An identification procedure for a transfer function model

The transfer function models generalize the multiple regression models. Namely, each independent variable (or input) can influence the dependent variable (or output) with a lag structure and the noise is an ARMA process. A transfer function model is characterized by a rational function of the lag operator for each input variable and the autoregressive and moving average polynomials of the noise model. The identification of such a model consists in determining the degrees of each polynomial. This paper proposes an identification procedure in the case of a single input or multiple uncorrelated inputs. It is based on the corner theorem of Hanssens and Liu which is a generalization of the identification theorem of Beguin, Gouriéroux and Monfort for an ARMA process. A complete proof is given. This theorem is using the cross-correlation structure between the output variable and one input variable. In practice, an observed cross-correlation structure is only obtained from time series.