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Lecture Notes in Statistics 154
Edited by P.Bickel, P.Diggle, S.Fienberg, K.Krickeberg,
I. Olkin,N. Wermuth,S. Zeger
Springer Science+Business Media, LLC
Regina Kaiser
Agustin Maravall
Measuring Business Cycles in
Economic Time Series
, Springer
Regina Kaiser Agustin Maravall
D. de Estadistica y Econometria Servicio de Estudios
Universidad Carlos III de Madrid Banco de Espaiia
Madrid,126 Alcală 50
28903 Getafe 28014 Madrid
Spain Spain
[email protected] [email protected]
The frrst author acknowledges support by the Spanish grant PB95-0299 of CICYT.
Library of Congress Cataloging-in-Publication Data
Kaiser, Regina.
Measuring business cycles in economic time series / Regina Kaiser, Agustin Maravall.
p. cm.-(Lecture notes in statistics; 154)
Includes bibliographical references and indexes.
ISBN 978-0-387-95112-6 ISBN 978-1-4613-0129-5 (eBook)
DOI 10.1007/978-1-4613-0129-5
1. Business cycles. 2. Time-series analysis. 1. Maravall, Agustin. Il. Title.
III. Lecture notes in statistics (Springer-Verlag); v. 154.
HB371 1 .K26 2001
388.5'42'OI51955-dc21 00-059552
Printed on acid-free paper.
© 2001 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, lnc. in 2001
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understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
byanyone.
Camera-ready copy provided by the authors.
9 8 7 6 5 4 3 2 1
ISBN 978-0-387-95112-6 SPIN 10777073
Contents
Figures VB
1 Introduction and Brief Summary 1
2 A Brief Review of Applied Time Series Analysis 5
2.1 Some Basic Concepts . . . . . . . . . . 5
2.2 Stochasti c Processes and Stationarity . . . . . . . . 7
2.3 Differencing .. . . . . . . . . . . . .. . . . . . . . 8
2.4 Linear Stationary Process, Wold Repr esentation , and Auto-
correl ation Function . . . . . . . . . . 13
2.5 The Spectrum . . . . . . . . . . . . . . 16
2.6 Linear Filter s and Their Squar ed Gain 27
3 ARIMA Models and Signal Extraction 31
3.1 ARIMA Models . . . . . . . . . . . . . . 31
3.2 Modeling Strategy, Diagnostics and Inference 39
3.2.1 Ident ifi cation . . . . . . . . 39
3.2.2 Estimation and Diagnosti cs . 40
3.2.3 Inference 41
3.2.4 A Particular Class of Models 43
3.3 Preadjustment 44
3.4 Unobserved Components and Signal Extra ction 48
3.5 ARIMA-Mod el-Based De compos ition of a Time eriesS 55
vi Contents
3.6 Short-Term and Long-Term Trends . 65
4 Detrending and the Hodrick-Prescott Filter 69
4.1 The Hedrick-Prescott Filter: Equivalent Representations . 69
4.2 Basic Characteristics of the Hodrick-Prescott Filter . . . 72
4.3 Some Criticisms and Discussionofthe Hedrick-Prescott Filter 77
4.4 The Hodrick-Prescott Filter as a Wiener-Kolmogorov Filter 80
4.4.1 An Alternative Representation 80
4.4.2 Derivation of the Filter . 82
4.4.3 The Algorithm . . . . . . 85
4.4.4 A Note on Computation . 86
5 Some Basic Limitations of the Hedrick-Prescott Filter 87
5.1 Endpoint Estimation and Revisions . 87
5.1.1 Preliminary Estimation and Revisions 87
5.1.2 An Example . 90
5.2 Spurious Results . 100
5.2.1 Spurious Crosscorrelation . 101
5.2.2 Spurious Autocorrelation ; Calibrat ion 103
5.2.3 Spurious Periodic Cycle 105
5.3 Noisy Cyclical Signal . 114
6 Improving the Hodrick-Prescott Filter 117
6.1 Reducing Revisions . 117
6.2 Improving the Cyclical Signal . . . . 121
7 Hodrick-Prescott Filtering Within a
Model-Based Approach 135
7.1 A Simple Model-Based Algorithm . 135
7.2 A Complete Model-Based Method;
Spuriousness Reconsidered . . . . . . . . . . . . . 137
7.3 Some Comments on Model-Based Diagnostics and Inference 156
7.4 MMSE Estimation of the Cycle: A Paradox . 167
Appendix 171
References 177
Author Index 184
Subject Index 186
Figures
2.1 Roots of unit circle page 10
2.2 Sine-cosine functions 12
2.3 Theoretical ACF /Sample ACF 15
2.4 Generated time series/Fourier series 17
2.5 Histogram of frequencies/power spectrum 18
2.6 Spectra of AR(2) process 22
2.7 Realization of AR(2) process 23
.82 Examples of spectra 24
2.9 Cyclical period and frequency 26
2.10 Series spectrum/gain of seasonal detrending filters 29
3.1 Stationarity region for AR(2) parameters 35
3.2 Spectra of IMA(l,l) process 38
3.3 Forecasts and 95%confidence interval 42
3.4 Preadjustment 46
3.5 Deterministic effects 47
.63 Canonical decomposition of a random walk 57
3.7 Spectral AMB decomposition 59
3.8 Wiener-Kolmogorov filter 60
.93 Squared gains 16
3.10 Series and estimated components 62
3.11 Standard error and estimators 63
3.12 Forecasts 64
3.13 Squared gain for trend-cycle filter 65
3.14 Squared gain for trend/trend estimators 67
