Topics in time series econometrics E-Book

Maria Grazia Zoia

• Herausgeber: EDUCatt
• Kategorie: Fachliteratur
• Sprache: Englisch
• Veröffentlichungsjahr: 2014

Beschreibung

The book is based on a series of lectures given at the University of Geneva during the spring semester of 2011 and more recently at the Catholic University of Milan. The main objective of the parent courses “Econometric Analysis of Time Series” and “Advanced Econometrics”, respectively, was to establish a sound statistical and mathematical toolkit for time series econometrics, while reappraising selected topics in the field of classical econometrics. The logical interdependence between the various chapters (and appendixes) of the book is indicated in the accompanying diagram.

Leseprobe

© 2014EDUCatt- Ente per il Diritto allo Studio Universitario dell’Università Cattolica

Largo Gemelli 1, 20123 Milano - tel. [phone] - fax [phone]

e-mail: [email protected] (produzione); [email protected] (distribuzione)

web: www.educatt.it/libri

Associato all’AIE – Associazione Italiana Editori

isbn edizione cartacea:978-88-6780-148-0

isbn edizione pdf:978-88-6780-565-5

isbn edizione ePub:978-88-6780-411-5

Contents

Preface

CHAPTER 1Review of Univariate Stochastic Processes

1.1Introduction

1.2Principal Univariate Stationary Processes

Appendix A - The autocovariance generating function

Appendix B- Forecasting

CHAPTER 2Review of Multivariate Stochastic Processes

2.1 Introduction

2.2 Principal Multivariate Stochastic Processes

2.3 Integrated Processes

2.4 The Notion and the Role of Cointegration86

Appendix C- The autocovariance functions of a VMQ(q)

Appendix D-The autocovariance generating function

CHAPTER 3The Estimation Problem in a VAR Model

3.1 Introduction

3.2 The impulse-response function

3.3The estimation problem in a stationary VAR model

3.4Tests for the correct specification of the model

3.5Vector-autoregressive vis-à-vis to structural econometric models

CHAPTER 4Inference about Non-Stationary Processes

4.1 Introduction

4.2 Processes with deterministic trend

4.3 Brownian motion and extensions

CHAPTER 5Testing for the Unit-Root Hypothesis

5.1 Introduction

5.2 Principal unit- root tests

5.3 Phillips-Perron and augmented Dickey- Fuller tests

CHAPTER 6The Statistical Analysis of Models with Co-Integrated Variables

6.1 Introduction

6.2 Unit roots and residuals

6.3 Cointegration tests

6.4 The Phillips triangular representation of a cointegrated system

6.5 Estimating the cointegrating vector

6.6 Dynamic ordinary least squares (DOLS)

6.7 Phillips-Hansen’s fully modified least squares

CHAPTER 7VAR Statistical Properties and Representations

7.1 Introduction

7.2 The algebraic framework of unit-root econometrics

7.3 Implications of cointegration

7.4 Stock and Watson’s common- trend representation

7.5 Error correction representation

7.6 Granger representation theorem

CHAPTER 8Full-Information Maximum Likelihood Analysis of Cointegrated Systems

8.1 Introduction

8.2 Maximum likelihood estimation

8.3 Testing the null hypothesis of h cointegrating relations

8.4 Likelihood ratio tests about the cointegrating vector

CHAPTER 9Structural versus VAR Model Building

9.1 Introduction

9.2 From the analytic backstage to the econometric foreground

Appendix E-Matrix polynomial inversion by Laurent inversion

References

The graphical representation of a Brownian motionon the front cover is due to Massimiliano Fava

Preface

The book is based on a series of lectures given at the University of Geneva during the spring semester of 2011 and more recently at the Catholic University of Milan. The main objective of the parent courses “Econometric Analysis of Time Series” and “Advanced Econometrics”, respectively, was to establish a sound statistical and mathematical toolkit for time series econometrics, while reappraising selected topics in the field of classical econometrics. The logical interdependence between the various chapters (and appendixes) of the book is indicated in the accompanying diagram.

Milan, Maria Grazia Zoia

May2014

CHAPTER 1Review of Univariate Stochastic Processes

1.1Introduction

This chapter aims to review univariate stochastic processes.I have divided it into four parts.

First of all I will outline the basic notions concerning univariate stochastic processes and their statistical properties, where stationarity, ergodicity and invertibility play a central role. As we’ll see the notion of stationarity can assume a plurality of facets. The most interesting of these in econometrics are the notions of stationarity in mean, in covariance, and in the wide (or weak) sense.

After that, I will bring in the principal stochastic processes namely the white noise process, the moving average process, the autoregressive process as well as the (mixed) autoregressive moving average process.

Then, we will focus our attention on a particular class ofnon-stationaryprocesses, called integrated processes and, in particular, we will examine random walks and stochastic trends.

Finally, I will set out the Wold theorem which provides a general representation for weakly stationary processes and the Beveridge and Nelson decomposition which provides a useful decomposition for first-order integrated processes.

1.2Principal Univariate Stationary Processes

Most data in economics and finance come in the form of time series

wherey1is taken as an outcome of a random variablesyt.

A time seriesis meant to be a samplefrom a realization of a stochastic process.

A stochastic (discrete) process is an ordered infinite countable sequence of random variables, where the order is given by the time.

If we were to imagine having observed the process for an infinite period of time we would get an infinite countable sequence of outcomes of the process, namely

which represents a single realizationor stochastic function of the stochastic process.

The sample space of a stochastic process is the set of all possible sample stochastic functions engendered by the process.

Let us now take a look at the following graph showingHdifferent realizations of a stochastic process

The twofold reading key of a stochastic process as ordered sequence of random variables and as set of realizations is well shown by the following graph+

If we select thet-thobservation of these different realizations we get a sample of (H) outcomes of thet-thrandom variableYt.

This random variable will have some probability law

The unconditional mean or expectation ofYtis defined asE(Yt) whereEdenotes the expectation or averaging operator.

Accordingly, the second order moments ofYtare defined as

,variance

,j-th autocovariance

The moments of the random variables of a stochastic process are objects of the statistical inference. However, since what we usually observe is only a sample of a single realization of a stochastic process, namely a set of outcomes of (possibly) different random variables (say, a time series), statistical inference can be tackled only under due qualifications. In order to apply effectively statistical inferential methods to a sample of a single realization of the process, some restrictions on the characteristics of the process must be introduced.

The restrictions involve mainly the extent to which the random variables of the process are heterogeneous, on the one hand, and the degree and structure of the memory of he process, on the other hand. Altogether the issues are ascribable to the evolutive vs the stationary (in some sense) structure of the process, on the hand and to the ergodiuc properties of the same, on the other. The previous graph show the concept of ergodicity for the momentE(g(Yt)).

Stationarity

A stochastic process is called stationary insofar as it exhibits (at least to some extent) characteristics of permanence and enjoys statistical properties which are not affected by a shift in the time origin. This in turn grants it some sort of temporal homogeneity.