Econometrics II

Given the previous we have (descriptively) been occupied with the notion of ergodicity and the subsequent corollary of Birkhoff's LLN in the framework of stationarity and ergodicity. Given the introduction of the ergodic property in ithe innovations process of our example resulted to the string consistency of the OLSE via the use of the aforementioned LLN and the CLT (remember that unbiasness is not generally the case due to the failure of the strong exogeneity condition).

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 sequence 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, the issue of strong consistency of the OLSE in the context of the aforementioned process was a simple corollary of our more general example.

We have seen that ergodicity amounts to asymptotic independence on average. This is not enough for the formulation of central limit theorems, hence further notions have to be introduced.

 You can find notes for the above here.