Econometrics II

Even in the context of an AR(1) process with iid white noise, we have seen that the Lindeberg-Levy CLT is not applicable for the derivation of the rate and limiting distribution of the OLSE for the relevant coefficient (why?). Hence in order to tackle such a problem we require the establishment of a CLT applicable in such-like cases. Stationarity and ergodicity are not sufficient for such an establishment without further control of the rate of asymptotic independence of the elements of the underlying process. In order to avoid the description of conditions involving complex mixing conditions, we moved on to the issue of establishing a CLT for stationary and dependent processes via the quite restrictive, yet easily describable, notion of a martingale difference. Hence, we have examined the notions of the filtration, of the adaptation to a filtration, of a martingale difference process w.r.t. a filtration, and of a square integrable martingale difference process w.r.t. a filtration. We have constructed examples that among others involved the construction and the examination of a strictly stationary, ergodic and square integrable martingale different process that in some cases it is also appropriately conditionally heteroskedastic.

We have examined the subsequent Martingale CLT, which is a generalization of the aforementioned Lindeberg-Levy CLT and involves stationary, ergodic s.m.d. (w.r.t. some filtration) processes.

We have used this in order to study the limit theory of the OLSE in the aforementioned framework by assuming that the innovation sequence is a stationary ergodic s.m.d. and several series are convergent. Those assumptions imply standard rates and asymptotic normality with the asymptotic variance crucially depending on the values of the aforementioned series which among others reflect dependence properties of the innovation process. In the iid case we have seen that the asymptotic variance assumes a simple (and somewhat interesting!) form for which it is easy to obtain a strongly consistent estimator, via the OLSE consistency and the CMT.  

 You can find notes for the above here.