Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 14, 2013

Real vs. nominal cycles: a multistate Markov-switching bi-factor approach

  • Danilo Leiva-Leon EMAIL logo


This paper proposes a probabilistic model based on comovements and nonlinearities useful to assess the type of shock affecting each phase of the business cycle. By providing simultaneous inferences on the phases of real activity and inflation cycles, contractionary episodes are dated and categorized into demand, supply and mix recessions. The impact of shocks originated in the housing market over the business cycle is also assessed, finding that recessions are usually accompanied by housing deflationary pressures, while expansions are mainly influenced by housing demand shocks, with the only exception occurred during the period surrounding the “Great Recession,” affected by expansionary housing supply shocks.

JEL classification: E32; C22; E27

Corresponding author: Danilo Leiva-Leon, International Economic Analysis Department, Bank of Canada, 234 Laurier Avenue West, Ottawa, ON, K1A 0G9, Canada, e-mail:


Thanks to Maximo Camacho, Gabriel Perez-Quiros, the editor and an anonymous referee for valuable comments and suggestions. Thanks to the participants at 20th Symposium of the Society for Nonlinear Dynamics and Econometrics, 31st Annual International Symposium on Forecasting, Quantitative Economics Doctorate Jamboree and internal seminar series at the University of Alicante for discussions that improved the content of this paper. The views expressed in this paper are those of the author and do not represent the views of the Bank of Canada.


It is assumed that m=2, and k=0. Focusing first on the real activity versus inflation model, according to the eleven-variable model used in the empirical application, the measurement equation, yt=t+et, with et~N(0,R), is

(A.1)[ΔGDPtΔINDtΔPINtΔSALtΔPAYtΔDEFtΔCPItΔPPItΔGSCtΔOILtΔHCNt]=[γ1bγ2bγ3bγ4bγ5bγ6bγ7bγ8bγ9bγ10bγ11bγ1pγ2pγ3pγ4pγ5pγ6pγ7pγ8pγ9pγ10pγ11p100000000000000000000001000000000000000000000010000000000000000000000100000000000000000000000000010000000000000000000000100000000000][ftbftpe1te1,t1e2te2,t1e11,t1]+[00000000000], (A.1)

where R is a matrix of zeroes. The notation of the variables is defined as: GDP is real GDP, IND is industrial production, PIN is real personal income less transfers, SAL is real manufacturing trading sales, PAY is total non-farm labor, DEF is deflator of GDP, CPI is consumer price index, PPI is producer price index, GSC is Standard and Poor’s GSCI non-energy commodities price index, OIL is spot oil price, and HCN hourly compensation in the non-farm business sector.

The transition equation, βt=μ˜Stb,Stp+Fβt1+υt, with υt~N(0,Q), is

(A.2)[ftbftpe1te1,t1e2te2,t1e11,t1]=[μStbbμStpp00000]+[0000000000000000ϕ11100000ϕ1200000000ϕ21100000ϕ220000000000ϕ11,1100000000ϕ11,20][ft1bft1pe1,t1e1,t2e2,t1e2,t2e11,t2]+[ωtbωtpε1t0ε2t00], (A.2)

where Q is a diagonal matrix in which the entries inside the main diagonal are collected in the vector

(A.3)(σb2,σp2,σ12,0,σ22,0,σ32,0,σ42,0,σ52,0,σ62,0,σ72,0,σ82,0,σ92,0,σ102,0,σ112,0) (A.3)

For the real activity versus housing prices model, the measurement and transition equations follow the same reasoning as in Equations (A.1) and (A.2), respectively. The main change, apart from adjusting the appropriate dimension of the matrices for 10 observed indicators, is that vector yt, as defined in Equation (A.1), is replaced by


The notation of the housing price indicators is defined as: PDI is Price deflator index of new single-family houses under construction, CMH is Conventional Mortgage Home Price Index, ATP is FHFA All-transactions Price Index, POI is FHFA Purchase-only Index, SPC is S&P Case-Shiller-10-cities Home Price Index.


Aruoba, B., and F. Diebold. 2010. “Real-Time Macroeconomic Monitoring: Real Activity, Inflation, and Interactions.” American Economic Review 100 (2): 20–24.10.1257/aer.100.2.20Search in Google Scholar

Banbura, M., and G. Rustler. 2007. “A Look into the Model Factor Black Box. Publication Lags and the Role of Hard and Soft Data in Forecasting GDP.” ECB working paper No. 751.10.2139/ssrn.984265Search in Google Scholar

Bengoechea, P., M. Camacho, and G. Perez-Quiros. 2006. “A Useful Tool for Forecasting the Euro-Area Business Cycle Phases.” International Journal of Forecasting 22 (4): 735–749.10.1016/j.ijforecast.2006.01.002Search in Google Scholar

Blanchard, J., and D. Quah. 1989. “The Dynamic Effects of Aggregate and Supply Disturbances.” American Economic Review 79 (4): 655–673.Search in Google Scholar

