Ιστολόγιο μαθήματος Econometrics II

Typos and Errata

Mon, 30 Jul 2018 21:39:25 +0300

You can find here the correction to the erratum appearing in the notes entitled “An Example: GARCH(1,1) Process” concerning part of the ARMA(1,1) representation.

You can find here and here corrections and derivations related to the notes entitled “Notes on the EGARCH(1,1) Model”. Those concern typos, as well as an erratum found in part of the derivation of the auto-covariance of the squares. Specifically, in the case where κ=1, the derivation should take into account that the product is empty and thereby by convention equal to one.

Note that whenever any of the above were met at the June exam papers, the grading of the relevant parts took into account in favour of the examinees, that they are not responsible for the reproduction of the erroneous parts of the derivations. Further typos and errata may remain in the notes. Please report any such typos and/or errata to stelios@aueb.gr or the course's e-class.

Synopsis Lectures 12th-13th

Sat, 26 May 2018 03:41:47 +0300

We have been occupied with further issues concerning the GARCH(1,1)processes, as for example that they cannot simultaneously have symmetric marginal distribution and exhibit phenomena of negative dynamic asymmetry (under the relevant moment existence assumptions), their extension to finite and arbitrary orders and the plausibility of the semi-parametric Gaussian QMLE in their context.

We have been occupied with 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 robust properties under both forms of conditional heteroskedasticity.

We have finally been occupied with another example of a conditionally heteroskedastic model, namely the EGARCH(1,1) one. We derived the properties required from our general definition of conditional heteroskedasticity (partially in the restricted context of conditional standard normality and of zero "magnitude" parameter), and established the model's comparative-to the GARCH case-flexibility w.r.t parameter restrictions for the establishment of suchlike properties as well as w.r.t. to the reproduction of the relevant financial returns stylized facts (in the aforementioned restricted context). You can find notes for the above here (as well as some similar derivations in a more general context here and here).

Synopsis 11th Lecture (2018)

Fri, 18 May 2018 22:45:13 +0300

We have been occupied with the definition of the GARCH(1,1) process, the existence and uniqueness of a stationary and ergodic solution to the relevant stochastic recurrence equation, the issue of the weak stationarity of the solution, and the issue of the ARMA(1,1) representation of the squared process under the appropriate restrictions. We thus obtained a non-linear process for which the properties of strict stationarity and ergodicity stem from weaker conditions compared to them that imply weak stationarity. Under stricter conditions we obtain a process that in addition to being an appropriate stationary and ergodic smd process, its squares constitute an ARMA(1,1) w.r.t. a relevant s.m.d. process. This is in partial accordance to some of the stylized facts of financial asset returns in some observation frequencies. You can find notes on this issue here.

Synopsis 10th Lecture

Sat, 12 May 2018 23:57:58 +0300

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 respectively. You can find notes on the above here. You can also find notes for our brief comment on the extension of the ARMA type models here.

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

Exercise

How would the definition of the aforementioned Indirect Estimator would alter if it waw known that the underlying MA(1) process were not invertible? Provide the details.

Synopsis: Almost 9th Lecture (2018)

Sun, 06 May 2018 20:30:50 +0300

We have examined issues concerning the semi-parametric estimation of ARMA type models introducing the Gaussian Quasi Maximum Likelihood Estimator (Gaussian QMLE), in cases where the MA component is non trivial (exercise: show that the Gaussian QMLE coincides with the OLSE when the MA component is trivial, the orders of the MA and AR components are known and this information is used). 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 framework concerning the MA(1) model. You can find notes on the above here.

Synopsis: 8th Lecture (2018)

Sat, 28 Apr 2018 23:43:22 +0300

Given a well defined ARMA process (e.g. the solution of the relevant recursion when the UDC holds for the Φ polynomial), we were occupied with the issue of what property is implied when the Θ polynomial satisfies the UDC, thus obtaining the notion of invertibility, which is equivalent to that the white noise process is specified as a linear causal process with respect to the ARMA process with absolutely summable coefficients that are absolutely bounded from above by the coefficients of a geometric series times a positive constant, that are obtained by the product of the inverse power series of Θ with Φ. This also implies that the white noise process is adapted to the filtration constructed from the history of the ARMA process at each time instance. When the first coefficient of the latter representation of the white noise process is not equal to zero (is this always the case or not?) then we also obtain a representation of the original ARMA process as an "AR infinity process". The invertibility concept can be important to the issue of statistical inference in ARMA models.

