Nonlinear time series analysis of business cycles volume 276 (contributions to economic analysis)
NONLINEAR TIME SERIES ANALYSIS OF BUSINESS CYCLES
CONTRIBUTIONS TO ECONOMIC ANALYSIS 276
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NONLINEAR TIME SERIES ANALYSIS OF BUSINESS CYCLES
Costas Milas Department of Economics, Keele University, UK Philip Rothman Department of Economics, East Carolina University, USA Dick van Dijk Econometric Institute, Erasmus University Rotterdam, The Netherlands
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Dedication 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INTRODUCTION TO THE SERIES This series consists of a number of hitherto unpublished studies, which are introduced by the editors in the belief that they represent fresh contributions to economic science.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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Introduction The notion of business cycle nonlinearity goes back a long time. For example, Mitchell (1927) and Keynes (1936) suggested that business cycles display asymmetric behavior in the sense that recessions are shorter and more volatile than expansions. Similarly, Hicks (1950) noted that business cycle troughs are sharper than peaks. Further, Friedman (1964) proposed his ‘‘plucking model’’ of economic fluctuations based upon the observation of asymmetry in correlations between successive phases of the business cycle, in the sense that the amplitude of a contraction is strongly correlated with the strength of the subsequent expansion, while the amplitude of an expansion is uncorrelated with the amplitude of the following contraction. Neftc- i (1984) initiated the modern econometric literature on business cycle nonlinearity with his study of U.S. unemployment rates using Markov chain techniques. His results implied that the U.S. unemployment rate displays ‘‘steepness’’-type business cycle asymmetry, following the taxonomy due to Sichel (1993). Neftc- i’s paper has been highly influential and since its publication roughly 20 years ago, a great deal of research has been done exploring the magnitude and economic significance of nonlinearity in business cycle fluctuations. For example, Hamilton (1989, p. 359) argued that the now very popular Markov-switching model he introduced is a natural generalization of Neftc- i’s framework. A useful survey of many important developments in this literature can be found in Clements and Krolzig (2003). To provide a comprehensive look at current work on this topic, for this book volume we solicited original contributions on business cycle nonlinearity from leading academics and practitioners in the field. Each chapter was subsequently reviewed by an ‘‘internal’’ referee (an author or coauthor of a different chapter in the book), and by an ‘‘external’’ referee. These external referees were Don Harding (University of Melbourne), Christopher Martin (Brunel University), Marcelo Medeiros (PUC Rio), Simon van Nordon (HEC Montre´al), Richard Paap (Erasmus University Rotterdam), Jean-Yves Pitarakis (University of Southampton), Tommaso Proietti (University of Udine), Pierre Siklos (Wilfred Laurier University), Peter Summers (Texas Tech University), Timo Tera¨svirta (Stockholm School of Economics), Gilles Teyssiere (Universite´ Paris 1), Greg Tkacz (Bank of Canada), Mark Wohar (University of Nebraska at Omaha), and Eric Zivot (University of Washington). We thank both our contributors and ix
referees for their cooperation in keeping to the ambitious time schedule we set at the 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 the role of nonlinearity in dating business cycle turning points and identifying business cycle regimes. Chauvet and Hamilton provide a detailed description of the Markov-switching approach to this issue, including not only the technicalities involved but also paying ample attention to the underlying intuition. They illustrate the promise of this approach by constructing a business cycle chronology 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 findings demonstrate that the resulting turning point dates closely match those of the business cycle dating committee of the National Bureau of Economic Research (NBER), but the model-based turning points typically become available much sooner than the NBER ones. Clements and Galva˜o use the context of predicting business cycle regime probabilities and output growth in the U.S. to consider the specific issue of combining forecasts versus combining information in modeling. The simple models whose forecasts they combine each use a single recession indicator, one of the components that comprise the Conference Board Composite Leading Indicator (CLI), as the explanatory variable to the model. Combining this information set in modeling is achieved by using a model selection strategy. For predicting output growth, their findings support pooling the forecasts of the single-indicator models, whilst the results are more mixed for predicting recessions and recession probabilities. Morley and Piger consider the ability of linear autoregressive integrated moving average (ARIMA) and nonlinear Markov-switching models to reproduce 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 reproduce business cycle features such as the average growth rate in recessions, the average length of recessions, and the total number of recessions. However, Markov-switching models are found to be better than linear models at reproducing the variability of growth rates in different business cycle phases. Furthermore, only Markov-switching specifications with three regimes or with a built-in ‘‘bounceback’’ effect are able to reproduce high-growth recoveries following recessions and a strong correlation between the severity of a recession and the strength of the subsequent recovery. The second topic analyzed, in a set of two papers, is the use of multivariate nonlinear models in econometric modeling of business cycles. Koop and Potter introduce a nonlinear extension of the Vector Autoregressive (VAR) model which they call the Vector Floor and Ceiling (VFC) model. The VFC model is also a multivariate extension of univariate nonlinear models the authors developed earlier with floor and ceiling effects; see Pesaran and Potter (1997) and
