Nonlinear time series analysis of business cycles volume 276 (contributions to economic analysis)

NONLINEAR TIME SERIES ANALYSIS OF BUSINESS CYCLESCostas MilasDepartment of Economics, Keele University, UK Philip RothmanDepartment of Economics, East Carolina University, USA Dick van D

NONLINEAR TIME SERIES ANALYSIS

OF BUSINESS CYCLES

i

TO ECONOMIC ANALYSIS

Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris

San Diego – San Francisco – Singapore – Sydney – Tokyo

ii

NONLINEAR TIME SERIES ANALYSIS OF BUSINESS CYCLES

Costas MilasDepartment of Economics, Keele University, UK

Philip RothmanDepartment of Economics, East Carolina University, USA

Dick van DijkEconometric Institute, Erasmus University Rotterdam, The Netherlands

Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris

San Diego – San Francisco – Singapore – Sydney – Tokyo

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DvD: To my nephews and nieces Judith, Gert-Jan, Suely, Matthijs, Ruben,Jacco, Nienke and Dani

CM: To my wife, Gabriella and my daughter Francesca

PR: To my mother, Laura Rothman

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INTRODUCTION TO THE SERIES

This series consists of a number of hitherto unpublished studies, which are introduced by the editors

in the belief that they represent fresh contributions to economic science.

The term ‘economic analysis’ as used in the title of the series has been adopted because it covers both the activities of the theoretical economist and the research worker.

Although the analytical methods used by the various contributors are not the same, they are nevertheless conditioned by the common origin of their studies, namely theoretical problems encountered in practical research Since for this reason, business cycle research and national accounting, research work on behalf of economic policy, and problems of planning are the main sources of the subjects dealt with, they necessarily determine the manner of approach adopted by the authors Their methods tend to be ‘practical’ in the sense of not being too far remote from application to actual economic conditions In addition they are quantitative.

It is the hope of the editors that the publication of these studies will help to stimulate the exchange of scientific information and to reinforce international cooperation in the field of economics.

The Editors

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The notion of business cycle nonlinearity goes back a long time For example,

Mitchell (1927)andKeynes (1936)suggested that business cycles display metric behavior in the sense that recessions are shorter and more volatile thanexpansions Similarly,Hicks (1950)noted that business cycle troughs are sharperthan peaks Further, Friedman (1964) proposed his ‘‘plucking model’’ of eco-nomic fluctuations based upon the observation of asymmetry in correlationsbetween successive phases of the business cycle, in the sense that the amplitude

asym-of a contraction is strongly correlated with the strength asym-of the subsequent pansion, while the amplitude of an expansion is uncorrelated with the amplitude

ex-of the following contraction

Neftc-i (1984)initiated the modern econometric literature on business cyclenonlinearity with his study of U.S unemployment rates using Markov chaintechniques His results implied that the U.S unemployment rate displays

‘‘steepness’’-type business cycle asymmetry, following the taxonomy due tohel (1993) Neftc-i’s paper has been highly influential and since its publicationroughly 20 years ago, a great deal of research has been done exploring themagnitude and economic significance of nonlinearity in business cycle fluctu-ations For example,Hamilton (1989, p 359)argued that the now very popularMarkov-switching model he introduced is a natural generalization of Neftc-i’sframework A useful survey of many important developments in this literaturecan be found inClements and Krolzig (2003)

Sic-To provide a comprehensive look at current work on this topic, for this bookvolume we solicited original contributions on business cycle nonlinearity fromleading academics and practitioners in the field Each chapter was subsequentlyreviewed by an ‘‘internal’’ referee (an author or coauthor of a different chapter

in the book), and by an ‘‘external’’ referee These external referees were DonHarding (University of Melbourne), Christopher Martin (Brunel University),Marcelo Medeiros (PUC Rio), Simon van Nordon (HEC Montre´al), RichardPaap (Erasmus University Rotterdam), Jean-Yves Pitarakis (University ofSouthampton), Tommaso Proietti (University of Udine), Pierre Siklos (WilfredLaurier University), Peter Summers (Texas Tech University), Timo Tera¨svirta(Stockholm School of Economics), Gilles Teyssiere (Universite´ Paris 1), GregTkacz (Bank of Canada), Mark Wohar (University of Nebraska at Omaha), andEric Zivot (University of Washington) We thank both our contributors and

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referees for their cooperation in keeping to the ambitious time schedule we set atthe start of this project.