We begun the examination of issues of statistical inference in ARMA models when the MA component is not trivial, i.e. q>0. We showed first that in this context the OLSE is computationally infeasible. We discussed an example of an inconsistent OLSE in such a context, when estimation is reduced to AR parameters (or the relevant linear model is accordingly misspecified). This showed us that the unknown parameters corresponding to MA components cannot be generaly ignored, since in such cases the feasible OLSE of the AR parameters can have poor asymptotic properties. Hence in such cases we need some feasible "generalization" of the OLSE as a semi-parametric estimation procedure, and this remark paves the way for the consideration of the Gaussian Quasi Maximum Likelihood Estimator (QMLE).

You can find notes on the above here and here.

Synopsis: 7th Lecture (2018)

Sat, 21 Apr 2018 22:57:01 +0300

We were occupied with algebraic properties of the ring of formal power series w.r.t. the lag operator as well as some analytic properties that emerge when given relevant properties of the sequences of coefficients. These along with the preparations in the previous lectures on the class of causal linear processes, 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 were further occupied with the example of the ARMA(1,1) process by deriving the solution and the relevant properties which were shown to comply with the results of our general theorem.

You can find notes for the above here and here.

Exercise. Derive the solutions and the autocovariance function of the ARMA(1,2) process under the relevant UDC.

Synopsis: 6th Lecture (2018)

Sat, 31 Mar 2018 19:16:54 +0300

Given the use of the afore-examined CLT in order to derive conditions that ensure that the OLSE in the particular context of the stationary and ergodic AR(1) model (i.e. stationarity, ergodicity, smd property, and moment and dependence properties that ensure the convergence of particular series for the white noise process) has the usual rate and it is asymptotically normal, we were occupied with the verification of the conditions and evaluated the asymptotic variance in the context of (a stricter form of) the afore-examined conditionally heterskedastic example. This showed that the presence of conditional heteroskedasticity may affect asymptotic properties of the estimator (in this case the form of the asymptotic variance), and thereby may also affect properties of inferential procedures based on the estimator. You can find notes for the above here.

In this context we were occupied with the consideration of a feasible semi-parametric estimator for the asymptotic variance that is consistent in a stricter form of the aforementioned context involving also the existence of fourth moments. We then used it to construct a feasible and asymptotically exact and consistent Wald-type testing procedure for the AR(1) coefficient. You can find notes for the above here.

Synopsis: 5th Lecture (2018)

Sun, 25 Mar 2018 00:11:50 +0300

Even in the context of an AR(1) process with iid white noise, we have seen that the Lindeberg-Levy CLT is not applicable for the derivation of the rate and limiting distribution of the OLSE for the relevant coefficient (why?). Hence in order to tackle such a problem we require the establishment of a CLT applicable in such-like cases. Stationarity and ergodicity are not sufficient for such an establishment without further control of the rate of asymptotic independence of the elements of the underlying process. In order to avoid the description of conditions involving complex mixing conditions, we moved on to the issue of establishing a CLT for stationary and dependent processes via the quite restrictive, yet easily describable, notion of a martingale difference. Hence, we have examined the notions of the filtration, of the 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 have constructed examples that among others involved the construction and the examination of a strictly stationary, ergodic and square integrable martingale different process that in some cases it is also appropriately conditionally heteroskedastic.

We have examined the subsequent Martingale CLT, which is a generalization of the aforementioned Lindeberg-Levy CLT and involves stationary, ergodic s.m.d. (w.r.t. some filtration) processes.

