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 alr

We examined an example of a Gaussian process, questioning whether or not it is strictly stationary and/or weakly stationary. You can find notes here.

We occupied ourselves with an example of a strictly stationary process which is not ergodic. This example is referred to as Example 4 here.

We further occupied ourselves with an example of a causal linear process w.r.t. a white noise and a sequence of square summable coefficients which is long-memory. You can find notes on the latter here.

We have proven that a causal linear process w.r.t. a white noise and a sequence of absolutely summable coefficients is always regular (is it also short memory?). We have been occupied with the issue of strict stationarity of a such a process. We have also provided with two simple short memory examples. Notes for the above can be found here. 

In the framework of stationarity we have (descriptively) been occupied with the notion of the invariant σ-algebra of the process, and examined the notion of

We defined a weakly stationary process as short memory iff its autocovariance function is absolutely summable, a condition that implies asymptotic uncorrelateness with sufficiently fast rate, and thereby implies regularity.

We examined the definition of a white noise process as a canonical example of weak stationarity. By an example we showed that such processes exist.

We occupied ourselves with comparisons, via examples in the framework of independence, of the notions of stationarity and weak s

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

We have been occupied with some introductory notions of unit root econometrics. Given some preparatory work, we have derived the limit theory of the OLSE for a unit root process with stationary, ergodic and s.m.d. innovations, and specified a "Dickey-Fuller type" of test for the relevant hypotheses structure. You can find notes on the above here.

We have been occupied with the issue of the ARMA(1,1) representation of the squared process in the context of appropriate GARCH(1,1) processes. You can find notes on this issue here. We have also been occupied with indicative further topics about conditional heteroskedasticity, notes of which you can find here.   

We have been occupied with the example of a GARCH(1,1) process. You can find notes here.

We have briefly studied the application of the Bayesian Information Criterion in the context of ARMA models of unknown order. We considered the example of a simple indirect inference estimator for an invertible MA(1) model, via the use of a stationary AR(1) model and the corresponding OLSE as auxiliary model and estimator repectively. You can also find notes on the above here.

We have begun the study of conditional heteroskedasticity by providing a general definition. You can also find notes on

We were occupied with further examples, properties and issues concerning ARMA models. We have also studied issues concerning the semi-parametric estimation of such models when the orders are known, introducing the Gaussian Quasi Maximum Likelihood Estimator (Gaussian QMLE), in cases where the MA component is non trivial. We also abstractly discussed the numerical nature of its derivation, and were briefly and not rigorously occupied with its strong consistency under the relevant assumption frame