4.1 Squared gain function: Butterworth filters 71
4.2 Hodrick-Prescott filter, trend 73
4.3 Hodrick-Prescott filter, cycle 75
4.4 XlI-HP cycle filter 76
4.5 Spectrum of the white noise component estimator 79
5.1 Short-term economic indicators : original series 92
5.2 XlI-SA series and HP trend 93
5.3 XlI and HP cycles 94
5.4 Concurrent versus final trend estimator 95
5.5 Concurrent versus final cycle estimator 96
5.6 Revisions in concurrent estimator 97
5.7 95% Confidence intervals for cycle (based on revisions) 99
5.8 Squared gain: convolution of HP and XlI filters 001
5.9 Density for correlation coefficient: white noise case 101
5.10 Density for correlation coefficient: random walk case 102
Figures viii
5.11 Density for correlation cient:coeffi airline model page103
5.12 Spectrum of AR(4) for the white noise case 106
5.13 Decomposition of a white neois eriess 701
5.14 Spectrum ofyeccl ocmponent in a random walk 801
5.15 Spectrum of ecycl in IMA(l ,l) as a function of the ta 901
5.16 Decomposition of a random walk series 110
5.17 Period ofcycle as a function of lambda 112
5.18 Spectrum of a ecycl in a random walk 113
5.19 Estimated trends 114
5.20 Estimated cycles 114
6.1 HP cycle based on X11 and SEATS-SA sserie 125
6.2 Spectrum ofcycle 126
6.3 Trend and trend-cycl e ocmponents 127
6.4 HP cycle based on SEATS trend and on X11-SA series 128
6.5 Spectrum ofcycle (SEATS trend and X11-SA series) 129
6.6 Difference between escycl (SEATS trend and X11-SA series) 130
6.7 HP cycles based on X11-SA sersei 131
6.8 HP syeccl based on SEATS trend 131
6.9 Standard deviation of revision from concurr ent to 132
final estimation 133
6.10 95% C.1.for HP cycle (based on revisions) 139
7.1 Spectra in the model-bas ed int erpretation 140
7.2 Spectra of the difference (original minus e)cycl 141
7.3 De composition of the eriess 148
7.4 Spectra for original series and components :sersie CC 149
7.5 Sp ectra for original series and omcponents: erisse IPI 150
7.6 Spectra for original series and mcoponents:erises CR 151
.77 Sp ectra for original series and ocmponnets:serise AP 152
7.8 Squared gain of filters for components: sserie CC 153
.97 Squared gain of filters for ocmponents:series IPI 154
7.10 Squared gain offilters for ocmponnets:series CR 155
7.11 Squared gain offilters for components :series AP 163
7.12 AMB and HP-X11: estimated components of CC 164
7.13 AMB and HP-X11: estimated components of IPI 165
7.14 AMB and HP-X1l: setimated components of CR 166
7.15 AMB and HP-X11: estimated components of AP 169
7.16 Spectra of the trend -cycle and cycle estimators
.1
Introduction and Brief Summary
This monograph addresses the problem of measuring economic cycles (also
called business cycles) in macroeconomic time series. In the decade that
followed the Great Depression, economists developed an interest in the
possible existence of (more or less systematic) cycles in the economy; see,
for example, Haberler (1944) or Shumpeter (1939). It became appar ent
that in order to identify economic cycles,one had to remove from the series
seasonal fluctuations, associated with short-term behavior , and the long
term secular trend , associated mostly with technological progress. Burns
and Mitchell (1946) provided perhaps the first main reference point for
much of the posterior research. Statistical measurement of the cycle was
broadly seen as capturing the variation of the series within a range of
frequencies, after the series has been seasonally adjusted and detrended .
(Burns and Mitchell suggested a range offrequencies associated with cycles
with a period between, roughly, two and eight earsy.)
Statistical methods weredevised to estimate cyclical variation, and these
gradually evolved towards methods fundamentally based on the application
of moving average filters to the series; see, for example, Bry and Boschan
(1971). These moving average filters were "ad hoc" filters,in the sense that
they were fixed, independent of the particular series being analyzed ; they
were designed as linear "band-pass" filters, that is, as filters aimed at cap
turing the series variation within a certain band offrequencies. The last 20
years have witnessed methodological research on two broad fronts:the first
dealt with further developments ofthe moving average type approach ;the
other wasthe development ofmore complex statistical approaches oriented
towards capturing cyclical features, such as asymmetries and varying period
R. Kaiser et al., Measuring Business Cycles in Economic Time Series
© Springer-Verlag New York, Inc. 2001
