The information matrix of multiple-input single-output time series models (Q1339357)

The authors consider the multiple-input single-output model defined by the equation \[ {\frac{A(q^{-1})}{F(q^{-1})}} y(t)= {\frac{B_ 1(q^{-1})}{E_ 1(q^{-1})}} u_ 1(t)+\cdots+{\frac{B_ m(q^{- 1})}{E_ m(q^{-1})}} u_ m(t)+ {\frac{C(q^{-1})}{D(q^{- 1})}}\varepsilon(t), \] where \(y(t)\) is the output variable, \(u_ 1(t),\dots, u_ m(t)\) are the input variables, \(\{\varepsilon(t)\}\) is a white noise process with zero mean and standard deviation \(\sigma_{\varepsilon}\), \(A(q^{-1})\), \(B_ 1(q^{-1}),\dots\), \(B_ m(q^{-1})\), \(C(q^{-1})\), \(F(q^{-1})\), \(E_ 1(q^{-1}),\dots, E_ m(q^{-1})\), \(D(q^{-1})\) are polynomials in the unit delay operator \(q^{-1}\), e.g. \(A(q^{-1})= a_ 0+ a_ 1 q^{-1}+ \cdots+ a_ k q^{-k}\). The input variables are driven by the ARMA processes. The model under consideration is a generalization of the multiple regression model with autocorrelated errors, the transfer model and the autoregressive moving average exogeneous (ARMAX) model. The main result of the paper are explicit expressions for the information matrix of the parameters of this model for correlated and uncorrelated inputs, involving lags between inputs.