Blanchard, J., and D. Quah. 1993. “The Dynamic Effects of Aggregate and Supply Disturbances: Reply.” American Economic Review 83 (3): 653–658.Search in Google Scholar

Burns, A., and W. Mitchell. 1946. “Measuring Business Cycles.” NBER Book Series Studies in Business Cycles. New York.Search in Google Scholar

Camacho, M., and G. Perez-Quiros. 2007. “Jump-and-Rest Effect of U.S. Business Cycles.” Studies in Nonlinear Dynamics and Econometrics 11 (4): 1558–3708.10.2202/1558-3708.1480Search in Google Scholar

Camacho, M., and G. Perez-Quiros. 2010. “Introducing the Euro-Sting: Short-Term Indicator of Euro Area Growth.” Journal of Applied Econometrics 25 (4): 663–694.10.1002/jae.1174Search in Google Scholar

Camacho, M., G. Perez-Quiros, and P. Poncela. 2010. “Green Shoots in the Euro Area. A Real Time Measure.” International Journal of Forecasting, In Press.10.2139/ssrn.1646848Search in Google Scholar

Chauvet, M. 1998. “An Econometric Characterization of Business Cycle Dynamics with Factor Structure and Regime Switches.” International Economic Review 39 (4): 969–996.10.2307/2527348Search in Google Scholar

Del Negro, M., and C. Otrok. 2007. “99 Luftballons: Monetary Policy and the House Price Boom Across U.S. States.” Journal of Monetary Economics 54 (7): 1962–1985.10.1016/j.jmoneco.2006.11.003Search in Google Scholar

Forni, M., and L. Gambetti. 2010. “Macroeconomic Shocks and the Business Cycle: Evidence from a Structural Factor Model.” Recent, Center for Economic Research. Working paper series. No. 40.Search in Google Scholar

Galí, J. 1989. “The Dynamic Effects of Aggregate Demand and Supply Disturbances.” American Economic Review 79 (4): 655–673.Search in Google Scholar

Galí, J. 1992. “How Well Does the IS-LM Model Fit Postwar U.S. Data?” Quarterly Journal of Economics 107 (2): 709–738.10.2307/2118487Search in Google Scholar

Galí, J. 1999. “Technology, Employment, and the Business Cycle: Do Technology Shocks Explain Aggregate Fluctuations?” American Economic Review 89 (1): 249–271.Search in Google Scholar

Hamilton, J. 1989. “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle.” Econometrica 57 (2): 357–384.10.2307/1912559Search in Google Scholar

Ireland, P. 2010. “A New Keynesian Perspective on the Great Recession.” Journal of Money, Credit and Banking, Blackwell Publishing 43 (1): 31–54.10.1111/j.1538-4616.2010.00364.xSearch in Google Scholar

Kholodilin, K., and V. Yao. 2005. “Measuring and Predicting Turning Points using A Dynamic bi-Factor Model.” International Journal of Forecasting 21 (3): 525–537.10.1016/j.ijforecast.2005.02.002Search in Google Scholar

Kim, C. 1994. “Dynamic Linear Models with Markov-Switching.” Journal of Econometrics 60 (1–2): 1–22.10.1016/0304-4076(94)90036-1Search in Google Scholar

Kim, C., and C. Nelson. 1998. “Business Cycle Turning Points, A New Coincident Index, and Tests of Duration Dependence Based on A Dynamic Factor Model with Regime Switching.” Review of Economics and Statistics 80 (2): 188–201.10.1162/003465398557447Search in Google Scholar

Kim, C., and C. Nelson. 1999. State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications, 1st edition, Vol. 1, No. 0262112388. MIT Press Books.Search in Google Scholar

Kim, C., and J. Yoo. 1995. “New Index of Coincident Indicators: A Multivariate Markov Switching Factor Model Approach.” Journal of Monetary Economics 36 (3): 607–630.10.1016/0304-3932(95)01229-XSearch in Google Scholar

Lippi M., and L. Reichlin. 1993. “The Dynamic Effects of Aggregate Demand and Supply Disturbances: Comment.” American Economic Review 83 (3): 644–652.Search in Google Scholar

Mariano, R., and Y. Murasawa. 2003. “A New Coincident Index of Business Cycles Based on Monthly and Quarterly Series.” Journal of Applied Econometrics 18 (4): 427–443.10.1002/jae.695Search in Google Scholar

Ng, S., and E. Moench. 2011. “A Hierarchical Factor Analysis of US Housing Market Dynamics.” Econometrics Journal 14 (1): C1–C24.10.1111/j.1368-423X.2010.00319.xSearch in Google Scholar

Stock, J., and M. Watson. 1991. “A Probability Model of the Coincident Economic Indicators.” Leading economic indicators: new approaches and forecasting records, edited by Lahiri, H., and Moore, G. Cambridge University Press. in Google Scholar

Published Online: 2013-12-14
Published in Print: 2014-12-1

©2014 by De Gruyter

Downloaded on 23.3.2023 from
Scroll Up Arrow