The papers in this volume can be classified into five groups, each focusing on

a particular topic The first question considered, in a group of three papers, is therole of nonlinearity in dating business cycle turning points and identifying busi-ness cycle regimes Chauvet and Hamilton provide a detailed description of theMarkov-switching approach to this issue, including not only the technicalitiesinvolved but also paying ample attention to the underlying intuition They il-lustrate the promise of this approach by constructing a business cycle chronol-ogy for the U.S based on real-time data for the post-World War II period, i.e.data as they were originally released at each historical date Their findingsdemonstrate that the resulting turning point dates closely match those of thebusiness cycle dating committee of the National Bureau of Economic Research(NBER), but the model-based turning points typically become available muchsooner than the NBER ones

Clements and Galva˜o use the context of predicting business cycle regimeprobabilities and output growth in the U.S to consider the specific issue ofcombining forecasts versus combining information in modeling The simplemodels whose forecasts they combine each use a single recession indicator, one

of the components that comprise the Conference Board Composite LeadingIndicator (CLI), as the explanatory variable to the model Combining this in-formation set in modeling is achieved by using a model selection strategy Forpredicting output growth, their findings support pooling the forecasts of thesingle-indicator models, whilst the results are more mixed for predicting reces-sions and recession probabilities

Morley and Piger consider the ability of linear autoregressive integratedmoving average (ARIMA) and nonlinear Markov-switching models to repro-duce business cycle-related features in U.S real Gross Domestic Product (GDP)data They find that both linear and Markov-switching models are able to re-produce business cycle features such as the average growth rate in recessions, theaverage length of recessions, and the total number of recessions However,Markov-switching models are found to be better than linear models at repro-ducing the variability of growth rates in different business cycle phases Fur-thermore, only Markov-switching specifications with three regimes or with abuilt-in ‘‘bounceback’’ effect are able to reproduce high-growth recoveries fol-lowing recessions and a strong correlation between the severity of a recessionand the strength of the subsequent recovery

The second topic analyzed, in a set of two papers, is the use of multivariatenonlinear models in econometric modeling of business cycles Koop and Potterintroduce a nonlinear extension of the Vector Autoregressive (VAR) modelwhich they call the Vector Floor and Ceiling (VFC) model The VFC model isalso a multivariate extension of univariate nonlinear models the authors devel-oped earlier with floor and ceiling effects; see Pesaran and Potter (1997) and

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Koop and Potter (2003) As a tightly restricted Threshold Autoregressive model,the authors argue that the VFC model provides a parsimonious framework forcapturing the type of business cycle nonlinearity suggested by economic theory.They use both classical and Bayesian methods to analyze the estimated models.Their results suggest strong nonlinearities in the contemporaneous relationshipsbetween the variables and weaker evidence of conditional mean nonlinearity.Camacho and Perez-Quiros propose a new framework to analyze pairwisebusiness cycle synchronization across a given set of countries The approach isbased on multivariate Markov-switching procedures, and essentially determinesthe relative position of two countries’ cycles in between the extreme cases ofcomplete independence and perfect synchronization An empirical application tothe G7 countries shows that these can be divided into two groups with distinctcommon business cycle dynamics, with one group consisting of Euro-zonecountries (France, Germany, and Italy) and the other including English-speak-ing countries (Canada, the U.K., and the U.S.).

Five of the papers explore a third topic, the extent to which nonlinearity canaccount for the well-documented instability and structural change which hasbeen observed in macroeconomic time series; see, e.g.Stock and Watson (1996).Marcellino’s paper is motivated by the many economic and political changeswhich have occurred in what is now called the Euro-zone since the early 1980s.Such changes, he argues, increase the difficulty of modeling macroeconomic timeseries for Euro-area countries with constant-parameter linear models To ex-plore this idea he carries out a simulated out-of-sample forecasting competitionusing linear, nonlinear, and time-varying models to predict the future values of

500 macroeconomic time series for these countries It turns out that, for roughlytwo-thirds of the series studied, nonlinear and time-varying models work best.These results lead him to conclude that use of such models should be stronglyconsidered by practitioners

Kapetanios and Tzavalis use a new model of structural breaks, one whichallows for parameter changes to be triggered by large economic shocks Incontrast to other structural break models in the literature, their approach allowsthem to examine such parameter changes without fixing either the number ormagnitude of the breaks The results support the view that the observed insta-bility in U.S macroeconomic time series is due to the oil-price shocks of the1970s and the changes in the Fed’s operating procedures in the late 1970s andearly 1980s

There are many nonparametric and model-based methods available for tracting the business cycle component from a macroeconomic time series Ko-opman, Lee, and Wong use a parametric trend-cycle decomposition procedure

ex-in which the parameters governex-ing the dynamics of these components are lowed to vary in a nonlinear but smooth manner They find substantial evidence

al-of smooth time variation in these parameters Of particular interest aretheir results suggesting that business cycle volatility for the U.S economy has

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decreased While these findings are consistent with results reported earlier in theliterature on the ‘‘great moderation,’’ it is the first to do so within the trend-cycledecomposition framework.