We have used this in order to study the limit theory of the OLSE in the aforementioned framework by assuming that the innovation sequence is a stationary ergodic s.m.d. and several series are convergent. Those assumptions imply standard rates and asymptotic normality with the asymptotic variance crucially depending on the values of the aforementioned series which among others reflect dependence properties of the innovation process. In the iid case we have seen that the asymptotic variance assumes a simple (and somewhat interesting!) form for which it is easy to obtain a strongly consistent estimator, via the OLSE consistency and the CMT.

You can find notes for the above here.

Synopsis: 4th Lecture (2018)

Sun, 18 Mar 2018 02:16:00 +0300

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.

Synopsis: 3rd Lecture (2018)

Sat, 10 Mar 2018 21:42:24 +0300

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 why the AR(1) process can be also characterized as a solution of a particular stochastic recurence equation. Notes for the above can be found here.

Via a linear regression example involving linear processes and the relevant OLSE, we have, in the framework of stationarity, descriptively been occupied with the notion of the invariant σ-algebra of the process, provided with the stationary version of Birkhoff's LLN and realized that stationarity alone may not be sufficient for the consistency of the OLSE. Hence we have set the stage for the examination of the notion of ergodicity and the stationary and ergodic version of Birkhoff's LLN. Notes for the above can be found here.

Exercises

Is the "geometric series upper bound" for series of the absolute values of the coefficients sequence necessary in order to obtain the short memory property for the relevant causal linear process?
If not, does absolute summability suffice?

Synopsis: 2nd Lecture (2018)

Sat, 03 Mar 2018 23:47:49 +0300

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 regularity.

We examined the definition of a white noise process as a canonical example of weak stationarity. By examples 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 stationarity. Hence homogeneity and enough moment existence conditions imply that both properties hold, heterogeneity and enough moment conditions imply that the first does not, yet the second holds, homogeneity with insufficient moment existence conditions imply that the first holds while the second does not, and heterogeneity with insufficient moment existence conditions imply that neither holds. Later in the course we will examine more complex examples involving dependence.

We also occupied ourselves with the question of invariance of strict and weak stationarity w.r.t. transformations. We saw that the former is invariant w.r.t. to measurable pointwise transformations while the latter is generally not since for example those might destroy moment existence conditions. Both are preserved by transformations w.r.t. powers of the lag operator.

We begun to examine the first method-in this course-of the construction of a process via some transformation of a given one. We defined the concept of a causal linear process w.r.t. to a white noise process and a sequence of absolutely summable coefficients. After some preparation involving series of such sequences we have proven that this process is weakly stationary, given some hidden to us results that imply that this process is well defined and in this framework the expectation with the series operator commute.

Notes for the above can be found here.

Exercise: Prove or disprove-If the process is Gaussian, then it is strictly stationary iff it is weakly stationary.

Synopsis: 1st Lecture (2018)

Sun, 25 Feb 2018 16:52:08 +0300

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 ordered) subset of the parameter set, then the finite dimensional distribution (fidi) of the process corresponding to is simply the joint distribution of the random vector consisting of the random variables of the process that are indexed by the elements of according to its ordering.

The (Daniell-) Kolmogorov Extension Theorem implies that such a process, or equivalently the probability measure that it defines on the the set of functions , is equivalently "described" by the set of all fidis, given some consistency conditions w.r.t. to permutations and integration.

A "natural class" of stochastic processes is that of the Gaussian ones. A Gaussian process is a stochastic process for which every fidi is a Normal distribution. A simple example of a Gaussian process can be found here.

When the parameter set is totally ordered, hence it could represent time, then the process is called time series (the term time series can be also used in order to characterize the sample paths of such a process-we will by convention use it for the process itself).

We will be occupied with time series examples with parameter sets that are usually subsets of the real line. When those subsets are discrete, then the time series is said to evolve in discrete time, while when they are continuous then it is said to evolve in continuous time.

When or then the time series is also called a double stochastic sequence and a stochastic sequence respectively, and usually denoted by , and respectively.

Typical examples of such stochastic sequences, and the more accessible to us at this early stage of the course are the iid ones, or more generally the ones comprised by independent random variables.