Koop and Potter (2003). As a tightly restricted Threshold Autoregressive model, the authors argue that the VFC model provides a parsimonious framework for capturing 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 relationships between the variables and weaker evidence of conditional mean nonlinearity. Camacho and Perez-Quiros propose a new framework to analyze pairwise business cycle synchronization across a given set of countries. The approach is based on multivariate Markov-switching procedures, and essentially determines the relative position of two countries’ cycles in between the extreme cases of complete independence and perfect synchronization. An empirical application to the G7 countries shows that these can be divided into two groups with distinct common business cycle dynamics, with one group consisting of Euro-zone countries (France, Germany, and Italy) and the other including English-speaking countries (Canada, the U.K., and the U.S.). Five of the papers explore a third topic, the extent to which nonlinearity can account for the well-documented instability and structural change which has been observed in macroeconomic time series; see, e.g. Stock and Watson (1996). Marcellino’s paper is motivated by the many economic and political changes which have occurred in what is now called the Euro-zone since the early 1980s. Such changes, he argues, increase the difficulty of modeling macroeconomic time series for Euro-area countries with constant-parameter linear models. To explore this idea he carries out a simulated out-of-sample forecasting competition using linear, nonlinear, and time-varying models to predict the future values of 500 macroeconomic time series for these countries. It turns out that, for roughly two-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 strongly considered by practitioners. Kapetanios and Tzavalis use a new model of structural breaks, one which allows for parameter changes to be triggered by large economic shocks. In contrast to other structural break models in the literature, their approach allows them to examine such parameter changes without fixing either the number or magnitude of the breaks. The results support the view that the observed instability in U.S. macroeconomic time series is due to the oil-price shocks of the 1970s and the changes in the Fed’s operating procedures in the late 1970s and early 1980s. There are many nonparametric and model-based methods available for extracting the business cycle component from a macroeconomic time series. Koopman, Lee, and Wong use a parametric trend-cycle decomposition procedure in which the parameters governing the dynamics of these components are allowed to vary in a nonlinear but smooth manner. They find substantial evidence of smooth time variation in these parameters. Of particular interest are their results suggesting that business cycle volatility for the U.S. economy has
decreased. While these findings are consistent with results reported earlier in the literature on the ‘‘great moderation,’’ it is the first to do so within the trend-cycle decomposition framework. Becker, Enders, and Hurn develop a methodology to model a time-varying intercept. The methodology relies on a Fourier approximation, which uses trigonometric functions to capture the unknown functional form of the intercept term. Two empirical applications illustrate the use of the methodology. The first example demonstrates how a time-varying intercept can be used to capture a structural 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 instability in the cointegrating vector among M3, income, prices and interest rates disappears 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 parameters follow random walks. The random walks in the parameters capture permanent structural change within a regime-switching framework, but in contrast to existing specifications, structural change in the random walk STAR (RW-STAR) setting follows a stochastic process rather than a deterministic function of time. Using industrial production data for several countries, they find 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 be able to capture different types of time-varying behavior of parameters. The fourth topic, the importance of nonlinearity for econometric analysis of monetary policy, is addressed in three of the papers in this volume. Kesriyeli, Osborn, and Sensier estimate smooth transition monetary policy rules for the U.S., U.K., and Germany. They find significant nonlinear structure in the monetary policy rules associated with interest rate changes rather than movements 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 rate reaction functions occurring around mid-1985. Dolado and Marı´ a-Dolores examine the issue of the asymmetric effects of monetary policy shocks on output in the Euro area. Assuming a nonlinear aggregate supply curve, they derive monetary policy shocks as the residuals from a nonlinear interest rate reaction function. The authors proceed by estimating a multivariate Markov-switching model for EU output and find that monetary policy shocks have a greater effect on output in recessions. Akram, Eitrheim, and Sarno adopt a different nonlinear model but reach similar conclusions on the effects of monetary policy on output. The authors use multivariate smooth transition models to characterize the behavior of output, money, and the real exchange rate in Norway over a period of almost two centuries. They find evidence of asymmetric effects of monetary policy on output. In particular, large contractionary monetary policy shocks tend to have significant effects on output, while small expansionary monetary policy shocks tend to have negligible effects on output.
