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

NONLINEAR TIME SERIES ANALYSIS OF BUSINESS CYCLES i CONTRIBUTIONS TO ECONOMIC ANALYSIS 276 Honorary Editors: D.W JORGENSON J TINBERGENy Editors: B Baltagi E Sadka D Wildasin Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris San Diego – San Francisco – Singapore – Sydney – Tokyo ii 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 Amsterdam – Boston – Heidelberg – London – New York – Oxford – Paris San Diego – San Francisco – Singapore – Sydney – Tokyo iii ELSEVIER B.V Radarweg 29 P.O Box 211 1000 AE Amsterdam The Netherlands ELSEVIER Inc 525 B Street, Suite 1900 San Diego CA 92101-4495 USA ELSEVIER Ltd The Boulevard, Langford Lane, Kidlington Oxford OX5 1GB UK ELSEVIER Ltd 84 Theobalds Road London WC1X 8RR UK r 2006 Elsevier B.V All rights reserved This work 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together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org iv 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Rothman v This page intentionally left blank vi INTRODUCTION TO THE SERIES This series consists of a number of hitherto unpublished studies, which are introduced by the editors in the belief that they represent fresh contributions to economic science The term ‘economic analysis’ as used in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 vii This page intentionally left blank viii 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 Teraăsvirta (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 396 Geetesh Bhardwaj and Norman R Swanson business-cycle turning points, we follow the chronology of the US business cycle as reported by National Bureau of Economic Research.13 To analyze the effect of business cycles on the forecasting performance of these models, we wanted to club together the data for all the recession (expansions) periods In order to justify doing so we carried out a small experiment, where we fitted simple linear autoregressive models with dummy variables for all the recession (expansion) periods The dummy variables for pre-World War II periods turned out to be significant, especially for the 1929 great depression and the subsequent recovery Given these results, we decided to also divide the data into pre- and post-World War II periods In an attempt to classify major recent global developments, two further periods were also considered The first period starts with the world oil shock of 1973 that began on October 17, 1973, when Arab members of the Organization of Petroleum Exporting Countries (OPEC), in the midst of the Yom Kippur War, announced that they would no longer ship petroleum to nations that had supported Israel in its conflict with Egypt and Syria; i.e to the United States and its allies in Western Europe At around the same time, OPEC member states agreed to use their leverage over the world price-setting mechanism for oil to quadruple world oil prices This period ended with a rapid decline in oil prices early in 1982, when OPEC appeared to lose control over world oil prices Finally, OPEC agrees to individual output quotas and cuts prices by $5 The other period considered is the most recent period starting in 1982 This period covers the two very long episodes of expansion in the 1980s and 1990s, the stock market crash of 1987, and the recent recession of 2001 Results are reported in Tables 2–7 It turns out that the single most important factor that affects the performance of these models, especially the ARFIMA models, is the sample size Based on the performance of long memory models, we can divide our six data groupings into two categories, i.e small sample size, and moderate or large sample size In the small sample size category, we have the pre-World War II recession (1387 data points), the pre-World War II expansion (1707 data points) the postWorld War II recession (2401 data points), and the 1973–1982 period of oil shocks (2273 data points) In the other category, we have the post-World War II expansion (12,141 data points) and the most recent data in the post 1982 period (5369 data points) The first thing to note is that while for the longer datasets, four estimates of d yield quite similar estimates, we have huge variations when the sample size is small For the smaller sample sizes, and based on the use of DM and encompassing tests, there is little to choose between ARFIMA and non-ARFIMA models, most of the time For example, for the pre-World War II expansion and the 13 For further details on actual dates and methodology see http://www.nber.org/cycles.html/ Predictive Comparison of Long- and Short Memory Models of Daily U.S Stock Returns Table 397 Analysis of U.S S&P500 daily absolute returns, pre-WWII recession NonARFIMA Model DM1 ENC-t1 0.62 (0.0028) ARMA (1,1) À0.24 1.00 RR (1,1) 0.21 (0.0001) ARMA (1,1) À0.55 RR (1,1) 0.21 (0.0001) ARMA (1,1) RR (1,1) 0.21 (0.0002) RR (1,1) WHI (1,1) Estimation Scheme and Forecast Horizon ARFIMA Model d day ahead, recursive day ahead, recursive 20 day ahead, recursive day ahead, rolling day ahead, rolling 20 day ahead, rolling GPH (1,1) STAR Model DM2 ENC-t2 LSTAR (4) À1.36 0.46 1.34 ESTAR (4) À0.86 1.12 À0.26 1.40 LSTAR (4) À0.64 0.81 ARMA (1,1) À0.50 1.33 LSTAR (4) À1.54 0.05 0.21 (0.0002) ARMA (1,1) À0.01 1.96 LSTAR (4) À1.18 0.80 0.62 (0.0397) MA (4) À1.40 1.46 LSTAR (4) À1.27 1.47 Note: See notes in Tables 1a Data for this table correspond to the two pre-World War II recessions from August 1929 to March 1933, and from May 1937 to June 1938 We have a total of 1387 data points Table Analysis of U.S S&P500 daily absolute returns, pre-WWII expansion ENC-t1 DM2 ENC-t2 LSTAR (2) À4.28 À2.24 1.86 LSTAR (2) À4.12 À2.00 À0.89 1.70 LSTAR (2) À3.86 À0.38 ARMA (1,1) 0.57 1.83 LSTAR (2) À4.86 À2.62 0.21 (0.0021) ARMA (1,1) 0.95 2.43 LSTAR (2) À4.85 À3.54 0.62 (0.0004) ARMA (1,1) À3.63 À0.81 LSTAR (2) À4.05 À0.81 NonARFIMA Model DM1 0.25 (0.0001) ARMA (1,1) 1.10 1.91 RR (1,1) 0.25 (0.0001) ARMA (1,1) 0.56 WHI (1,1) 0.58 (0.0005) ARMA (1,1) RR (1,1) 0.21 (0.0021) RR (1,1) WHI (1,1) Estimation Scheme and Forecast Horizon ARFIMA Model d day ahead, recursive day ahead, recursive 20 day ahead, recursive day ahead, rolling day ahead, rolling 20 day ahead, rolling RR (1,1) STAR Model Note: See notes in Tables 1a Data for this table correspond to the pre-World War II expansion period.We have a total of 1707 data points pre-World War II recession, in all but two cases DM test statistics fails to significantly distinguish between ARFIMA and non-AFIMA models However, based on point MSFE, ARFIMA models seem to outperform non-ARFIMA models more than half of the time Note, however, that for one of the smaller samples (i.e the pre-World War II expansion), the non-ARFIMA model outperforms the ARFIMA model based on point MSFE for all the cases expect the longest horizon forecasting (i.e 20 days ahead) With regard to the STAR models, they are clearly outperformed by the ARFIMA models for the larger sample sizes, although based on the smaller sample sizes there is little difference 398 Table Geetesh Bhardwaj and Norman R Swanson Analysis of U.S S&P500 daily absolute returns, post-WWII recession NonARFIMA Model DM1 0.49 (0.0001) ARMA (1,1) À0.63 0.65 WHI (2,2) 0.49 (0.0001) ARMA (1,1) À2.07 WHI (2,2) 0.49 (0.0001) ARMA (1,1) WHI (2,2) 0.54 (0.0008) WHI (2,2) RR (1,1) Estimation Scheme and Forecast Horizon ARFIMA Model d day ahead, recursive day ahead, recursive 20 day ahead, recursive day ahead, rolling day ahead, rolling 20 day ahead, rolling WHI (2,2) ENC-t1 STAR Model DM2 ENC-t2 LSTAR (2) À4.90 À0.04 À0.82 LSTAR (2) À3.17 1.68 À3.27 À1.46 LSTAR (2) À3.20 0.76 ARMA (1,1) 1.11 2.07 LSTAR (2) À4.37 0.86 0.54 (0.0008) ARMA (1,1) À0.09 1.25 LSTAR (2) À3.98 1.04 0.21 (0.0015) ARMA (1,1) À0.60 1.27 LSTAR (2) À3.05 À1.89 Note: See notes in Tables 1a Data for this table correspond to the post-World War II recessions We have a total of 2401 data points Table Analysis of U.S S&P500 daily absolute returns, post-WWII expansion NonARFIMA Model DM1 0.38 (0.0002) ARMA (1,1) À1.04 0.50 WHI (1,1) 0.38 (0.0002) ARMA (1,1) À3.09 WHI (1,1) 0.38 (0.0002) ARMA (1,1) WHI (1,1) 0.40 (0.0009) WHI (1,1) WHI (1,1) Estimation Scheme and Forecast Horizon ARFIMA Model d day ahead, recursive day ahead, recursive 20 day ahead, recursive day ahead, rolling day ahead, rolling 20 day ahead, rolling WHI (1,1) ENC-t1 STAR Model DM2 ENC-t2 ESTAR (2) À7.69 À0.29 À1.09 ESTAR (2) À5.19 À0.38 À5.49 À1.21 LSTAR (2) À5.87 À1.28 ARMA (1,1) À0.34 1.01 LSTAR (2) À7.45 À0.23 0.40 (0.0009) ARMA (1,1) À2.31 À0.48 ESTAR (2) À4.69 0.35 0.40 (0.0009) ARMA (1,1) À4.57 À0.62 ESTAR (2) À5.44 À0.40 Note: See notes in Tables 1a Data for this table correspond to the post-World War II expansion We have a total of 12,141 data points between their respective forecasting performances It should be further noted that as reported in Table 7, for the post-1982 period and for one day ahead forecasts, non-ARFIMA models clearly have lower point MSFE compared to ARFIMA models Since one day ahead forecasts are important to practitioners, this is a notewothy observation.14 14 We thank the editors for pointing this out Predictive Comparison of Long- and Short Memory Models of Daily U.S Stock Returns Table 399 Analysis of U.S S&P500 daily absolute returns, the period of oil shocks 1973–1982 NonARFIMA Model DM1 ENC-t1 0.51 (0.0021) ARMA (2,2) À0.92 0.84 WHI (1,1) 0.51 (0.0021) ARMA (2,2) À0.74 WHI (1,1) 0.51 (0.0021) ARMA (2,2) WHI (1,1) 0.50 (0.0030) WHI (1,1) WHI (1,1) Estimation Scheme and Forecast Horizon ARFIMA Model d day ahead, recursive day ahead, recursive 20 day ahead, recursive day ahead, rolling day ahead, rolling 20 day ahead, rolling WHI (1,1) STAR Model DM2 ENC-t2 ESTAR (2) À2.82 1.13 0.94 ESTAR (2) À2.90 1.28 À0.52 1.26 LSTAR (2) À1.38 1.40 ARMA (2,2) À1.71 0.12 LSTAR (2) À3.60 0.61 0.50 (0.0030) ARMA (2,2) À0.77 0.97 ESTAR (2) À3.78 0.47 0.50 (0.0030) ARMA (2,2) À1.38 0.07 ESTAR (2) À2.37 0.91 Note: See notes in Tables 1a Data for this table correspond to the period of oil shocks The starting date is October 17, 1973 when Arab members of OPEC restricted shipment of petroleum While the ending data for this period corresponds to the 1982 reduction in oil prices by OPEC We have a total of 2273 data points Table Analysis of post-1982 S&P500 daily absolute returns ENC-t1 DM2 ENC-t2 LSTAR (3) À4.23 À0.01 1.67 LSTAR (3) À4.29 À0.35 À2.80 0.32 LSTAR (3) À4.20 0.44 ARMA (1,1) 0.58 3.18 ESTAR (3) À6.41 À2.30 0.24 (0.0001) ARMA (1,1) À0.51 2.18 LSTAR (3) À5.41 À3.02 0.46 (0.0003) ARMA (1,1) À4.37 À2.51 ESTAR (3) À5.01 0.40 NonARFIMA Model DM1 0.21 (0.0004) ARMA (1,1) 1.77 3.76 GPH (2,3) 0.64 (0.0009) ARMA (1,1) À1.29 GPH (2,3) 0.64 (0.0009) ARMA (1,1) RR (2,2) 0.24 (0.0001) RR (2,2) WHI (1,2) Estimation Scheme and Forecast Horizon ARFIMA Model d day ahead, recursive day ahead, recursive 20 day ahead, recursive day ahead, rolling day ahead, rolling 20 day ahead, rolling RR (2,2) STAR Model Note: See notes in Tables 1a Data for this table correspond to the post-1982 period We have a total of 5369 data points Of further note it is clear that when the sample size increases, ARFIMA models significantly outperform the STAR model Finally, though the motivation for the data groupings was to capture the possible effects of the business cycle on model performance, what we have found is that the most important factor seems to be the sample size and that ARFIMA models clearly improve their performance as the sample size becomes large; suggesting, at least in part, the importance of estimating d as precisely as possible when constructing ARFIMA-based prediction models 400 Geetesh Bhardwaj and Norman R Swanson Concluding remarks We have presented the results of an empirical study of the usefulness of ARFIMA models in a practical prediction-based application where returns data are the object of interest, and find evidence that such models may yield reasonable approximations to unknown underlying DGPs, in the sense that the models often significantly outperform a fairly wide class of the benchmark nonARFIMA models, including AR, ARMA, ARIMA, random walk, GARCH, and STAR models This finding is particularly apparent with longer samples of data, underscoring the importance of estimating d as precisely as possible when constructing ARFIMA-type forecasting models Interestingly, there appears little to choose between various estimators of d when samples are as large as often encountered in financial economics Overall, and in support of the finding of Bhardwaj and Swanson (2004), we conclude that long memory processes, and in particular ARFIMA processes, might not fall into the ‘‘empty box’’ category after all, although much further research is needed before overwhelmingly conclusive evidence in either direction can be given Acknowledgements The authors are grateful to the editors, Valentina Corradi and Clive W.J Granger for stimulating discussions, and Zhuanxin Ding, Steve Leybourne, and Mark Watson for providing the financial and macroeconomic datasets used in the empirical section of the paper Swanson has benefited from the support of Rutgers University in the form of a Research Council grant References Agiakloglou, C., P Newbold and M Wohar (1992), ‘‘Bias in an estimator of the fractional difference parameter’’, Journal of Time Series Analysis, Vol 14, pp 235–246 Andrews, D.W.K and Y Sun (2002) ‘‘Adaptive local whittle estimation of long-range dependence’’, Working Paper, Yale University Baillie, R.T (1996), ‘‘Long memory processes and fractional integration in econometrics’’, Journal of Econometrics, Vol 73, pp 5–59 Bank of Sweden (2003), ‘‘Time-series econometrics: cointegration and autoregressive conditional heteroskedasticity’’, Advanced information on the Bank of Sweden Prize in Economic Sciences in memory of Alfred Nobel, The Royal Swedish Academy of Sciences Beran, J (1995), ‘‘Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models’’, Journal of the Royal Statistical Society B, Vol 57, pp 659–672 Predictive Comparison of Long- and Short Memory Models of Daily U.S Stock Returns 401 Bhansali, R.J and P.S Kokaszka (2002), ‘‘Computation of the forecast coefficients for multistep prediction of long-range dependent time series’’, International Journal of Forecasting, Vol 18, pp 181–206 Bhardwaj, G and N.R Swanson (2004), ‘‘An empirical investigation of the usefulness of ARFIMA models for predicting macroeconomic and financial time series’’ Journal of Econometrics, (forthcoming) Bos, C.S., P.H Franses and M Ooms (2002), ‘‘Inflation, forecast intervals and long memory regression models’’, International Journal of Forecasting, Vol 18, pp 243–264 Breitung, J and U Hassler (2002), ‘‘Inference on the cointegration rank in fractionally integrated processes’’, Journal of Econometrics, Vol 110, pp 167–185 Brockwell, P.J and R Dahlhaus (2004), ‘‘Generalized Levinson-Durbin and Burg algorithms’’, Journal of Econometrics, Vol 118, pp 129–149 Chao, J.C., V Corradi and N.R Swanson (2001), ‘‘An out of sample test for Granger causality’’, Macroeconomic Dynamics, Vol 5, pp 598–620 Cheung, Y.-W (1993), ‘‘Tests for fractional integration: a Monte Carlo investigation’’, Journal of Time Series Analysis, Vol 14, pp 331–345 Cheung, Y.-W and F.X Diebold (1994), ‘‘On maximum likelihood estimation of the difference parameter of fractionally integrated noise with unknown mean’’, Journal of Econometrics, Vol 62, pp 301–316 Choi, K and E Zivot (2002), ‘‘Long memory and structural changes in the forward discount: an empirical investigation’’, Working Paper, University of Washington Christoffersen, P.F (1998), ‘‘Evaluating interval forecasts’’, International Economic Review, Vol 39, pp 841–862 Christoffersen, P.F and F.X Diebold (1997), ‘‘Optimal prediction under asymmetric loss’’, Econometric Theory, Vol 13, pp 808–817 Clark, T.E and M.W McCracken (2001), ‘‘Tests of equal forecast accuracy and encompassing for nested models’’, Journal of Econometrics, Vol 105, pp 85–110 Clements, M.P and J Smith (2000), ‘‘Evaluating the forecast densities of linear and nonlinear models: applications to output growth and unemployment’’, Journal of Forecasting, Vol 19, pp 255–276 Clements, M.P and J Smith (2002), ‘‘Evaluating multivariate forecast densities: a comparison of two approaches’’, International Journal of Forecasting, Vol 18, pp 397–407 Corradi, V and N.R Swanson (2002), ‘‘A consistent test for out of sample nonlinear predictive ability’’, Journal of Econometrics, Vol 110, pp 353–381 Corradi, V and N.R Swanson (2003), ‘‘The block bootstrap for parameter estimation error in recursive estimation schemes, with applications to predictive evaluation’’, Working Paper, Rutgers University Corradi, V and N.R Swanson (2004a), ‘‘Predictive density accuracy tests’’, Working Paper, Rutgers University 402 Geetesh Bhardwaj and Norman R Swanson Corradi, V and N.R Swanson (2004b), ‘‘Predictive density evaluation’’, in: G Elliott, C.W.J Granger and A Timmerman, editors, Handbook of Economic Forecasting, Amsterdam: Elsevier forthcoming Corradi, V., N.R Swanson and C Olivetti (2001), ‘‘Predictive ability with cointegrated variables’’, Journal of Econometrics, Vol 104, pp 315–358 Diebold, F.X., T.A Gunther and A.S Tay (1998), ‘‘Evaluating density forecasts with applications to finance and management’’, International Economic Review, Vol 39, pp 863–883 Diebold, F.X., J Hahn and A.S Tay (1999), ‘‘Multivariate density forecast evaluation and calibration in financial risk management: high frequency returns on foreign exchange’’, Review of Economics and Statistics, Vol 81, pp 661–673 Diebold, F.X and A Inoue (2001), ‘‘Long memory and regime switching’’, Journal of Econometrics, Vol 105, pp 131–159 Diebold, F.X and R.S Mariano (1995), ‘‘Comparing predictive accuracy’’, Journal of Business and Economic Statistics, Vol 13, pp 253–263 Diebold, F.X and G.D Rudebusch (1989), ‘‘Long memory and persistence in aggregate output’’, Journal of Monetary Economics, Vol 24, pp 189–209 Diebold, F.X and G.D Rudebusch (1991a), ‘‘Is consumption too smooth? Long memory and the Deaton paradox’’, Review of Economics and Statistics, Vol 73, pp 1–9 Diebold, F.X and G.D Rudebusch (1991b), ‘‘On the power of the Dickey–Fuller test against fractional alternatives’’, Economics Letters, Vol 35, pp 155–160 Ding, Z., C.W.J Granger and R.F Engle (1993), ‘‘A long memory property of stock returns and a new model’’, Journal of Empirical Finance, Vol 1, pp 83–106 Dittman, I and C.W.J Granger (2002), ‘‘Properties of nonlinear transformations of fractionally integrated processes’’, Journal of Econometrics, Vol 110, pp 113–133 Doornik, J.A and M Ooms (2003), ‘‘Computational aspects of maximum likelihood estimation of autoregressive fractionally integrated moving average models’’, Computational Statistics and Data Analysis, Vol 42, pp 333–348 Engle, R.F and A.D Smith (1999), ‘‘Stochastic permanent breaks’’, Review of Economics and Statistics, Vol 81, pp 553–574 Geweke, J and S Porter-Hudak (1983), ‘‘The estimation and application of long memory time series models’’, Journal of Time Series Analysis, Vol 4, pp 221–238 Granger, C.W.J (1969), ‘‘Investigating causal relations by econometric models and cross-spectral methods’’, Econometrica, Vol 37, pp 424–438 Granger, C.W.J (1980), ‘‘Long memory relationships and the aggregation of dynamic models’’, Journal of Econometrics, Vol 14, pp 227–238 Predictive Comparison of Long- and Short Memory Models of Daily U.S Stock Returns 403 Granger, C.W.J (1999), ‘‘Aspects of research strategies for time series analysis’’, Presentation to the conference on New Developments in Time Series Economics, Yale University Granger, C.W.J and A.P Andersen (1978), Introduction to Bilinear Time Series Models, Gottingen: Vanden-hoeck and Ruprecht Granger, C.W.J and Z Ding (1996), ‘‘Varieties of long memory models’’, Journal of Econometrics, Vol 73, pp 61–77 Granger, C.W.J and M Hatanaka (1964), Spectral Analysis of Economic Time Series, Princeton: Princeton University Press Granger, C.W.J and N Hyung (2004), ‘‘Occasional structural breaks and long memory with application to the S&P 500 absolute stock returns’’, Journal of Empirical Finance, Vol 11, pp 399–421 Granger, C.W.J and R Joyeux (1980), ‘‘An introduction to long memory time series models and fractional differencing’’, Journal of Time Series Analysis, Vol 1, pp 15–30 Granger, C.W.J and P Newbold (1986), Forecasting Economic Time Series, San Diego: Academic Press 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 Hansen, P.R., Lunde, A and J.M Nason (2005), ‘‘Model confidence sets for forecasting models’’, Federal Reserve Bank of Atlanta Working Paper No 2005–7 Harvey, D.I., S.J Leybourne and P Newbold (1998), ‘‘Tests for forecast encompassing’’, Journal of Business and Economic Statistics, Vol 16, pp 254–259 Hassler, U and J Wolters (1995), ‘‘Long memory in inflation rates: International evidence’’, Journal of Business and Economic Statistics, Vol 13, pp 37–45 Henry, M (2001), ‘‘Robust automatic bandwidth for long memory’’, Journal of Time Series Analysis, Vol 22, pp 293–316 Henry, M and P.M Robinson (1996), ‘‘Bandwidth choice in Gaussian semiparametric estimation of long range dependence’’, pp 220–232 in: P.M Robinson and M Rosenblatt, editors, Athens Conference on Applied Probability and Time Series Analysis, Vol II: Time Series Analysis, Springer, New York: Springer (In Memory of E J Hannan) Hosking, J (1981), ‘‘Fractional differencing’’, Biometrika, Vol 68, pp 165–176 Hurst, H.E (1951), ‘‘Long-term storage capacity of reservoirs’’, Transactions of the American Society of Civil Engineers, Vol 116, pp 770–799 Hurvich, H., R Deo and J Brodsky (1998), ‘‘The mean squared error of Geweke and Porter-Hudak’s estimator of memory parameter of a long memory time series’’, Journal of Time Series Analysis, Vol 19, pp 19–46 Hyung, N and P.H Franses (2001), ‘‘Structural breaks and long memory in US inflation rates: Do they matter for forecasting?’’, Working Paper, Erasmus University 404 Geetesh Bhardwaj and Norman R Swanson Inoue, A and L Kilian (2003), ‘‘On the selection of forecasting models’’, Journal of Econometrics, (forthcoming) Kim, C.-J and C.R Nelson (1999), State Space Models with Regime Switching, Cambridge, MA: MIT Press Kuănsch, H.R (1987), Statistical aspects of self-similar processes’’, in: Y Prohorov and V.V Sasanov, editors, Proceedings of the First World Congress of the Bernoulli Society, Utrecht: VNU Science Press Lo, A.W (1991), ‘‘Long-term memory in stock market prices’’, Econometrica, Vol 59, pp 1279–1313 McCracken, M.W (1999) ‘‘Asymptotics for out of sample tests of causality’’, Working Paper, Louisiana State University Newey, W.K and K.D West (1987), ‘‘A simple positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix’’, Econometrica, Vol 55, pp 703–708 Phillips, P.C.B (1987), ‘‘Time series regression with a unit root’’, Econometrica, Vol 55, pp 277–301 Robinson, P.M (1995a), ‘‘Log-periodogram regression of time series with long range dependence’’, Annals of Statistics, Vol 23, pp 1048–1072 Robinson, P.M (1995b), ‘‘Gaussian semiparametric estimation of long range dependence’’, Annals of Statistics, Vol 23, pp 1630–1661 Robinson, P.M (2003), Time Series with Long Memory, Oxford: Oxford University Press Rossi, B (2005), ‘‘Testing long-horizon predictive ability with high persistence, and the Meese–Rogoff puzzle’’, International Economic Review, Vol 46, pp 61–92 Shimotsu, K and P.C.B Phillips (2002), ‘‘Exact local Whittle estimation of fractional integration’’, Annals of Statistics, Vol 33(4), pp 1890–1933 Sowell, F.B (1992a), ‘‘Maximum likelihood estimation of stationary univariate fractionally integrated time series models’’, Journal of Econometrics, Vol 53, pp 165–188 Sowell, F.B (1992b), ‘‘Modelling long-run behavior with the fractional ARIMA model’’, Journal of Monetary Economics, Vol 29, pp 277–302 Swanson, N.R and H White (1995), ‘‘A model selection approach to assessing the information in the term structure using linear models and artificial neural networks’’, Journal of Business and Economic Statistics, Vol 13, pp 265–279 Swanson, N.R and H White (1997), ‘‘A model selection approach to real-time macroeconomic forecasting using linear models and artificial neural networks’’, Review of Economics and Statistics, Vol 79, pp 540–550 Taqqu, M and V Teverovsky (1995), ‘‘Estimators for long range dependence: an empirical study’’, Fractals, Vol 3, pp 785–789 Taqqu, M and V Teverovsky (1997), ‘‘Robustness of Whittle-type estimators for time series with long-range dependence’’, Stochastic Models, Vol 13, pp 723–757 Predictive Comparison of Long- and Short Memory Models of Daily U.S Stock Returns 405 van Dijk, D., P.H Franses and R Paap (2002), ‘‘A nonlinear long memory model, with an application to US unemployment’’, Journal of Econometrics, Vol 110, pp 135–165 van Dijk, D., T Terasvirta and P.H Franses (2002), ‘‘Smooth transition autoregressive models – A survey of recent developments’’, Econometric Reviews, Vol 21, pp 1–47 West, K.D (1996), ‘‘Asymptotic inference about predictive ability’’, Econometrica, Vol 64, pp 1067–1084 White, H (2000), ‘‘A reality check for data snooping’’, Econometrica, Vol 68, pp 1097–1126 This page intentionally left blank 406 CHAPTER 15 Nonlinear Modeling of the Changing Lag Structure in U.S Housing Construction Christian M Dahlà and Tamer Kulaksizog˘lu Abstract In this paper, we reconsider the nature of the changing lag structure in housing construction We assume that the U.S housing construction industry is subject to two regimes determined by the de-trended level of housing units under construction: recession and expansion The econometric model we employ is a generic nonlinear autoregressive distributed lag model The results imply that builders seem to change the speed of construction depending on whether the construction industry is in a recession or expansion The mean lag between housing completions and housing starts is significantly shorter in recession periods than in expansion periods This conclusion is in line with the so-called accordion effect JEL classifications: C22, L74 Introduction This paper is concerned with the empirical modeling of the supply of housing units, in particular, with the relationship between housing starts and completions This apparently modest problem has received ample attention in the literature The main contribution of this paper is to provide new insights by applying flexible nonlinear time-series models The duration between the time construction of a housing unit begins and the time the housing unit is completed depends on a number of factors like the size of the housing unit, e.g., Merkies and Bikker (1981), the complexity of the ÃCorresponding author r 2006 ELSEVIER B.V ALL RIGHTS RESERVED CONTRIBUTIONS TO ECONOMIC ANALYSIS VOLUME 276 ISSN: 0573-8555 DOI:10.1016/S0573-8555(05)76015-6 407 408 Christian M Dahl and Tamer Kulaksizog˘lu structure, weather