Becker, Enders, and Hurn develop a methodology to model a time-varyingintercept The methodology relies on a Fourier approximation, which uses trig-onometric functions to capture the unknown functional form of the interceptterm Two empirical applications illustrate the use of the methodology The firstexample demonstrates how a time-varying intercept can be used to capture astructural break in the U.S inflation rate The second example relates to the U.S.long-run money demand function The authors show that the apparent insta-bility in the cointegrating vector among M3, income, prices and interest ratesdisappears once a time-varying intercept is taken into account

Anderson and Low extend the family of smooth transition autoregressive(STAR) models by proposing a specification in which the autoregressive pa-rameters follow random walks The random walks in the parameters capturepermanent structural change within a regime-switching framework, but in con-trast to existing specifications, structural change in the random walk STAR(RW-STAR) setting follows a stochastic process rather than a deterministicfunction of time Using industrial production data for several countries, theyfind evidence of nonconstant parameters in a setting where there is also evidence

of regime-switching In addition, they find that RW-STAR models seem to beable to capture different types of time-varying behavior of parameters

The fourth topic, the importance of nonlinearity for econometric analysis ofmonetary policy, is addressed in three of the papers in this volume Kesriyeli,Osborn, and Sensier estimate smooth transition monetary policy rules for theU.S., U.K., and Germany They find significant nonlinear structure in themonetary policy rules associated with interest rate changes rather than move-ments in the inflation rate or the output gap The nonlinear models also identify

a significant shift in the parameter values of the U.S and U.K interest ratereaction functions occurring around mid-1985

Dolado and Marı´a-Dolores examine the issue of the asymmetric effects ofmonetary policy shocks on output in the Euro area Assuming a nonlinearaggregate supply curve, they derive monetary policy shocks as the residuals from

a nonlinear interest rate reaction function The authors proceed by estimating amultivariate Markov-switching model for EU output and find that monetarypolicy shocks have a greater effect on output in recessions

Akram, Eitrheim, and Sarno adopt a different nonlinear model but reachsimilar conclusions on the effects of monetary policy on output The authors usemultivariate smooth transition models to characterize the behavior of output,money, and the real exchange rate in Norway over a period of almost twocenturies They find evidence of asymmetric effects of monetary policy on out-put In particular, large contractionary monetary policy shocks tend to havesignificant effects on output, while small expansionary monetary policy shockstend to have negligible effects on output

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Finally, two of the papers study the statistical and economic impact of lowing for business cycle regime-dependent behavior in models of importantmacroeconomic and financial time series Bhardwaj and Swanson compare theability of fractional ARIMA (ARFIMA), non-ARFIMA, and other nonlinearmodels to forecast U.S daily stock returns in recessions versus expansions andfor larger versus smaller samples The findings of their paper suggest thatARFIMA models do not predict better or worse than any other model acrossthe business cycle On the other hand, the forecasting ability of ARFIMAmodels increases with larger samples.

al-Dahl and Kulaksızog˘lu use a nonlinear autoregressive distributed lag model

to study the relationship between housing completions and housing starts in theU.S economy Their results suggest that builders change the speed of construc-tion depending upon whether the home construction industry is in a recession orexpansion In particular, the mean lag between housing completions and hous-ing starts is significantly shorter in recessionary than in expansionary periods.This finding is consistent with what has been called the ‘‘accordion effect’’ in theliterature; seevan Alphen and Merkies (1976)

References

Clements, M.P and H.-M Krolzig (2003), ‘‘Business cycle asymmetries: acterization and testing based on Markov-switching autoregressions’’, Jour-nal of Business and Economic Statistics, Vol 21, pp 196–211

char-Friedman, M (1964) ‘‘Monetary studies of the National Bureau’’, in: The tional Bureau Enters its 45th Year, 44th Annual Report, pp 7–25 Reprinted in

Na-M Friedman, The Optimum Quantity of Money and Other Essays, Chicago:Aldine pp 261–284

Hamilton, J.D (1989), ‘‘A new approach to the economic analysis of tionary time series and the business cycle’’, Econometrica, Vol 57, pp.357–384

nonsta-Hicks, J.R (1950), A Contribution to the Theory of the Trade Cycle, Oxford:Clarendon Press

Keynes, J.M (1936), The General Theory of Employment, Interest and Money,London: Macmillan

Koop, G and S Potter (2003), ‘‘Bayesian analysis of endogenous delay old models’’, Journal of Business and Economic Statistics, Vol 21, pp 93–103.Mitchell, W.C (1927), Business Cycles: The Problem and its Setting, New York:NBER

thresh-Neftc-i, S.N (1984), ‘‘Are economic time series asymmetric over the businesscycle’’, Journal of Political Economy, Vol 92, pp 307–328

Pesaran, M.H and S Potter (1997), ‘‘A floor and ceiling model of US output’’,Journal of Economic Dynamics and Control, Vol 21, pp 661–695

Sichel, D.E (1993), ‘‘Business cycle asymmetry: a deeper look’’, Economic quiry, Vol 31, pp 224–236

In-xiii

Stock, J.H and M.W Watson (1996), ‘‘Evidence on structural instability inmacroeconomic time series relations’’, Journal of Business and EconomicStatistics, Vol 14, pp 11–30.

