Econometrics II

We were occupied with properties of formal power series w.r.t. the lag operator. These along with the preparations in the previous lectures allowed us to easily describe conditions for the existence and properties of ARMA models as linear processes with absolutely summable coefficients, their weak stationarity, regularity and short memory, and further conditions for their strict stationarity and ergodicity. 

We used the aforeexamined CLT in order to derive conditions that ensure that the OLSE in the aforementioned context of the stationary and ergodic AR(1) model has the usual rate and it is asymptotically normal. We verified the conditions and evaluated the asymptotic variance in the context of the aforementioned example, which showed that the presence of conditional heteroskedasticity may affect the asymptotic properties of the estimator, and thereby pmay also affect properties of inferential proc

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

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

We have been further occupied with strict stationarity, defined weak (or second order, or covariance) stationarity, examined examples and counter-examples, and defined the autocovariance (and autocorrelation) function(s) of a weakly stationary process, and the subsequent notions of regularity and absolute summability of those functions. We considered the basic example of a causal linear process defined on a strictly stationary white noise, and examined the stationarity (strict and weak) properti

An imprecise definition of a stochastic process was given: An -valued stochastic process is a collection of random variables parameterized by an index (or parameter) set , that satisfies some consistency conditions (see (Daniell)-Kolmogorov Extension Theorem), which imply that it is equivalently a random element (an appropriatele measurable function) with values a the set of functions , and thereby it defines a probability measure on the latter set. Any such function is termed as a sample path o