Introduction to Modern Time Series Analysis · Gebhard Kirchgässner Jürgen Wolters Introduction to Modern Time Series Analysis With43Figuresand17Tables 123 ProfessorDr.GebhardKirchgässner UniversityofSt.Gallen SIAW-HSG Bodanstrasse8 CH-9000St.Gallen Switzerland [email protected] ProfessorDr.JürgenWolters FreieUniversitätBerlin InstituteforStatisticsandEconometrics Boltzmannstraße20 14195Berlin Germany [email protected] LibraryofCongressControlNumber:2007932230 ISBN978-3-540-73290-7 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerial is concerned, specificallythe rights of translation, reprinting, reuseof illustrations, recitation, broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtained fromSpringer.ViolationsareliabletoprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Production:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Cover-design:WMXDesignGmbH,Heidelberg SPIN12071654 42/3180YL-543210 Printedonacid-freepaper Preface Econometrics has been developing rapidly over the past four decades. This is not only true for microeconometrics which more or less originated during this period, but also for time series econometrics where the cointegration revolution influenced applied work in a substantial manner. Economists have been using time series for a very long time. Since the 1930s when econometrics became an own subject, researchers have mainly worked with time series. However, economists as well as econometricians did not really care about the statistical properties of time series. This attitude started to change in 1970 with the publication of the textbook Time Series Analysis, Forecasting and Control by GEORGE E.P. BOX and GWILYM M.JENKINS. The main impact, however, stems from the work of CLIVEW.J.GRANGER starting in the 1960s. In 2003 together with ROBERT W.ENGLE, he received the Nobel Prize in Economics for his work. This textbook provides an introduction to these recently developed methods in time series econometrics. Thus, it is assumed that the reader is familiar with a basic knowledge of calculus and matrix algebra as well as of econometrics and statistics at the level of introductory textbooks. The book aims at advanced Bachelor and especially Master students in economics and applied econometrics but also at the general audience of economists using empirical methods to analyse time series. For these readers, the book is intended to bridge the gap between methods and applications by also presenting a lot of empirical examples. A book discussing an area in rapid development is inevitably incomplete and reflects the interests and experiences of the authors. We do not include, for example, the modelling of time-dependent parameters with the Kalman filter as well as Marcov Switching Models, panel unit roots and panel cointegration. Moreover, frequency domain methods are not treated either. Earlier versions of the different chapters were used in various lectures on time series analysis and econometrics at the Freie Universität Berlin, Germany, and the University of St. Gallen, Switzerland. Thus, the book has developed over a number of years. During this time span, we also learned a lot from our students and we do hope that this has improved the presentation in the book. VI Preface We would like to thank all those who have helped us in producing this book and who have critically read parts of it or even the whole manuscript. It is our pleasure to mention, in particular, MICHAEL-DOMINIK BAUER, ANNA CISLAK,LARSP.FELD,SONJALANGE,THOMAS MAAG,ULRICHK. MÜLLER, GABRIELA SCHMID, THORSTEN UEHLEIN, MARCEL R. SAVIOZ, and ENZO WEBER. They have all made valuable contributions towards improving the presentation but, of course, are not responsible for any remaining deficiencies. Our special thanks go to MANUELA KLOSS- MÜLLER who edited the text in English. Moreover, we are indebted to Dr. WERNER A. MÜLLER and MANUELA EBERT from Springer for their kind collaboration. St Gallen/Berlin, April 2007 GEBHARDKIRCHGÄSSNER JÜRGENWOLTERS Contents Preface..................................................................................................V 1 Introduction and Basics........................................................................1 1.1 The Historical Development of Time Series Analysis...................2 1.2 Graphical Representations of Economic Time Series....................5 1.3 Ergodicity and Stationarity...........................................................12 1.4 The Wold Decomposition.............................................................21 References............................................................................................22 2 Univariate Stationary Processes........................................................27 2.1 Autoregressive Processes..............................................................27 2.1.1 First Order Autoregressive Processes....................................27 2.1.2 Second Order Autoregressive Processes...............................40 2.1.3 Higher Order Autoregressive Processes................................49 2.1.4 The Partial Autocorrelation Function....................................52 2.1.5 Estimating Autoregressive Processes....................................56 2.2 Moving Average Processes...........................................................57 2.2.1 First Order Moving Average Processes.................................58 2.2.2 Higher Order Moving Average Processes.............................64 2.3 Mixed Processes...........................................................................67 2.3.1 ARMA(1,1) Processes...........................................................67 2.3.2 ARMA(p,q) Processes...........................................................73 2.4 Forecasting....................................................................................75 2.4.1 Forecasts with Minimal Mean Squared Errors......................75 2.4.2 Forecasts of ARMA(p,q) Processes.......................................80 2.4.3 Evaluation of Forecasts.........................................................84 2.5 The Relation between Econometric Models and ARMA Processes..........................................................................87 References............................................................................................88 3 Granger Causality...............................................................................93 3.1 The Definition of Granger Causality............................................95 3.2 Characterisations of Causal Relations in Bivariate Models..........97 VIII Contents 3.2.1 Characterisations of Causal Relations using the Autoregressive and Moving Average Representations.........97 3.2.2 Characterising Causal Relations by Using the Residuals of the Univariate Processes....................................................99 3.3 Causality Tests............................................................................102 3.3.1 The Direct Granger Procedure.............................................102 3.3.2 The Haugh-Pierce Test........................................................106 3.3.3 The Hsiao Procedure...........................................................110 3.4 Applying Causality Tests in a Multivariate Setting....................114 3.4.1 The