van Alphen, H.J and A.H.Q.M Merkies (1976), ‘‘Distributed lags in tion: an empirical study’’, International Economic Review, Vol 17, pp.411–430

construc-xiv

Marcelle Chauvet and James D Hamilton

4 Using multiple indicators to identify turning points 22

5 Empirical performance of the monthly recession probability index 32

CHAPTER 2 COMBINING PREDICTORS & COMBINING

INFORMATION IN MODELLING: FORECASTING

US RECESSION PROBABILITIES AND OUTPUT

CHAPTER 3 THE IMPORTANCE OF NONLINEARITY IN

James Morley and Jeremy Piger

2 An algorithm for establishing business cycle turning points 79

3 Business cycle features in U.S real GDP data 81

4 Business cycle features in simulated data from time-series

4.2 Business cycle features from linear models 87

4.3 Business cycle features from regime-switching

Gary Koop and Simon Potter

2 A nonlinear VAR with floor and ceiling effects 99

3.2 A comparison of Bayesian and classical results 109

Appendix B: Bayesian analysis of the VFC model 122

Appendix C: Classical analysis of the VFC model 128

Appendix D: Further details on impulse response analysis 131

Maximo Camacho and Gabriel Perez-Quiros

2 A framework to analyze business cycle synchronization 135

2.2 Multivariate Markov-switching approach 136xvi

3 Empirical results 139

3.2 Comparative analysis of business cycle synchronization 143

3.3 Business cycle synchronization across G7 countries 145

5.2 Forecast evaluation for unstable series 165

6 Forecasting industrial production, unemployment

George Kapetanios and Elias Tzavalis

2 Modelling structural breaks in autoregressive coefficients 177

CHAPTER 8 TREND-CYCLE DECOMPOSITION MODELS

WITH SMOOTH-TRANSITION PARAMETERS:

EVIDENCE FROM U.S ECONOMIC TIME

Siem Jan Koopman, Kai Ming Lee andSoon Yip Wong

4 Empirical evidence from U.S economic time series 206

DEMAND USING A FOURIER-SERIES

Ralf Becker, Walter Enders and Stan Hurn

4 Selecting the optimal number of terms in the

CHAPTER 10 RANDOM WALK SMOOTH TRANSITION

3.1 Performance of the nonlinearity tests 253

4 Modelling industrial production of selected OECD

CHAPTER 11 NONLINEARITY AND STRUCTURAL CHANGE IN

INTEREST RATE REACTION FUNCTIONS FOR

Mehtap Kesriyeli, Denise R Osborn andMarianne Sensier

CHAPTER 12 STATE ASYMMETRIES IN THE EFFECTS OF

MONETARY POLICY SHOCKS ON OUTPUT:

SOME NEW EVIDENCE FOR THE EURO-AREA 311

Juan J Dolado and Ramo´n Marı´a-Dolores

3 Estimation of a monetary policy reaction function 315

4 Markov Switching Models for real output growth 320

4.1 Extended Markov Switching model including

CHAPTER 13 NON-LINEAR DYNAMICS IN OUTPUT, REAL

EXCHANGE RATES AND REAL MONEY

5.1 STR models of output, the real exchange rate and

5.6 Dynamics of the linear versus the non-linear systems of

CHAPTER 14 A PREDICTIVE COMPARISON OF SOME SIMPLE

LONG- AND SHORT MEMORY MODELS OFDAILY U.S STOCK RETURNS, WITH EMPHASIS

Geetesh Bhardwaj and Norman R Swanson

CHAPTER 15 NONLINEAR MODELING OF THE CHANGING

LAG STRUCTURE IN U.S HOUSING

Christian M Dahl and Tamer Kulaksizog˘lu

5 Nonlinear autoregressive distributed lag models 419

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List of Contributors

Q Farooq Akram Norges Bank, Norway

Heather Anderson School of Economics, Australian National

University, AustraliaRalf Becker Center for Growth and Business Cycle Research,

University of Manchester, UKGeetesh Bhardwaj Department of Economics, Rutgers University,

USAMaximo Camacho Departamento de Metodos Cuantitativos,

Universidad de Murcia, SpainMarcelle Chauvet Department of Economics, University of

California Riverside, USAMichael P Clements Department of Economics, University of

Warwick, UKChristian M Dahl Department of Economics, Purdue University,

USAJuan Jose´ Dolado Department of Economics, Universidad Carlos

III de Madrid, SpainØyvind Eitrheim Norges Bank, Norway

Walter Enders Department of Economics and Finance,

University of Alabama, USAAna Beatriz Galva˜o Ibmec Sa˜o Paulo, Brazil

James D Hamilton Department of Economics, University of

California San Diego, USAStan Hurn School of Economics and Finance, Queensland

University of Technology, AustraliaGeorge Kapetanios Department of Economics, Queen Mary,

University of London, UKMehtap Kesriyeli Central Bank of Turkey, Turkey

xxiii

Gary Koop Department of Economics, University of

Strathclyde, UKSiem Jan Koopman Department of Econometrics, Vrije Universiteit

Amsterdam, The NetherlandsTamer Kulaksızog˘lu Department of Economics, Purdue University,

USAKai Ming Lee Department of Econometrics, Vrije Universiteit

Amsterdam, The NetherlandsChin Nam Low Department of Econometrics and Business

Statistics, Monash University, AustraliaRamo´n Marı´a-Dolores Universidad de Murcia, Departament of

Economic Analysis, SpainMassimiliano Marcellino IGIER-Universita` Bocconi, Italy

James Morley Department of Economics, Washington

University at St Louis, USADenise R Osborn Centre for Growth and Business Cycle Research,