Finally, two of the papers study the statistical and economic impact of allowing for business cycle regime-dependent behavior in models of important macroeconomic and financial time series. Bhardwaj and Swanson compare the ability of fractional ARIMA (ARFIMA), non-ARFIMA, and other nonlinear models to forecast U.S. daily stock returns in recessions versus expansions and for larger versus smaller samples. The findings of their paper suggest that ARFIMA models do not predict better or worse than any other model across the business cycle. On the other hand, the forecasting ability of ARFIMA models increases with larger samples. Dahl and Kulaksızog˘lu use a nonlinear autoregressive distributed lag model to study the relationship between housing completions and housing starts in the U.S. economy. Their results suggest that builders change the speed of construction depending upon whether the home construction industry is in a recession or expansion. In particular, the mean lag between housing completions and housing starts is significantly shorter in recessionary than in expansionary periods. This finding is consistent with what has been called the ‘‘accordion effect’’ in the literature; see van Alphen and Merkies (1976). References Clements, M.P. and H.-M. Krolzig (2003), ‘‘Business cycle asymmetries: characterization and testing based on Markov-switching autoregressions’’, Journal of Business and Economic Statistics, Vol. 21, pp. 196–211. Friedman, M. (1964). ‘‘Monetary studies of the National Bureau’’, in: The National Bureau Enters its 45th Year, 44th Annual Report, pp. 7–25. Reprinted in 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 nonstationary time series and the business cycle’’, Econometrica, Vol. 57, pp. 357–384. 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 threshold 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. Neftc- i, S.N. (1984), ‘‘Are economic time series asymmetric over the business cycle’’, 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 Inquiry, Vol. 31, pp. 224–236.
Stock, J.H. and M.W. Watson (1996), ‘‘Evidence on structural instability in macroeconomic time series relations’’, Journal of Business and Economic Statistics, Vol. 14, pp. 11–30. van Alphen, H.J. and A.H.Q.M. Merkies (1976), ‘‘Distributed lags in construction: an empirical study’’, International Economic Review, Vol. 17, pp. 411–430.
LIST OF CONTRIBUTORS CHAPTER 1
1. 2. 3. 4. 5. 6.
DATING BUSINESS CYCLE TURNING POINTS Marcelle Chauvet and James D. Hamilton
Introduction What can we infer from U.S. GDP growth rates? Parametric representation Using multiple indicators to identify turning points Empirical performance of the monthly recession probability index Alternative approaches to monthly inference Acknowledgements References Appendix
COMBINING PREDICTORS & COMBINING INFORMATION IN MODELLING: FORECASTING US RECESSION PROBABILITIES AND OUTPUT GROWTH Michael P. Clements and Ana Beatriz Galva˜o
Introduction Models and data 2.1. Logit models 2.2. Models of output growth 2.3. Non-linear models of output growth Out-of-sample forecasting exercise 3.1. Forecast combination schemes 3.2. Forecast evaluation 3.3. Empirical results Conclusions Acknowledgements References xv
2 3 10 22 32 48 50 51 53
55 58 58 61 62 63 63 63 64 69 70 70
1. 2. 3. 4.
Introduction An algorithm for establishing business cycle turning points Business cycle features in U.S. real GDP data Business cycle features in simulated data from time-series models 4.1. Model description and estimation 4.2. Business cycle features from linear models 4.3. Business cycle features from regime-switching models 4.4. Business cycle features and heteroskedasticity Conclusions Acknowledgements References
1. 2. 3.
THE VECTOR FLOOR AND CEILING MODEL Gary Koop and Simon Potter
Introduction A nonlinear VAR with floor and ceiling effects Empirical results 3.1. Model comparison results 3.2. A comparison of Bayesian and classical results Impulse response analysis Conclusions Acknowledgements References Appendix A: Sample information Appendix B: Bayesian analysis of the VFC model Appendix C: Classical analysis of the VFC model Appendix D: Further details on impulse response analysis
THE IMPORTANCE OF NONLINEARITY IN REPRODUCING BUSINESS CYCLE FEATURES James Morley and Jeremy Piger
A NEW FRAMEWORK TO ANALYZE BUSINESS CYCLE SYNCHRONIZATION Maximo Camacho and Gabriel Perez-Quiros
Introduction A framework to analyze business cycle synchronization 2.1. Univariate Markov-switching approach 2.2. Multivariate Markov-switching approach
75 79 81 83 83 87 88 90 92 93 93 97
97 99 105 106 109 113 118 118 119 121 122 128 131
133 135 135 136
Empirical results 3.1. Preliminary analysis of data 3.2. Comparative analysis of business cycle synchronization 3.3. Business cycle synchronization across G7 countries Conclusions Acknowledgements References
1. 2. 3.
