Skip to main content Accessibility help


  • Federico Poloni (a1) and Giacomo Sbrana (a2)


The Hodrick–Prescott filter represents one of the most popular methods for trend–cycle extraction in macroeconomic time series. In this paper we provide a multivariate generalization of the Hodrick–Prescott filter, based on the seemingly unrelated time series approach. We first derive closed-form expressions linking the signal–noise matrix ratio to the parameters of the VARMA representation of the model. We then show that the parameters can be estimated using a recently introduced method, called “Moment Estimation Through Aggregation (META).” This method replaces traditional multivariate likelihood estimation with a procedure that requires estimating univariate processes only. This makes the estimation simpler, faster, and better behaved numerically. We prove that our estimation method is consistent and asymptotically normal distributed for the proposed framework. Finally, we present an empirical application focusing on the industrial production of several European countries.


Corresponding author

Address correspondence to: Giacomo Sbrana, NEOMA Business School, 1 Rue du Marchal Juin, 76130 Mont-Saint-Aignan, France; e-mail:


Hide All

F. Poloni is partially supported by INDAM (Istituto Nazionale di Alta Matematica) and by a PRA project of the University of Pisa.



Hide All
Akaike, H. (1980) Seasonal adjustment by a Bayesian modeling. Journal of Time Series Analysis 1 (1), 113.
Baxter, M. and King, R.G. (1999) Measuring business cycles: Approximate band-pass filters for economic time series. Review of Economics and Statistics 81 (4), 575593.
Beveridge, S. and Nelson, C.R. (1981) A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle. Journal of Monetary Economics 7 (2), 151174.
Box, G.E.P. and Jenkins, G.M. (1976) Time Series Analysis: Forecasting and Control. San Francisco: Holden-Day.
Canova, F. (1998) Detrending and business cycle facts: A user's guide. Journal of Monetary Economics 41 (3), 533540.
Gohberg, I., Lancaster, P., and Rodman, L. (1982) Matrix Polynomials. New York: Academic Press.
Harvey, A.C. (1989) Forecasting Structural Time Series and the Kalman Filter. Cambridge, UK: Cambridge University Press.
Harvey, A.C. and Jaeger, A. (1993) Detrending, stylized facts and the business cycle. Journal of Applied Econometrics 3, 231247.
Harvey, A.C. and Trimbur, T.M. (2003) General model-based filters for extracting cycles and trends in economic time series. Review of Economics and Statistics 85 (2), 244255.
Harvey, A.C. and Trimbur, T.M. (2008) Trend estimation and the Hodrick–Prescott filter. Journal of the Japan Statistical Society 85 (2), 244255.
Hodrick, R.J. and Prescott, E.C. (1997) Postwar U.S. business cycle: An empirical investigation. Journal of Money, Credit and Banking 29 (1), 116.
Kaiser, R. and Maravall, A. (2001) Measuring Business Cycles in Economic Statistics, Lecture Notes in Statistics, Vol. 154. NewYork: Springer.
Kaiser, R. and Maravall, A. (2005) Combining filter design with model-based filtering (with an application to business-cycle estimation). International Journal of Forecasting 22 (4), 691710.
Koopman, S.J., Harvey, A.C., Doornik, J.A., and Shephard, N. (2007) STAMP 8.2 Structural Time Series Analyzer, Modeller and Predictor. London: Timberlake Consultants Ltd.
Kozicki, S. (1999) Multivariate detrending under common trend restrictions: Implications for business cycle research. Journal of Economic Dynamics and Control 23 (7), 9971028.
Leser, C.E.V. (1961) A simple method of trend construction. Journal of the Royal Statistical Society Series B 23 (1), 91107.
Ling, S.Q. and McAleer, M. (2010) A general asymptotic theory for time-series models. Statistica Neerlandica 64 (1), 97111.
Maravall, A. and Del-Rio, A. (2007) Temporal aggregation, systematic sampling, and the Hodrick–Prescott filter. Computational Statistics and Data Analysis 52, 975998.
McElroy, T. (2008) Exact formulas for the Hodrick–Prescott filter. Econometrics Journal 11 (1), 209217.
McElroy, T. and Trimbur, T. (2015) Signal extraction for non-stationary multivariate time series with illustrations for trend inflation. Journal of Time Series Analysis 36 (2), 209227.
Mills, T. (2009) Modelling trends and cycles in economic time series: Historical prospective and future development. Cliometrica 3, 221244.
Morley, J.C., Nelson, C., and Zivot, E. (2003) Why are the Beveridge–Nelson and unobserved-components decomposition of GDP so different? Review of Economics and Statistics 85 (2), 235243.
Oh, K.H., Zivot, E., and Creal, D. (2008) The relationship between the Beveridge–Nelson decomposition and other permanent–transitory decompositions that are popular in economics. Journal of Econometrics 146, 207219.
Organisation for Economic Co-operation and Development (2014) Main Economic Indicators. Paris: Organisation for Economic Co-operation and Development. Available at
Poloni, F. and Sbrana, G. (2014) A note on forecasting demand using multivariate exponential smoothing framework. International Journal of Production Economics 162, 143150.
Ravn, M.O. and Uhlig, H. (2002) On adjusting the Hodrick–Prescott filter for the frequency of observations. Review of Economics and Statistics 84 (2), 371376.
Sbrana, G. (2011) Structural time series models and aggregation: Some analytical results. Journal of Time Series Analysis 32 (3), 315316.
Stock, J. and Watson, M.W. (1988) Testing for common trends. Journal of the American Statistical Society 83 (404), 10971107.
Watson, M.W. (1986) Univariate detrending methods with stochastic trends. Journal of Monetary Economics 18 (1), 4975.



  • Federico Poloni (a1) and Giacomo Sbrana (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed