Empirical Economics

, Volume 47, Issue 2, pp 495–522 | Cite as

Measuring the natural rates, gaps, and deviation cycles



One definition of the natural rate is the (time-varying) steady state equilibrium rate. Then the gap is the difference between the actual and natural rates, or the forecastable movement. Although modern business cycle theories study deviation cycles (cycles in the gap), the NBER business cycle reference dates measure classical cycles (cycles in the actual rate) in the US. Measuring deviation cycles requires detrending, and this motivated the invention of the Beveridge–Nelson (B–N) decomposition. This paper considers multivariate detrending, and proposes a Bayesian approach to the multivariate B–N decomposition. An application of the method to US data gives (i) a joint estimate of the natural rates and gaps of output, inflation, interest, and unemployment with reliable error bands, and (ii) the posterior probabilities of positive gap, recession, and revival. These results may help us to identify the four phases of deviation cycles: expansion, recession, contraction, and revival.


Beveridge–Nelson decomposition Bayesian Business cycle  Growth cycle Turning point 

JEL classification

C11 C32 C53 C82 E32 



I thank Andrew Harvey, Chengsi Zhang, and two referees for useful comments. This work was supported by KAKENHI (16730113, 19530185, 23530255).


  1. Abadir KM, Magnus JR (2002) Notation in econometrics: a proposal for a standard. Econom J 5:76–90CrossRefGoogle Scholar
  2. Apel M, Jansson P (1999a) System estimates of potential output and the NAIRU. Empir Econ 24:373–388CrossRefGoogle Scholar
  3. Apel M, Jansson P (1999b) A theory-consistent system approach for estimating potential output and the NAIRU. Econ Lett 64:271–275CrossRefGoogle Scholar
  4. Artis M, Marcellino M, Proietti T (2004) Dating business cycles: a methodological contribution with an application to the euro area. Oxf Bull Econ Stat 66:537–565CrossRefGoogle Scholar
  5. Bańbura M, Giannone D, Reichlin L (2010) Large Bayesian vector auto regressions. J Appl Econom 25:71–92CrossRefGoogle Scholar
  6. Basistha A, Startz R (2008) Measuring the NAIRU with reduced uncertainty: a multiple indicator-common component approach. Rev Econ Stat 90:805–811CrossRefGoogle Scholar
  7. Benati L, Vitale G (2007) Joint estimation of the natural rate of interest, the natural rate of unemployment, expected inflation, and potential output. Working Paper 797. European Central Bank, FrankfurtGoogle Scholar
  8. Berger T (2011) Estimating Europe’s natural rates. Empir Econ 40:521–536CrossRefGoogle Scholar
  9. Beveridge S, Nelson CR (1981) A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’. J Monet Econ 7:151–174CrossRefGoogle Scholar
  10. Bry G, Boschan C (1971) Cyclical analysis of time series: Selected procedures and computer programs. Technical Paper 20. National Bureau of Economic Research, CambridgeGoogle Scholar
  11. Burns AF, Mitchell WC (1946) Measuring business cycles. National Bureau of Economic Research, CambridgeGoogle Scholar
  12. Chib S, Greenberg E (1994) Bayes inference in regression models with ARMA \((p, q)\) errors. J Econom 64:183–206CrossRefGoogle Scholar
  13. Chib S, Greenberg E (1995a) Hierarchical analysis of SUR models with extensions to correlated serial errors and time-varying parameter models. J Econom 68:339–360CrossRefGoogle Scholar
  14. Chib S, Greenberg E (1995b) Understanding the Metropolis–Hastings algorithm. Am Stat 4:327–335Google Scholar
  15. Cogley T, Sargent TJ (2001) Evolving post-World War II U.S. inflation dynamics. NBER Macroecon Annu 16:331–373CrossRefGoogle Scholar
  16. Cogley T, Sargent TJ (2005) Drifts and volatilities: monetary policies and outcomes in the post WWII US. Rev Econ Dyn 8:262–302CrossRefGoogle Scholar
  17. Cogley T, Morozov S, Sargent TJ (2005) Bayesian fan charts for U.K. inflation: forecasting and sources of uncertainty in an evolving monetary system. J Econ Dyn Control 29:1893–1925CrossRefGoogle Scholar
  18. Del Negro M, Schorfheide F (2011) Bayesian macroeconometrics. In: Geweke J, Koop G, van Dijk H (eds) The Oxford handbook of Bayesian econometrics, chap 7. Oxford University Press, Oxford, pp 293–389Google Scholar
  19. Doménech R, Gómez V (2006) Estimating potential output, core inflation, and the NAIRU as latent variables. J Bus Econ Stat 24:354–365CrossRefGoogle Scholar
  20. Garnier J, Wilhelmsen BR (2009) The natural rate of interest and the output gap in the euro area: a joint estimation. Empir Econ 36:297–319CrossRefGoogle Scholar