Given the fidi characterization of a process, several potential properties can be described via fdis.

In this respect, for such a sequence and a fidi of it, the latter is said to be invariant w.r.t. time translations iff the fidi remains the same when the time indices that define it are arbitrarily translated.

This implies the following property. A time series is called (strictly-) stationary iff every fidi remains invariant w.r.t. time translations-try to extend the definition when the parameter space is the real line. We have also defined weak (or second order, or covariance) stationarity, via a set of (potentially joint) moment conditions.

Notes for the above can be found here.

Lecture 0: Acquaintance

Sun, 18 Feb 2018 01:22:34 +0300

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 here some preparatory material.

Some more Introductory material and guidance - to get you started

Wed, 14 Feb 2018 20:14:35 +0300

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/gb/academic/subjects/economics/econometrics-statistics-and-mathematical-economics/statistical-foundations-econometric-modelling?format=PB&isbn=9780521269124

The book is 30 years old but still one of the best for this type of introductory educational review on these notions. What interests us here is "Part II - Probability Theory", namely, chapters 3-10, and especially, ch, 3, 4, 8, 9. The other chapters in the 3-10 collection contain extensions and more specific theoretical tools (say, about conditional expectation, modes of convergence, etc), and they are certainly also useful as review helpers.

If you cannot find this book, as an imperfect substitute (regarding our goal here) search for the book

"Probability Theory and Statistical Inference: Econometric Modeling with Observational Data" (1999) by the same author

http://www.cambridge.org/gb/academic/subjects/economics/econometrics-statistics-and-mathematical-economics/probability-theory-and-statistical-inference-econometric-modeling-observational-data?format=PB&isbn=9780521424080

and look up chapters 2,3,8,9, in there.

B) e-Class Material to start with

I suggest you go over the documents "Notes on Introductory Notions of Stochastic Processes 1" and "Notes on Introductory Notions of Stochastic Processes 2".

that can be found in the Documents section of e-class. The basic goal here is mainly to realize and understand what parts you don't understand.

C) THE MATLAB ISSUE

As we have said, we will be inspecting code in the MATLAB environment later on, so it would be good if you could browse the subject and get a little bit familiar with the logic of creating Matlab scripts either to estimate actual data, or to run "Monte Carlo"(simulation) experiments.

Closing, don't hesitate to contact me with absolutely any question you may have related to the class and the material. I check my e-mail many times a day, and usually sit down to answer them late at night on the same day, so you will get quick feedback. I cannot promise that the feedback will be what you have imagined/hoped for, but feedback it will be.

-- 
Alecos Papadopoulos
PhD Candidate - TA for Econometrics II 2017-2018
papadopalex@aueb.gr
https://alecospapadopoulos.wordpress.com/

Synopsis: Lecture 13th (2017)

Sat, 27 May 2017 16:18:26 +0300

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 finally derived the limit theory of the OLSE for a unit root AR(1)-type of process, where among others we have found super-consistency and a non-normal limiting distribution, and specified a "Dickey-Fuller type" of test for the relevant hypotheses structure of a unit root test. You can find notes on the above here.

Synopsis: 8th Tutorial

Wed, 24 May 2017 22:56:55 +0300

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.

Synopsis: 7th Tutorial

Mon, 22 May 2017 16:09:17 +0300

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.

Synopsis: Lecture 12th (2017)

Fri, 19 May 2017 23:55:47 +0300

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 robust properties under both forms of conditional heteroskedasticity.

We begun our examination of introductory notions of unit root econometrics. We initiated into some preparatory work involving the notion of the Wiener process. You can find notes on the above here.

Synopsis: 6th Tutorial

Sun, 14 May 2017 11:40:44 +0300

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.