Economics, School of Social Sciences, University

of Manchester, UKGabriel Perez-Quiros Dpto Politica Economica del Pte del Gobierno,

Madrid, SpainJeremy Piger Federal Reserve Bank of Saint Louis, USASimon Potter Federal Reserve Bank of New York, USALucio Sarno Warwick Business School, University of Warwick,

UKMarianne Sensier Centre for Growth and Business Cycle Research,

Economics, School of Social Sciences, University

of Manchester, UKNorman R Swanson Department of Economics, Rutgers University,

USAElias Tzavalis Department of Economics, Queen Mary,

University of London, UKSoon Yip Wong Department of Econometrics, Vrije Universiteit

Amsterdam, The Netherlands

xxiv

CHAPTER 1

Dating Business Cycle Turning Points

Marcelle Chauvet and James D Hamilton

of GDP growth is reported or one extra month of the monthly indicators releasedbefore making a call of a business cycle turning point We introduce two newmeasures for dating business cycle turning points, which we call the ‘‘quarterly real-time GDP-based recession probability index’’ and the ‘‘monthly real-time multiple-indicator recession probability index’’ that incorporate these principles Bothindexes perform quite well in simulation with real-time data bases We also discusssome of the potential complicating factors one might want to consider for such ananalysis, such as the reduced volatility of output growth rates since 1984 and thechanging cyclical behavior of employment Although such refinements can improvethe inference, we nevertheless recommend the simpler specifications which performvery well historically and may be more robust for recognizing future business cycleturning points of unknown character

JEL classifications: E32

1 Introduction

The National Bureau of Economic Research (NBER) is a private researchorganization that, among other activities, identifies dates at which the U.S.would be said to be experiencing an economic recession These dates, reported at

http://www.nber.org/cycles/cyclesmain.html, are regarded as authoritative byboth academic researchers and the public at large For example, in July 2003, theNBER announced that the most recent recession had finally ended Remarkably,what the NBER announced in July 2003 was that the recession had actuallyended in November 2001 There had been a similar two-year delay in theprevious recession, for which the NBER announced in December 1992 that therecession had ended in March 1991

These quasi-official dates are the outcome of discussions of the NBER’sBusiness Cycle Dating Committee, a group of highly respected academics whoreview a variety of economic indicators to form a qualitative judgment about thestate of the economy The delays are explained by the fact that the Committeewants to be quite confident about its assessment before making a publicdeclaration There is nevertheless a cost to this accuracy, in that many members

of the public can continue to believe that the economy is in a recession long after

a solid recovery is under way For example, in the 1992 election, the oppositionparty declared that the U.S was experiencing the worst economic downturnsince the Great Depression A look at most of the facts would lead one todismiss this claim as political hyperbole However, if it had been the case that therecession beginning in July 1990 was still persisting as of November 1992, asone might have legitimately inferred from the failure of the NBER to announcethe recession as over, it indeed would have qualified as the longest economicdownturn since the Depression More recently, the widespread belief by theAmerican public that the U.S was still in recession in 2003 may have played arole in tax cuts approved by the U.S Congress, the outcome of a special electionfor the governor of California, and a host of other policy and planning decisions

by government bodies, private firms, and individual households

During the last decade, academic researchers have come to treat the question

of whether the economy is experiencing a recession as a formal statistical issuerather than a subjective qualitative assessment This approach started with

Hamilton (1989)and has since been adopted in hundreds of academic studies.1Given the importance to the public at large of identifying where the economy is

at any given point in time, it seems worthwhile to investigate whether theseformal quantitative methods could be used to produce announcements thatmight be useful to the public in real time The purpose of this chapter is to review

the performance of several such methods We begin in Section 2 with abackground discussion of this approach in a very simple application that usesonly data on U.S real Gross Domestic Product (GDP) growth and minimaldistributional assumptions In Section 3, we implement a parametric version ofthis approach to GDP data Section 4 describes a method for combining theinference from a number of different economic indicators.2Section 5 presentsresults from such multivariate inference, while Section 6 explores the robustness

of these multivariate inferences to several alternative specifications.3

2 What can we infer from U.S GDP growth rates?

Figure 1plots quarterly growth rates (quoted at an annual rate) of U.S realGDP since 1947, with dates of economic recessions as determined by the NBERindicated with shaded regions Consider what we can say from this GDP dataalone about the broad properties of NBER’s classifications Forty-five of the 229quarters between 1947:II and 2004:II were classified as ‘‘recession’’ and theremaining 184 as ‘‘expansion.’’ First consider the 45 recession quarters asrepresentatives of a certain population, namely, what GDP growth looks likewhen the economy is in recession The average quarterly growth rate in recession

is 1.23% (expressed at an annual rate), with a standard deviation of 3.55 Thetop panel of Figure 2 plots a nonparametric kernel estimate of the density ofthese 45 quarters.4One is more likely to see GDP falling than rising during arecession, but this is by no means certain; in fact, 15 of the 45 recession quartersare associated with positive GDP growth

