Econometrics II

We have provided the definition of a stochastic process as a solution of a (particular form of a) stochastic recurrence equation (or a stochastic difference equation), and loosely described conditions, some involving the properties of the Lipschitz coefficient of the recursion, under which a unique strictly stationary and ergodic solution process is obtained. Using this, we have seen that if the white noise process is strictly stationary and ergodic and the relevant coefficient satisfies the already examined condition, then the AR(1) linear process is the unique strictly stationary and ergodic solution of the relevant AR(1) recursion. Given this, we have become the examination of the asymptotic properties of the OLSE in the context of the aforementioned process, by first noting that unbiasness is not generally the case due to the failure of the strong exogeneity condition. You can find notes for the above here. 

Using the Birkhoff's LLN and the CMT we have derived the strong consistency of the OLSE in the context of a stationary and ergodic AR(1) process. Moving on to the issue of establishing a CLT for stationary and dependent processes, we have examined the notions of the filtration, 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 begun 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. You can find notes for the above here.