  21. Garratt A, Robertson D, Wright S (2006) Permanent vs transitory components and economic fundamentals. J Appl Econom 21:521–542CrossRefGoogle Scholar
  22. Hamilton JD (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57:357–384CrossRefGoogle Scholar
  23. Harding D, Pagan A (2000) Knowing the cycle. In: Backhouse RE, Salanti A (eds) Macroeconomics and the real world, chap 2, vol 1. Oxford University Press, Oxford, pp 23–41Google Scholar
  24. Harding D, Pagan A (2002) Dissecting the cycle: a methodological investigation. J Monet Econ 49:365–381CrossRefGoogle Scholar
  25. Harding D, Pagan A (2005) A suggested framework for classifying the mode of cycle research. J Appl Econom 20:151–159CrossRefGoogle Scholar
  26. Harvey AC, Trimbur TM, van Dijk HK (2007) Trends and cycles in economic time series: a Bayesian approach. J Econom 140:618–649CrossRefGoogle Scholar
  27. Kadiyala KR, Karlsson S (1997) Numerical methods for estimation and inference in Bayesian VAR-models. J Appl Econom 12:99–132CrossRefGoogle Scholar
  28. Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Assoc 90:773–795CrossRefGoogle Scholar
  29. Kilian L (1998) Small-sample confidence intervals for impulse response functions. Rev Econ Stat 80:218–230CrossRefGoogle Scholar
  30. King TB, Morley J (2007) In search of the natural rate of unemployment. J Monet Econ 54:550–564CrossRefGoogle Scholar
  31. Koop G (2003) Bayesian econometrics. Wiley, HobokenGoogle Scholar
  32. Laubach T, Williams JC (2003) Measuring the natural rate of interest. Rev Econ Stat 85:1063–1070CrossRefGoogle Scholar
  33. Litterman RB (1986) Forecasting with Bayesian vector autoregressions—five years of experience. J Bus Econ Stat 4:25–38Google Scholar
  34. Mariano RS, Murasawa Y (2010) A coincident index, common factors, and monthly real GDP. Oxf Bull Econ Stat 72:27–46CrossRefGoogle Scholar
  35. Mésonnier JS, Renne JP (2007) A time-varying “natural” rate of interest for the euro area. Eur Econ Rev 51:1768–1784CrossRefGoogle Scholar
  36. Mintz I (1969) Dating postwar business cycles: methods and their application to Western Germany, 1950–1967. Occasional Paper 107. National Bureau of Economic Research, CambridgeGoogle Scholar
  37. Mintz I (1972) Dating American growth cycles. In: Zarnowitz V (ed) The business cycle today, economic research: retrospect and prospect, vol 1. National Bureau of Economic Research, Cambridge, pp 39–88Google Scholar
  38. Morley JC (2002) A state-space approach to calculating the Beveridge–Nelson decomposition. Econ Lett 75:123–127CrossRefGoogle Scholar
  39. Morley JC, Piger J (2008) Trend/cycle decomposition of regime-switching processes. J Econom 146:220–226CrossRefGoogle Scholar
  40. Morley JC, Nelson CR, Zivot E (2003) Why are the Beveridge–Nelson and unobserved-components decompositions of GDP so different? Rev Econ Stat 85:235–243CrossRefGoogle Scholar
  41. Nelson CR (2008) The Beveridge–Nelson decomposition in retrospect and prospect. J Econom 146:202–206CrossRefGoogle Scholar
  42. Orphanides A, van Norden S (2002) The unreliability of output-gap estimates in real time. Rev Econ Stat 84:569–583CrossRefGoogle Scholar
  43. Phelps ES (1995) The origins and further development of the natural rate of unemployment. In: Cross R (ed) The natural rate of unemployment: reflections on 25 years of the hypothesis, chap 2. Cambridge University Press, Cambridge, pp 15–31CrossRefGoogle Scholar
  44. R Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. http://www.R-project.org
  45. Rotemberg JJ, Woodford M (1996) Real-business-cycle models and the forecastable movements in output, hours, and consumption. Am Econ Rev 86:71–89Google Scholar
  46. Sims CA, Zha T (1999) Error bands for impulse responses. Econometrica 67:1113–1155CrossRefGoogle Scholar
  47. Stock JH, Watson MW (2010) Indicators for dating business cycles: cross-history selection and comparisons. Am Econ Rev 100:16–19CrossRefGoogle Scholar
  48. Stock JH, Watson MW (2013) Estimating turning points using large data sets. J Econom forthcomingGoogle Scholar
  49. Verdinelli I, Wasserman L (1995) Computing Bayes factors using a generalization of the Savage–Dickey density ratio. J Am Stat Assoc 90:614–618CrossRefGoogle Scholar
  50. Woodford M (2003) Interest and prices. Princeton University Press, PrincetonGoogle Scholar
  51. Zarnowitz V, Ozyildirim A (2006) Time series decomposition and measurement of business cycles, trends, and growth cycles. J Monet Econ 53:1717–1739CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of EconomicsOsaka Prefecture UniversitySakaiJapan

Personalised recommendations