be used for cross-spectral estimation