Econometrics II

We were further occupied with the example of the ARMA(1,1) process by deriving the solution and the relevant properties which were shown to comply with the results of our general theorem.

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 begun the examination of issues of statistical inference in ARMA models with the overall remark that when the MA component is not trivial, i.e. q>0, then 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). You can find notes on the above here and here.