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Gebhard Kirchgässner • Jürgen Wolters
Uwe Hassler
Introduction to
Modern Time
Series Analysis
Second Edition
Gebhard Kirchgässner Jürgen Wolters
SIAW-HSG Institute for Statistics and Econometrics
University of St. Gallen FU Berlin
St. Gallen Berlin
Switzerland Germany
Uwe Hassler
Applied Econometrics and
International Economic Policy
Goethe University Frankfurt
Frankfurt
Germany
ISSN2192-4333 ISSN2192-4341 (electronic)
ISBN978-3-642-33435-1 ISBN978-3-642-33436-8 (eBook)
DOI10.1007/978-3-642-33436-8
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012950003
© Springer-VerlagBerlinHeidelberg2013
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Preface to the Second Edition
In preparing this second and enlarged edition, a third author has joined the
team. Still, the scope of the book has not changed. We try to provide a rig-
orous understanding of the theory and methods of univariate and multivar-
iate time series analysis. At the same time, the main objective is the devel-
opment of empirical skills with a special emphasis on the link to economic
applications. Therefore, we strengthened the specific feature of our book
that now contains 63 examples, most of them using real data sets. The
computations for the empirical examples were performed by means of
EViews, Version 7.2. Note that previous versions partly result in (slightly)
different numbers for parameters, standard errors and test statistics. The
same is likely to hold true with other computer programmes or future ver-
sions of EViews. Since the empirical examples are central to the book, we
now provide all data sets contained in EViews files on the homepage of
UWE HASSLER.
For this second edition we have updated some of the time series ana-
lysed in the examples, while other data sets containing historical series
taken from the literature remain unchanged. The major change of this en-
larged edition, however, consists of additional material. First, the new
Chapter 7 covers nonstationary panel data analysis. This accommodates
that during the last decade many of the time series techniques treated in
our book have been carried to the panel situation where series from sever-
al, possibly correlated units are investigated. Second, the final chapter on
conditional heteroscedasticity has been supplemented by a section on mul-
tivariate ARCH models accounting for time-varying conditional correla-
tion. Third, some subsections have been added (see Section 2.2.2 on tem-
poral aggregation), while others have been enlarged (see Section 5.5.1 on
fractional integration). Finally, we removed typos from the first edition
and improved the exposition where this seemed necessary.
We wish to thank all those who have helped us with this second edition.
It is our pleasure to mention, in particular, FLORIAN HABERMACHER.
TERESA KÖRNER, and GABRIELA SCHMID. They have made valuable con-
tributions towards improving the presentation but, of course, are not re-
V
VI Preface
sponsible for any remaining deficiencies. Moreover, we are indebted to Dr.
MARTINA BIHN and RUTH MILEWSKI from Springer for their kind collabo-
ration.
St Gallen/Berlin/Frankfurt, August 2012
GEBHARD KIRCHGÄSSNER JÜRGEN WOLTERS UWE HASSLER
Preface to the First Edition
Econometrics has been developing rapidly over the past four decades. This
is not only true for microeconometrics which more or less originated dur-
ing this period, but also for time series econometrics where the cointegra-
tion revolution influenced applied work in a substantial manner. Econo-
mists 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 atti-
tude 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
CLIVE W.J. GRANGER starting in the 1960s. In 2003 together with ROBERT
F. 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 eco-
nomics and applied econometrics but also at the general audience of econ-
omists 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 in-
clude, for example, the modelling of time-dependent parameters with the
Kalman filter as well as Markov 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.
VII
VIII 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, LARS P. FELD, SONJA LANGE, THOMAS MAAG, ULRICH K.
MÜLLER, GABRIELA SCHMID, THORSTEN UEHLEIN, MARCEL R. SAVIOZ,
and ENZO WEBER. They have all made valuable contributions towards im-
proving the presentation but, of course, are not responsible for any remain-
ing 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
GEBHARD KIRCHGÄSSNER JÜRGEN WOLTERS
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 The Lag Operator .......................................................................... 10
1.4 Ergodicity and Stationarity ........................................................... 12
1.5 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 ........................................................... 58
2.2.1 First Order Moving Average Processes ................................. 58
2.2.2 MA(1) and Temporal Aggregation ........................................ 62
2.2.3 Higher Order Moving Average Processes ............................. 65
2.3 Mixed Processes ........................................................................... 68
2.3.1 ARMA(1,1) Processes ........................................................... 69
2.3.2 ARMA(p,q) Processes ........................................................... 75
2.4 Forecasting .................................................................................... 78
2.4.1 Forecasts with Minimal Mean Squared Errors ...................... 78
2.4.2 Forecasts of ARMA(p,q) Processes ....................................... 81
2.4.3 Evaluation of Forecasts ......................................................... 85
2.5 The Relation between Econometric Models and
ARMA Processes .......................................................................... 89
References ............................................................................................ 90
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