Econometrics II

We have proven that a causal linear process w.r.t. a white noise and a sequence of absolutely summable coefficients is always regular. Under a particular restriction on the coefficients sequence we have also proven it to have the property of short memory. We have also been occupied with the issue of strict stationarity of a such a process. We have also provided with two simple short memory examples, namely the AR(1) and the MA(1) processes. We have left for the moment unanswered the question of

For weakly stationary processes we defined the autcovariance and autocorrelation functions that codify part of the dependence inherent in the process. Given this such a process is termed regular iff it is comprised of random variables that are asymptotically uncorrelated. Furthermore,  a weakly stationary process is termed short memory iff its autocovariance function is absolutely summable, a condition that implies asymptotic uncorrelateness with sufficiently fast rate, and thereby implies regul

We were occupied with an imprecise definition of a stochastic process:

An -valued stochastic process is a collection of random variables parameterized by an index (or parameter) set , that satisfies some conditions, 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 of the process.

If is a non-empty finite (and

In this "introductory partial lecture" we have established the course's pedagogical and informational structure, as well as its aim, scope and material to be covered as they appear in the relevant synopsis. We also noted that a set of optional exercises will be timely available at the students' disposal for potential grade improvement. You can obtain some idea for this procedure in the relevant list concerning the previous academic year here.

Given the scope of the course material you can find h

Good afternoon class. A long message, following up on my promises:

A) For you to review fundamental/introductory notions of Probability Spaces, Random Variables, LLN's and CLT's in an accessible way, as well as some hints about what is at stake with Stochastic Processes (remember, it's essentially about memory and heterogeneity and how we can (or cannot) restrict them), I suggest you find the book

"Statistical Foundations of Econometric Modelling" by Aris Spanos (1986).

http://www.cambridge.org

We have continued our occupation with introductory notions of unit root econometrics. We have concluded our preparatory work by examining the Functional Central Limit Theorem (FCLT the link concerns a special case for iid innovations), concerning the convergence in distribution to a standard Wiener process of a properly scaled partial sum process constructed by stationary, ergodic and s.m.d. (w.r.t. some filtration) innovations, and the relevant Continuous Mapping Theorem. Given this, we have fi

We were occupied with the regression of one random walk onto another independent random walk. This is called a spurious regression and most likely indicates a non-existing relationship between two variables. In this context we examined the asymptotic behavior of the OLS estimator. We finally discussed briefly the notion of cointegration. You can find notes here.

We have defined and been occupied with the definition of the EGARCH(1,1) (exponential generalized autoregressive conditional heteroskedastic) process, the existence and uniqueness of a stationary and ergodic solution to the relevant recursion and the evaluation of several moments of the conditional variance process. You can find notes here.

We have completed the examination of the limit theory of the OLSE for the autoregressive parameter for an appropriate form of an AR(1)-ARCH(1) process. You can find notes for this here and compare our derivations with the analogous ones we have derived in previous lectures, for an AR(1) process with a different form of conditional hereoskedasticity (see for example here and here). We have essentially seen that the asymptotic variance estimator and the subsequent Wald-type testing procedure have

We examined issues concerning the GARCH(1,1) process using Matlab. Semi-parametric estimation was addressed evaluating the Gaussian QMLE and discussing the numerical nature of its derivation. A Matlab code can be found here.

We have also been occupied with the example of an EGARCH(1,1) process. You can find notes on the latter here.