Synopsis: Lecture 9th+1/3 -11 (2017)

Sat, 13 May 2017 20:12:38 +0300

We have defined and been occupied with the definition of the GARCH(1,1) process, the existence and uniqueness of a stationary and ergodic solution to the relevant system of recursion, the issue of the weak stationarity of the solution, and the issue of the ARMA(1,1) representation of the squared process under the appropriate restrictions. You can find notes on this issue here. We have also begun our occupation on the limit theory of the OLSE for the autoregressive parameter in the context of an appropriate AR(1)-ARCH(1) model, notes for which you can find here.

Synopsis: 5th Tutorial

Wed, 10 May 2017 20:23:51 +0300

We were occupied with a linear model with instrumental variables where the orthogonality conditions involved the residuals and a matrix of instrumental variables. Under appropriate assumptions we derived the GMM estimator and its asymptotic properties in two cases. Firstly, in the case where the errors are uncorrelated and secondly in the case where the errors are serially correlated. We observed that the asymptotic variance of the GMME depends on the choice of the weighting matrix, so we addressed the issue of the optimal selection of this matrix. We further occupied ourselves with the consistent estimation of the asymptotic variance. You can find notes here.

Synopsis: Lecture 9th+1/3 (2017)

Sat, 06 May 2017 18:01:11 +0300

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

Synopsis: 4th Tutorial

Tue, 02 May 2017 16:39:19 +0300

We studied issues concerning the ARMA(1,1) process using Matlab. Semi-parametric estimation of ARMA models when the orders are known was addressed, evaluating the Gaussian QMLE and discussing the numerical nature of its derivation. In the case where the order of the process was unknown we were occupied with the use of the Bayesian Information Criterion (BIC) in order to select between candidate models. You can find notes on the latter here. A Matlab code for all the above can be found here.

Synopsis: 8th Lecture (2017)

Fri, 28 Apr 2017 14:31:01 +0300

We were occupied further with issues concerning statistical inference in ARMA models in the framework of correct statistical specification and of known unit variance for the white noise process. We pointed out that in the context of general AR models the extraction of the asymptotic properties of the OLSE can be similar to the one we have taken in the case of the AR(1) model modulo technical details of essentially multivariate nature that are not present in the latter case. We have studied issues concerning the semi-parametric estimation of such models introducing the Gaussian Quasi Maximum Likelihood Estimator (Gaussian QMLE), in cases where the MA component is non trivial (exe: show that the Gaussian QMLE coincides with the OLSE when the MA component is trivial, the orders of the MA and AR components are known and this information is used). 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 framework concerning the MA(1) model. You can find notes on the above here.

Synopsis: 3rd Tutorial

Wed, 12 Apr 2017 12:38:27 +0300

We examined the MA(1) process built on a stationary ergodic white noise. We were occupied with a GMM estimator using the sample first order autocovariance. In the case that the white noise process is comprised by i.i.d. variables with unit variance and finite fourth moment we examined the asymptotic properties of the GMM estimator. We finally noticed that in the case that we can't prove that the process we are interested in, is a s.m.d. in order to use the appropriate CLT for s.m.d. processes, we may use a generalization (Gordin's CLT). You can find notes here.

Synopsis: 7th Lecture (2017)

Sat, 08 Apr 2017 18:15:01 +0300

We were further occupied with the example of the ARMA(1,1) process by deriving the solution and the relevant properties which were shown to comply with the results of our general theorem.

We begun the examination of issues of statistical inference in ARMA models with the overall remark that when the MA component is not trivial, i.e. q>0, then the OLSE is computationally infeasible. We discussed an example of an inconsistent OLSE in such a context, when estimation is reduced to AR parameters (or the relevant linear model is accordingly misspecified). You can find notes on the above here and here.

Synopsis: 6th Lecture (2017)

Fri, 31 Mar 2017 15:10:17 +0300

We were occupied with algebraic properties of the ring of formal power series w.r.t. the lag operator as well as some analytic properties that emerge when given relevant properties of the sequences of coefficients. 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. You can find notes for the above here and here.

Synopsis: 2nd Tutorial

Mon, 27 Mar 2017 18:20:08 +0300

We occupied ourselves with an example of a linear model with endogeneity problem, in the context of which we examined the inconsistency of the OLS estimator and the consistency and asymptotic distribution of the IV estimator. You can find notes here.

