ON THE INVERTIBILITY OF MULTIVARIATE LINEAR PROCESSES
Saı̈d Nsiri, Roch Roy · Journal of Time Series Analysis · 1993
Abstract.It is shown that a multivariate linear stationary process whose coefficients are absolutely summable is invertible if and only if its spectral density is regular everywhere. This general characterization of invertibility is applied later to the case of a linear process having an autoregressive moving‐average (ARMA) representation. Under the usual assumptions, it is deduced that a processYdescribed by an ARMA(φ, TH) model is invertible if and only if the polynomial detTH(z) has no roots on the unit circle. Given an invertible processYwhich has an ARMA representation, it is finally shown that the processYT, whereYT, =εi=0lSiYt‐i, is invertible if and only if the matrixS(z) =εi=0lSiziis of full rank for allzof modulus 1. It follows, in particular, that any subprocess of an invertible ARMA process is also invertible.