Econometrics II

We have proven that a causal linear process w.r.t. a white noise and a sequence of absolutely summable coefficients is always regular. Under a particular restriction on the coefficients sequence we have also proven it to have the property of short memory. We have also been occupied with the issue of strict stationarity of a such a process. We have also provided with two simple short memory examples, namely the AR(1) and the MA(1) processes. We have left for the moment unanswered the question of why the AR(1) process can be also characterized as a solution of a particular stochastic recurence equation. Notes for the above can be found here. 

Via a linear regression example involving linear processes and the relevant OLSE, we have, in the framework of stationarity, descriptively been occupied with the notion of the invariant σ-algebra of the process, provided with the stationary version of Birkhoff's LLN and realized that stationarity alone may not be sufficient for the consistency of the OLSE. Hence we have set the stage for the examination of the notion of ergodicity and the stationary and ergodic version of Birkhoff's LLN. Notes for the above can be found here.

Exercises

1. Is the "geometric series upper bound" for series of the absolute values of the coefficients sequence necessary in order to obtain the short memory property for the relevant causal linear process?
2. If not, does absolute summability suffice?