Synopsis: 5th Lecture (2017)

Fri, 24 Mar 2017 18:14:07 +0300

We used the afore-examined CLT in order to derive conditions that ensure that the OLSE in the particular 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 may also affect properties of inferential procedures based on the estimator. You can find notes for the above here.

In this context we were occupied with the consideration of a feasible semi-parametric estimator for the asymptotic variance that is consistent. We then used it to construct a feasible and asymptotically exact and consistent Wald-type testing procedure for the AR(1) coefficient. You can find notes for the above here.

Synopsis: 4th Lecture (2017)

Fri, 17 Mar 2017 16:29:10 +0300

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. You can find notes for the above here.

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 stationary, ergodic and square integrable martingale different process that in some cases it is also appropriately conditionally heteroskedastic. You can find notes for the above here.

Synopsis: 1st Tutorial

Fri, 10 Mar 2017 18:25:26 +0300

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.

Synopsis: 3rd Lecture (2017)

Fri, 10 Mar 2017 02:28:50 +0300

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 ergodicity and Birkhoff's LLN. Notes for the above can be found here.

Synopsis: 2nd Lecture (2017)

Sat, 04 Mar 2017 05:55:54 +0300

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 also occupied ourselves with the question of invariance of strict and weak stationarity w.r.t. transformations. We saw that the former is invariant w.r.t. to measurable pointwise transformations while the latter is generally not since those might destroy moment existence conditions. Both are preserved by transformations w.r.t. powers of the lag operator.

Synopsis: Lecture 1 (2017)

Fri, 24 Feb 2017 05:51:46 +0300

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 of the process.

A Gaussian process is a stochastic process for which every fidi is a Normal distribution. A simple example of a Gaussian process can be found here.

When or then the time series is also called a double stochastic sequence and a stochastic sequence respectively, and usually denoted by , and respectively.

Typical examples of such stochastic sequences are the iid ones.

Given such a sequence and a fidi of it, the latter is said to be invariant w.r.t. time translations iff the fidi remains the same when the time indices that define it are arbitrarily translated.

A time series is called (strictly-) stationary iff every fidi remains invariant w.r.t. time translations. We have also defined weak (or second order, or covariance) stationarity, via a set of moment conditions. Notes for the above can be found here.

Synopsis: 13th Lecture

Fri, 27 May 2016 01:24:47 +0300

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.

Synopsis: 11th-12th Lectures

Sun, 22 May 2016 12:46:14 +0300

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.

Synopsis: 10th Lecture

Mon, 16 May 2016 01:35:58 +0300

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

Synopsis: 9th Lecture

Fri, 22 Apr 2016 01:05:22 +0300

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 the above here.

Synopsis: 7th-8th Lectures

Fri, 15 Apr 2016 03:53:02 +0300

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 framework concerning the MA(1) model. You can find notes on the above here.

Synopsis: 6th Lecture

Fri, 01 Apr 2016 03:26:18 +0300

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.

Synopsis: 5th Lecture

Fri, 01 Apr 2016 03:23:25 +0300

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 procedures based on the estimator.

Synopsis: 4th Lecture

Thu, 17 Mar 2016 17:57:31 +0300

Synopsis: 3rd Lecture

Mon, 14 Mar 2016 20:41:51 +0300

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.

Synopsis: 2nd Lecture

Fri, 04 Mar 2016 00:07:38 +0300

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) properties of such a process.

Synopsis: 1st Lecture

Fri, 26 Feb 2016 18:18:19 +0300

A Gaussian process is a stochastic process for which every fidi is a Normal distribution.

When or then the time series is also called a double stochastic sequence and a stochastic sequence respectively, and usually denoted by , and respectively.

Typical examples of such stochastic sequences are the iid ones.

Given such a sequence and a fidi of it, the latter is said to be invariant w.r.t. time translations iff the fidi remains the same when the time indices that define it are arbitrarily translated.

A time series is called (strictly-) stationary iff every fidi remains invariant w.r.t. time translations.