Econometrics II

Given a well defined ARMA process (e.g. the solution of the relevant recursion when the UDC holds for the Φ polynomial), we were occupied with the issue of what property is implied when the Θ polynomial satisfies the UDC, thus obtaining the notion of invertibility, which is equivalent to that the white noise process is specified as a linear causal process with respect to the ARMA process with absolutely summable coefficients that are absolutely bounded from above by the coefficients of a geometric series times a positive constant, that are obtained by the product of the inverse power series of Θ with Φ. This also implies that the white noise process is adapted to the filtration constructed from the history of the ARMA process at each time instance. When the first coefficient of the latter representation of the white noise process is not equal to zero (is this always the case or not?) then we also obtain a representation of the original ARMA process as an "AR infinity process". The invertibility concept can be important to the issue of statistical inference in ARMA models.

We were occupied further with issues concerning statistical inference in ARMA models in the framework of correct statistical specification and of known unit variance for the white noise process. We pointed out that in the context of general AR models the extraction of the asymptotic properties of the OLSE can be similar to the one we have taken in the case of the AR(1) model modulo technical details of essentially multivariate nature that are not present in the latter case.

We begun the examination of issues of statistical inference in ARMA models when the MA component is not trivial, i.e. q>0. We showed first that in this context the OLSE is computationally infeasible. We discussed an example of an inconsistent OLSE in such a context, when estimation is reduced to AR parameters (or the relevant linear model is accordingly misspecified). This showed us that the unknown parameters corresponding to MA components cannot be generaly ignored, since in such cases the feasible OLSE of the AR parameters can have poor asymptotic properties. Hence in such cases we need some feasible "generalization" of the OLSE as a semi-parametric estimation procedure,  and this remark paves the way for the consideration of the Gaussian Quasi Maximum Likelihood Estimator (QMLE). 

You can find notes on the above here and here.