The bottom panel ofFigure 2 plots the corresponding density for the 184postwar quarters classified as economic expansion These are characterized by amean annualized growth rate of 4.49% with a standard deviation of 3.24 Thisdistribution is overwhelmingly dominated by positive growth rates, though thereagain is some small probability of observing a negative growth rate during what

is considered to be an economic expansion

If one simply selects a postwar quarterly growth rate at random, there is a 20%probability it would be one of the 45 quarters classified as a recession and an 80%probability of falling in an expansion The unconditional distribution of GDPgrowth rates can be viewed as a mixture of the two distributions inFigure 2 This

2 More specifically, we use a dynamic factor model with regime switching, as in Chauvet (1998) , which is a nonlinear state space model This class of models is very popular in several fields Some of the important work in this area includes Gordon and Smith (1990) , Carlin et al (1992) , Kitagawa (1987) , Fridman and Harris (1998) , Kim and Nelson (1999a) , Durbin and Koopman (1997) , among others.

mixture is represented in the top panel ofFigure 3, in which the height of the dashed line is found by multiplying the height of the top panel ofFigure 2by 0.2.The short-dashed line represents 0.8 times the bottom curve ofFigure 2 The sum

long-of these two curves (the solid line in the top panel long-of Figure 3) represents theunconditional density of one quarter’s growth rate without knowing whether ornot the quarter would be classified as recession

From the top panel ofFigure 3, one could make an intelligent prediction as

to what classification NBER will eventually arrive at (expansion or recession)

as soon as the GDP figures are released If GDP falls by more than 6%, most

of the height of the solid line is coming from the long-dashed density,suggesting that it is overwhelmingly likely that the quarter will be classified asrecession If GDP rises by more than 6%, almost none of the density comesfrom the short-dashed line, leading us to expect NBER to classify that quarter

as expansion Intuitively, we might use the ratio of the height of the dashed line to the height of the solid line as a measure of the likelihood thatNBER would classify a quarter with GDP growth of an amount specified onthe horizontal axis as being part of a recession This ratio is plotted in thebottom panel of Figure 3

long-Using this ratio in this way is more than intuitively appealing It turns out to

be precisely an application of Bayes Law for this setting Specifically, let St ¼ 1

1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 -12

Figure 1 U.S real GDP growth rates, 1947–2004

Marcelle Chauvet and James D Hamilton 4

if the NBER ends up classifying quarter t as an expansion and St¼ 2 ifrecession Let ytdenote the quarter t GDP growth rate Then f ðytjSt¼ 2Þ is thedensity of GDP growth rates in recession, a nonparametric estimate of which isgiven by the top panel of Figure 2, while the expansion density f ðytjSt ¼ 1Þcorresponds to the bottom panel Let Pr(St¼ 2) ¼ 0.20 be the probability thatany given quarter is classified as recession Bayes Law states that the probabilitythat NBER will declare a recession given that the GDP growth for the quarter isknown to be ytcan be calculated from

PrðSt¼ 2jytÞ ¼ f ðytjSt ¼ 2ÞPrðSt ¼ 2Þ

f ðytjSt¼ 1ÞPrðSt¼ 1Þ þ f ðytjSt¼ 2ÞPrðSt¼ 2Þ: ð1ÞBut f ðytjSt ¼ 2ÞPrðSt ¼ 2Þ is simply the height of the long-dashed line inFigure

3, while f ðytjSt¼ 1ÞPrðSt¼ 1Þ is the height of the short-dashed line Hence, theratio plotted in the bottom panel ofFigure 3,

recessions Bottom panel: density for expansions

is indeed the optimal prediction PrðSt ¼ 2jytÞ about what NBER will declare ifthe quarter’s GDP growth is yt.

Predicting NBER’s declaration if we get growth rates as extreme as 76% isobviously quite robust and sensible Unfortunately, it is not particularly useful,since the vast majority of GDP growth rates are not this extreme, and for typicaldata the prediction about what NBER will declare in the bottom panel ofFigure

3is not very precise Fortunately, there is another piece of information about theNBER’s classifications that can be extremely helpful here, which is the fact thatthe Committee usually makes the same declaration in t+1 that it made in t Ofthe 45 quarters characterized as recession, 35 or 78% were followed by anotherquarter of recession Of the 183 expansion quarters between 1947:II and 2004:I,

173 or 95% were followed by another quarter of expansion

Suppose we observe a particular GDP growth rate for quarter t of yt; perhapsthis is a value like yt¼ 6, which we are reasonably confident will be described

as a recession Given this information, the probability that next quarter t+1

specified value for GDP growth

Marcelle Chauvet and James D Hamilton 6

will also be classified as a recession is no longer 0.20 but is much higher.Specifically,

of the bottom panel of Figure 3, but instead from a mixture whoserecession probability is 0.71 rather than 0.20, that is, Equation (1) would bereplaced with

PrðStþ1¼ 2jytþ1; ytÞ ¼ Pf ðy2 tþ1jStþ1¼ 2; ytÞPrðStþ1¼ 2jytÞ

j¼1f ðytþ1jStþ1¼ j; ytÞ PrðStþ1¼ jjytÞ

¼ 0:71 f ðytþ1jStþ1¼ 2; ytÞ0:29 f ðytþ1jStþ1¼ 1; ytÞ þ 0:71 f ðytþ1jStþ1¼ 2; ytÞ

ð2Þ

If we assume that recessions are the only source of GDP dynamics, so that

f ðytþ1jstþ1; ytÞ ¼ f ðytþ1jstþ1Þ, we could again use the height of the top panel of

