Econometrics II

We have occupied ourselves with the fact that strict stationarity is preserved by transformations such as pointwise sums, products, scalar multiplication, transformations by polynomials w.r.t. the lag operator. We pointed that scalar multiplication, or transformations by polynomials also preserve weak stationarity.

Given Doob's LLN we defined ergodicity as the nessecary and sufficient property for which the relevant σ-algebra appearing in the limiting conditional expectation, and contains events associated with the action of the lag operator, becomes "trivial" in the sense that it contains only events of probability 0 or 1, and thereby the conditional expectation coincides with the unconditional one. We pointed out that ergodicity is equivalent to a form of asymptotic independence. In this respect by the addition of ergodicity , Birkhoff's LLN is obtained as a corrolary of Doob's LLN.

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.