Introduction The data Forecasting methods 3.1. Linear methods 3.2. Time-varying methods 3.3. Non-linear methods Forecast evaluation Measuring the extent of instability 5.1. Instability tests 5.2. Forecast evaluation for unstable series Forecasting industrial production, unemployment and inflation Conclusions Acknowledgements References
1. 2. 3. 4. 5.
NON-LINEARITY AND INSTABILITY IN THE EURO AREA Massimiliano Marcellino
NONLINEAR MODELLING OF AUTOREGRESSIVE STRUCTURAL BREAKS IN SOME US MACROECONOMIC SERIES George Kapetanios and Elias Tzavalis
Introduction Modelling structural breaks in autoregressive coefficients A Monte Carlo study Empirical application Conclusions Acknowledgements References Data Appendix
Introduction Trend-cycle decomposition model 2.1. Fixed parameter specification 2.2. Time-varying parameter specification State space representation Empirical evidence from U.S. economic time series 4.1. Data 4.2. Basic decompositions 4.3. Smooth transitions over time Discussion and conclusion References Appendix
3. 4. 5.
TREND-CYCLE DECOMPOSITION MODELS WITH SMOOTH-TRANSITION PARAMETERS: EVIDENCE FROM U.S. ECONOMIC TIME SERIES Siem Jan Koopman, Kai Ming Lee and Soon Yip Wong
MODELING INFLATION AND MONEY DEMAND USING A FOURIER-SERIES APPROXIMATION Ralf Becker, Walter Enders and Stan Hurn
Introduction Modeling with a Fourier approximation 2.1. Dependent error structures 2.2. Power A structural break in the inflation rate Selecting the optimal number of terms in the Fourier expansion Structural breaks in the demand for money 5.1. The bootstrap 5.2. The error-correction model 5.3. The restricted model 5.4. Integer frequencies 5.5. Missing variables Conclusions Acknowledgements References
Introduction The RW-STAR Model 2.1. The model Modelling procedure 3.1. Performance of the nonlinearity tests Modelling industrial production of selected OECD countries 4.1. The data 4.2. Linearity tests 4.3. Development of baseline models 4.4. Estimation of RW-STAR models 4.5. Forecast performance Conclusions Acknowledgements References Appendix : DGPs for the power simulations
RANDOM WALK SMOOTH TRANSITION AUTOREGRESSIVE MODELS Heather M. Anderson and Chin Nam Low
NONLINEARITY AND STRUCTURAL CHANGE IN INTEREST RATE REACTION FUNCTIONS FOR THE US, UK AND GERMANY Mehtap Kesriyeli, Denise R. Osborn and Marianne Sensier
Introduction Interest rate models 2.1. The models 2.2. Selection of explanatory and transition variables 2.3. Sample periods and data Results 3.1. Linear models 3.2. Nonlinear models Concluding remarks Acknowledgements References Appendix : Modelling methodology and additional results
Introduction Related literature Estimation of a monetary policy reaction function Markov Switching Models for real output growth 4.1. Extended Markov Switching model including interest-rate shocks Effects of monetary policy on state switches Conclusions Acknowledgements References
3. 4. 5.
STATE ASYMMETRIES IN THE EFFECTS OF MONETARY POLICY SHOCKS ON OUTPUT: SOME NEW EVIDENCE FOR THE EURO-AREA Juan J. Dolado and Ramo´n Marı´a-Dolores
NON-LINEAR DYNAMICS IN OUTPUT, REAL EXCHANGE RATES AND REAL MONEY BALANCES: NORWAY, 1830–2003 Q. Farooq Akram, Øyvind Eitrheim and Lucio Sarno
Introduction STR models 2.1. Testing for non-linearity and its form 2.2. Evaluation of STR models Data and its properties Multivariate linear models 4.1. Linear dynamic models Non-linear conditional models 5.1. STR models of output, the real exchange rate and real money 5.2. The STR models 5.3. LSTR model of output 5.4. STR model of the real exchange rate 5.5. LSTR model of real money 5.6. Dynamics of the linear versus the non-linear systems of equations Concluding remarks Acknowledgements References Appendix : Data