Figure 2at the given value of yt+1as our estimate of f ðytþ1jStþ1¼ 2; ytÞ, in whichcase we just replace the mixture in the top panel ofFigure 3 (which assumed a20% weight on the recession density and 80% on the expansion density), with amixture that puts 71% weight on the recession density and 29% on the expansiondensity, as in the top panel ofFigure 4 The ratio of the height of the long-dashedcurve to the solid curve in the top panel ofFigure 4gives inference (2), plotted inthe bottom panel ofFigure 4 If we were reasonably confident that quarter t was arecession, we are much more prone to call t+1 a recession as well

Another perspective on this form of inference is obtained as follows Supposethat GDP growth for quarter t is given by yt¼ y, from which we calculatePrðSt¼ 2jyt ¼ yÞ as in the bottom panel of Figure 3 We can then use thismagnitude PrðSt¼ 2jyt¼ yÞ in the place of constant 0.20 to weight the recessiondistribution The ratio of the heights of the recession curve to the combineddistribution would then correspond to PrðSt¼ 2jyt ¼ y; yt¼ yÞ, that is, it is theprobability of recession if we happened to observe GDP growth equal to y fortwo quarters in a row This quantity is plotted in the bottom panel ofFigure 5,which is substantially steeper than the plot of PrðStþ1¼ 2jytþ1¼ yÞ shown in thetop panel For example, if we had only a single quarter’s observation of GDP,

we would not have 50% confidence in predicting a recession unless GDP growthwas below 3.4% By contrast, two consecutive quarters GDP growth of1.8% would also give us 50% confidence that the economy had entered arecession.

We could use the same principle to get a better picture of whether theeconomy was in a recession in quarter t once we know the economic growth rate

in quarter t+1 Specifically, we first make a prediction about both Stand St+1based on ytalone

t+1

Marcelle Chauvet and James D Hamilton 8

desired inference about the economy at date t based on information observed atdate t+1 is then

Pr S
t¼ ijytþ1; yt

 ¼X2 j¼1

Pr S
tþ1¼ j; St ¼ ijytþ1; yt

We have thus seen how, given nonparametric knowledge of how the distribution

of GDP growth is different between expansions and contractions,

at hand, since it appears fromFigure 2that a Gaussian distribution works quite

5 In the parametric application of this approach described in the next section, we tested this assumption by using several alternative specifications of the Markov switching model, including higher autoregressive processes or allowing the variance and mean to follow the same or distinct Markov processes We find that the simplest representation describes the data quite well and is most robust on a recursive sample of real-time data.

Marcelle Chauvet and James D Hamilton 10

well to describe these densities The fact that the recession distribution has astandard deviation very similar to that for the expansion distribution impliesthat we would also lose little by assuming that the two distributions differ only intheir means and share the same standard deviation s The suggestion is then that

we replace the arbitrary density f(yt|St¼ 2) in the top panel ofFigure 2with theN(m2, s2) distribution,

p11 ¼ Pr Sð tþ1¼ 1jSt¼ 1Þ,

and p22the analogous probability for recessions

p22 ¼ Pr Sð tþ1¼ 2jSt¼ 2Þ

Again, the historical experience would lead us to expect that p11¼ 0.95 and

p22¼ 0.78 Let h ¼ (m1,m2, s, p11, p22)0denote the various unknown parameters

A two-state Markov chain with transition probabilities piihas unconditionaldistribution given by6

PrðSt¼ 2Þ ¼2 1 p11

p11 p22¼ p2.The likelihood of the first observation in the sample (ytfor t ¼ 1) is then given bythe mixture

f ðy1;hÞ ¼X

2

i¼1

piffiffiffiffiffiffiffiffi2ps

Pr S
1¼ ijy1;h

 ¼ f y
1;h 1 pi

ffiffiffiffiffiffiffiffi2ps

See, for example, Hamilton (1994, p 683)

probability for the second observation of

Pr S
2¼ jjy1;h

 ¼X2 i¼1

1ffiffiffiffiffiffiffiffi2ps

or the kind of calculation that produced the solid curve in the top panel of

Figure 4 From this, we obtain as in the bottom panel ofFigure 4the filteredprobabilities for the second observation

Pr S
2¼ ijy2; y1;h

 ¼ f y
2jy1;h 1 1

ffiffiffiffiffiffiffiffi2ps

log f y
1;h þ XT

t¼2log f y
tjyt 1; yt 2; ; y1;h ð11Þ

We motivated this way of thinking about the data by taking the NBER’sconclusions as given and trying to characterize what the NBER has done.7However, no aspect of the NBER’s dating appears in the final result (11), which issolely a function of observed GDP growth rates and the unknown parameters h.One could accordingly choose as an estimate of h the value that maximizes the

7 An alternative approach developed by Bry and Boschan (1971) attempts to formalize and elaborate

on the rule of thumb that two quarters of falling GDP constitute a recession However, this rule of thumb does not describe the decisions of the NBER Business Cycle Dating Committee, which defines

a recession as ‘‘a significant decline in economic activity spread across the economy, lasting more than a few months, normally visible in real GDP, real income, employment, industrial production, and wholesale–retail sales’’ ( http://www.nber.org/cycles.html/ ) We view our approach, unlike Bry and Boschan, as a direct statistical formalization of the NBER’s stated method for qualitative evaluation.