conditions, e.g Coulson and Richard (1996) and Fergus (1999), and economic conditions, e.g van Alphen and Merkies (1976), Borooah (1979), van Alphen and de Vos (1985), Merkies and Steyn (1994), and Coulson (1999) While each of these factors are important, we will focus entirely on those related to the underlying economic conditions Thus, the questions of particular interest to us are: Does the lag structure between housing starts and housing completions change when the construction industry goes into a recession?; Is the lag structure longer or shorter when the industry is in a recession or expansion?; and How the changing economic conditions affect the lag structure in the construction industry? In the existing literature one can find convincing arguments for at least two opposing views regarding a possible asymmetric relationship between housing starts and completions First, the lag structure between housing starts and housing completions is expected to be shorter when the construction industry is in a recession because contractors have to allocate all of their resources to a few existing projects at hand and they need money flows to survive hard times Alternatively, it can be argued that since economic expansions typically result in more opportunities, contractors might want to complete existing projects as soon as possible in order to be ready to meet increased demand Another supporting argument for this second view is that during economic recessions construction companies may have to lay off workers to decrease their costs and to be able to compete better with other companies Since this implies a reduced capacity, it will take longer to complete existing projects During economic expansions, however, construction companies are likely to increase their capacity by hiring additional workers and equipment, thus increasing their ability to complete construction projects relatively quickly Merkies and Steyn (1994) address the effect of economic conditions on the lag structure in the construction process Their motivation is an empirical observation by van Alphen and Merkies (1976) who showed that the lag pattern between starts and completions has a tendency to contract during slow periods and expand during boom periods, reflecting a given production capacity over projects at hand as business slows down or speeds up They nickname this observed phenomenon the accordion effect1 Note that the accordion effect supports the first view described above Merkies and Steyn (1994) were the first to attempt to explicitly model the relation between construction starts and actual production to provide empirical evidence for the accordion effect This paper addresses the same issues but in a different and more formal way Specifically, we apply flexible nonlinear time-series models and provide a statistical hypothesis test for the possible existence of the accordion effect Unlike our approach, Merkies and Steyn (1994) employ an econometric model which is a Merkies and Steyn (1994, p 501) Changing Lag Structure in U.S Housing Construction 409 variant of the Almon’s polynomial lag pattern model and is nonlinear only in the coefficients (which are time varying) They also allow for a time-dependent variance However, within their framework they are not able to derive a formal test for the existence of the accordion effect Alternatively, we suggest using regime switching autoregressive distributed lag (ARDL) models Within this class of models we show that identification and estimation of the accordion effect becomes very straightforward Namely, we propose a simple Wald test based on comparing the estimated ‘‘mean lags’’ under the alternative economic regimes We also depart from Merkies and Steyn (1994) by using U.S data and real variables In particular, we use the number of housing units started in past quarters as the explanatory variable and the number of completed houses during a given quarter as the response variable The main finding of the paper is that builders actually seem to change the pace of construction under different economic regimes Builders speed up (slow down) the construction process if the industry is in a recession (an expansion) This empirical finding is in strong support of the accordion effect and has important implications for the supply side of the housing market In particular, our results indicate that the supply side in a more realistic housing market model should be specified as ( if rX ị ẳ 1; h St ; Á Á Á ; S tÀpr ; s À Á C ¼ h St ; Á Á Á ; S tpe ; if rX ị ẳ 0; where Cs and S are the supply of housing completions and starts respectively, r(X) denotes a binary variable such that rX ị ẳ indicates recession, e (or rX ị ẳ 0) denotes expansion, and X contains economic factors determining whether the economy is in a recession or expansion The paper is organized as follows Section presents the data Section starts our empirical analysis with a simple unrestricted finite distributed lag model Section introduces an autoregressive distributed lag model, which addresses the shortcomings of the unrestricted finite distributed lag model Section presents two regime-switching ARDL models, which assume that the construction industry is subject to two regimes: recession/contraction and expansion Section concludes The data The data we use consist of monthly observations on the number of new privately owned housing units in the U.S measured in thousands.2 We consider the The data source is the U.S Census of Bureau and can be obtained at http://www.census.gov/const/ www/newresconstindex.html 410 Christian M Dahl and Tamer Kulaksizog˘lu following three series:  New privately owned housing units started (i.e., starts), from January 1959 to December 2003  New privately owned housing units completed (i.e., completions), from Jan- uary 1968 to December 2003  New privately owned housing units under construction (i.e., construction), from January 1970 to December 2003 We aggregate the raw data series to obtain quarterly observations by simply adding the monthly observations within each quarter This aggregation is performed to ensure parsimony in the estimating equations, as the lag/dependence structure between starts and completions turns out to be relatively long There is a pronounced seasonality in the series, which we remove using the seasonal adjustment method advocated by Lovell (1963) and discussed in Davidson and MacKinnon (1993) This seasonal adjustment method produces the same results as using seasonal dummies in a linear regression equation Apart from its simplicity (i.e., transparancy), this method is desirable since it does not affect the mean level of the series Thus, the method makes it possible to interprete the constant term and its significance as an important indicator of the ‘‘correctness’’ of the model.3 Another desirable feature is that the filter is linear and does not introduce ‘‘generated’’ nonlinearities in the data Figure illustrates the seasonally adjusted series and Table presents some summary statistics Several features should be noticed First, there does not seem to be any apparent trend present in starts and completions while the construction series perhaps exhibits a slight downward trend Second, starts and completions seem to move closely together Third, there appears to be a lead-lag relation between starts and completions, with starts leading completions Finally, as shown in Table 1, the mean and the median values for starts and completions are very similar However, starts are more volatile as shown by the standard deviations and the minimum and maximum values To investigate the stationarity of the series further we conduct Augmented Dickey–Fuller (ADF) unit root tests Although, as previously mentioned, there is no apparent trend in starts and completions series, we include an intercept as well as an intercept + trend in the tests Further, we include a sufficiently high number of lags to remove any serial correlation in the error terms in the ADF regressions Table presents the results of the tests for the seasonally adjusted quarterly series Note that the augmentation in all cases consists of four lagged differences The ADF tests strongly reject the unit root hypothesis for all three series at the percent level in the case of intercept and intercept + trend It should be noted that these results are in opposition to some previous results on construction data For instance, Coulson (1999), using the ADF tests, This will become clearer when the empirical models are introduced .. .NONLINEAR TIME SERIES ANALYSIS OF BUSINESS CYCLES i CONTRIBUTIONS TO ECONOMIC ANALYSIS 276 Honorary Editors: D.W JORGENSON J TINBERGENy Editors: B Baltagi E Sadka D Wildasin Amsterdam – Boston... – Singapore – Sydney – Tokyo ii NONLINEAR TIME SERIES ANALYSIS OF BUSINESS CYCLES Costas Milas Department of Economics, Keele University, UK Philip Rothman Department of Economics, East Carolina... 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