Asymptotic distribution of least squares estimators for purely unstable ARMAX(p,s,q) process

We establish the limiting distribution of least squares estimators (phi) over cap (n), of the vector f of the autoregressive parameters in a purely unstable ARMAX( r,s,q). This extends the results obtained by Boutahar (1991) for the unstable ARX model. More precisely, it is shown that the asymptotic distribution (phi) over cap (n) - phi which are stochastic integrals with respect to Brownian motion differs from the purely unstable ARX by a decentring term due to the correlation between the (yt)'s and their MA part.