Direct Granger Procedure with More Than Two Variables.............................................................................114 3.4.2 Interpreting the Results of Bivariate Tests in Systems With More Than Two Variables.........................................117 3.5 Concluding Remarks..................................................................118 References..........................................................................................120 4 Vector Autoregressive Processes.....................................................125 4.1 Representation of the System.....................................................127 4.2 Granger Causality.......................................................................136 4.3 Impulse Response Analysis........................................................138 4.4 Variance Decomposition............................................................144 4.5 Concluding Remarks..................................................................149 References..........................................................................................150 5 Nonstationary Processes...................................................................153 5.1 Forms of Nonstationarity............................................................153 5.2 Trend Elimination......................................................................159 5.3 Unit Root Tests...........................................................................163 5.3.1 Dickey-Fuller Tests.............................................................165 5.3.2 The Phillips-Perron Test......................................................171 5.3.3 Unit Root Tests and Structural Breaks................................176 5.3.4 A Test with the Null Hypothesis of Stationarity.................178 5.4 Decomposition of Time Series...................................................180 5.5 Further Developments................................................................187 5.5.1 Fractional Integration..........................................................187 5.5.2 Seasonal Integration............................................................189 5.6 Deterministic versus Stochastic Trends in Economic Time Series.................................................................................191 References..........................................................................................194 6 Cointegration.....................................................................................199 6.1 Definition and Properties of Cointegrated Processes.................203 Contents IX 6.2 Cointegration in Single Equation Models: Representation, Estimation and Testing...............................................................205 6.2.1 Bivariate Cointegration.......................................................205 6.2.2 Cointegration with More Than Two Variables....................208 6.2.3 Testing Cointegration in Static Models...............................209 6.2.4 Testing Cointegration in Dynamic Models..........................213 6.3 Cointegration in Vector Autoregressive Models........................218 6.3.1 The Vector Error Correction Representation.......................219 6.3.2 The Johansen Approach.......................................................222 6.3.3 Analysis of Vector Error Correction Models.......................229 6.4 Cointegration and Economic Theory..........................................234 References..........................................................................................235 7 Autoregressive Conditional Heteroskedasticity.............................241 7.1 ARCH Models............................................................................245 7.1.1 Definition and Representation.............................................245 7.1.2 Unconditional Moments......................................................248 7.1.3 Temporal Aggregation.........................................................249 7.2 Generalised ARCH Models........................................................252 7.2.1 GARCH Models..................................................................252 7.2.2 The GARCH(1,1) process...................................................254 7.2.3 Nonlinear Extensions...........................................................257 7.3 Estimation and Testing...............................................................259 7.4 ARCH/GARCH Models as Instruments of Financial Market Analysis..........................................................................261 References..........................................................................................263 Index of Names and Authors................................................................267 Subject Index..........................................................................................271 1 Introduction and Basics A time series is defined as a set of quantitative observations arranged in chronological order. We generally assume that time is a discrete variable. Time series have always been used in the field of econometrics. Already at the outset, JAN TINBERGEN (1939) constructed the first econometric model for the United States and thus started the scientific research programme of empirical econometrics. At that time, however, it was hardly taken into ac- count that chronologically ordered observations might depend on each other. The prevailing assumption was that, according to the classical linear regression model, the residuals of the estimated equations are stochasti- cally independent from each other. For this reason, procedures were ap- plied which are also suited for cross section or experimental data without any time dependence DONALD COCHRANE and GUY H. ORCUTT (1949) were the first to no- tice that this practice might cause problems. They showed that if residuals of an estimated regression equation are positively autocorrelated, the vari- ances of the regression parameters are underestimated and, therefore, the values of the F and t statistics are overestimated. This problem could be solved at least for the frequent case of first order autocorrelation by trans- forming the data adequately. Almost at the same time, JAMES DURBIN and GEOFFREY S. WATSON (1950/51) developed a test procedure which made it possible to identify first order autocorrelation. The problem seemed to be solved (more or less), and, until the 1970’s, the issue was hardly ever raised in the field of empirical econometrics. This did not change until GEORGE E.P. BOX and GWILYM M. JENKINS (1970) published a textbook on time series analysis that received consider- able attention. First of all, they introduced univariate models for time se- ries which simply made systematic use of the information included in the observed values of time series. This offered an easy way to predict the fu- ture development of this variable. Today, the procedure is known as Box- Jenkins Analysis and is widely applied. It became even more popular when CLIVE W.J. GRANGER and PAUL NEWBOLD (1975) showed that simple forecasts which only considered information given by one single time se- ries often outperformed the forecasts based on large econometric models which sometimes consisted of many hundreds of equations.