Marcelle Chauvet and James D Hamilton 12

sample log likelihood of GDP growth rates (11) This maximum likelihoodestimate is compared with the values we would have expected on the basis of theNBER inferences inTable 1.8The two sets of parameter values, although arrived

at by different methods, are remarkably similar This similarity is veryencouraging for two different reasons First, it enhances the intellectuallegitimacy of the perspective that the economy can be classified as being in anexpansion or recession at any point in time, and that whether or not the economy

is in recession can account for much of the variability and serial dependence ofGDP growth rates We did not impose any kind of conditions on the two means

m1and m2, and one could imagine the data being better described by all sorts ofchoices, such as ‘‘very rapid growth’’ versus ‘‘normal growth,’’ or ‘‘normalgrowth’’ versus ‘‘slow growth.’’Table 1implies that, using just GDP data alonewithout any reference to what NBER may have said, we would come up with avery similar conceptual scheme to the one that economists and the NBER havetraditionally relied on

A second reason that the correspondence between the two columns inTable 1

is encouraging is that it raises the promise that we might be able to use GDPgrowth rates alone to arrive at classifications in a more timely and objectivefashion than the NBER The top panel ofFigure 6plots the filtered recessionprobabilities PrðSt¼ 2jyt; yt 1; ; y1; ^hÞ implied by the maximum likelihoodestimate of the parameter vector h For any date t, this is the probability that theeconomy is in recession based on observations of GDP growth rates at the time.The dates of economic recessions as determined after the fact by NBER areindicated by shaded regions on the graph It seems clear that the twomethodologies are identifying the same series of events over the postwar period,with the filter probabilities rising above 75% at some point during every postwarrecession and typically remaining below 30% in times of expansions There aresome minor differences, with the two consecutive quarters of falling GDP in1947:II–III and the 1.9% growth in 1956:I temporarily pushing the filter

Table 1 Parameter estimates based on (1) characteristics of expansions andrecessions as classified by NBER, and (2) values that maximize the observed

sample log likelihood of postwar GDP growth rates

8

Maximum likelihood estimates were found using the EM algorithm described in Hamilton (1990)

probabilities a little over 50% in episodes that the NBER did not characterize asrecessions Also, in the 1990–1991 recession, the filter probabilities did not comeback below 50% until 1991:IV, although the NBER says that the recessionended in 1991:I Overall, though, the correspondence seems quite strong.The bottom panel ofFigure 6plots the smoothed probabilities, for which thefull sample of observations through 2004:II was used to form an inference aboutthe state of the economy at any given date Using the full sample substantiallysmooths out a number of the minor temporary blips evident in the filter

Current filter probabilities and NBER recessions

1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 -0.25

NBER up to two years after the fact

Marcelle Chauvet and James D Hamilton 14

estimates, and brings the 1947 and 1956 inferences just under 50%, ever soslightly favoring the NBER final call Dates at which recessions began and endedaccording to the NBER are compared with the dates for which the smoothedprobabilities are above 50% in Table 2 The smoothed probabilities date the

1980 recession as beginning three quarters earlier than the date assigned bythe NBER The two methods never differ by more than a quarter for either thestarting date or ending date for any other recession

This suggests that using a mechanical algorithm to identify business cycleturning points holds considerable promise However, even the filter probabilities

in the top panel ofFigure 6 do not accurately capture the predictions that onecould actually make with this framework in real time, for two reasons First, thecomplete sample of data through 2004 was used to estimate the values of theparameter vector h This perhaps is not an overwhelming concern, since, as wesaw in Table 1, one would have arrived at very similar magnitudes for h justbased on the properties that one expects expansions and recessions should have.The second, more serious, problem is that the GDP figures as originally released

by the Bureau of Economic Analysis can differ substantially from the historicalseries now available Croushore and Stark (2003) have established that thesecond issue can be extremely important in practice, and have helped develop anextensive data set archived at the Federal Reserve Bank of Philadelphia(available athttp://www.phil.frb.org/econ/forecast/reaindex.html) This data setincludes the history of GDP values that would have actually been available to aresearcher or forecaster at any given point in time The database consists of oneset of GDP levels for 1947:I–1965:III that would have been reported as of themiddle of 1965:IV, a second set of GDP levels for 1947:I–1965:IV reported as ofthe middle of 1966:I, and so on, ending with a data set of GDP levels from1947:I to 2004:II as reported in the middle of 2004:III, with the latter data setbeing the one on whichFigure 6was based There are a few gaps in this series,

Table 2 Dates of recessions as determined by (1) NBER and (2) properties of

GDP growth alone