Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 3, 2021

Bayesian analysis of structural correlated unobserved components and identification via heteroskedasticity

  • Mengheng Li ORCID logo EMAIL logo and Ivan Mendieta-Muñoz ORCID logo


We propose a structural representation of the correlated unobserved components model, which allows for a structural interpretation of the interactions between trend and cycle shocks. We show that point identification of the full contemporaneous matrix which governs the structural interaction between trends and cycles can be achieved via heteroskedasticity. We develop an efficient Bayesian estimation procedure that breaks the multivariate problem into a recursion of univariate ones. An empirical implementation for the US Phillips curve shows that our model is able to identify the magnitude and direction of spillovers of the trend and cycle components both within-series and between-series.

JEL classification: C11; C32; E31; E32; E52

Corresponding author: Mengheng Li, Economics Discipline Group, University of Technology Sydney, UTS Business School, Ultimo, Sydney, NSW 2007, Australia; and Centre for Applied Macroeconomic Analysis, Australian National University, Canberra, Australia, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.


Anufriev, M., and V. Panchenko. 2015. “Connecting the Dots: Econometric Methods for Uncovering Networks with an Application to the Australian Financial Institutions.” Journal of Banking & Finance 61: S241–55. in Google Scholar

Athanasopoulos, G., D. S. Poskitt, F. Vahid, and W. Yao. 2016. “Determination of Long-Run and Short-Run Dynamics in EC-VARMA Models via Canonical Correlations.” Journal of Applied Econometrics 31 (6): 1100–19. in Google Scholar

Basistha, A. 2007. “Trend-Cycle Correlation, Drift Break and the Estimation of Trend and Cycle in Canadian gdp.” Canadian Journal of Economics/Revue canadienne d’économique 40 (2): 584–606. in Google Scholar

Basistha, A., and R. Startz. 2008. “Measuring the Nairu with Reduced Uncertainty: A Multiple-Indicator Common-Cycle Approach.” The Review of Economics and Statistics 90 (4): 805–11. in Google Scholar

Beveridge, S., and C. R. Nelson. 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): 151–74. in Google Scholar

Blanchard, O. J., and D. Quah. 1989. “The Dynamic Effects of Aggregate Demand and Supply Disturbances.” The American Economic Review 79 (4): 655–73.10.3386/w2737Search in Google Scholar

Blanchard, O., E. Cerutti, and L. Summers. 2015. Inflation and Activity–Two Explorations and Their Monetary Policy Implications. Cambridge, USA: National Bureau of Economic Research. No. w21726.10.3386/w21726Search in Google Scholar

Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31 (3): 307–27. in Google Scholar

Canova, F., and F. J. Pérez Forero. 2015. “Estimating Overidentified, Nonrecursive, Time-Varying Coefficients Structural Vector Autoregressions.” Quantitative Economics 6 (2): 359–84. in Google Scholar

Charemza, W. W., and E. M. Syczewska. 1998. “Joint Application of the Dickey-Fuller and KPSS Tests.” Economics Letters 61 (1): 17–21. in Google Scholar

Chen, M.-H. 1994. “Importance-Weighted Marginal Bayesian Posterior Density Estimation.” Journal of the American Statistical Association 89 (427): 818–24. in Google Scholar

Chib, S. 2001. “Markov Chain Monte Carlo Methods: Computation and Inference.” In Handbook of Econometrics, Vol. 5, 3569–649. Oxford, UK: Elsevier.10.1016/S1573-4412(01)05010-3Search in Google Scholar

Davis, S. J., and J. A. Kahn. 2008. “Interpreting the Great Moderation: Changes in the Volatility of Economic Activity at the Macro and Micro Levels.” The Journal of Economic Perspectives 22 (4): 155–80. in Google Scholar

Delle Monache, D., I. Petrella, and F. Venditti. 2016. “Adaptive State Space Models with Applications to the Business Cycle and Financial Stress.” In CEPR Discussion Paper DP(11599).Search in Google Scholar

Doan, T., R. Litterman, and C. Sims. 1984. “Forecasting and Conditional Projection Using Realistic Prior Distributions.” Econometric Reviews 3 (1): 1–100. in Google Scholar

Dufour, J.-M., and D. Pelletier. 2005. “Practical Methods for Modelling Weak VARMA Processes: Identification, Estimation and Specification with a Macroeconomic Application.” Preprint.Search in Google Scholar

Dungey, M., J. P. Jacobs, J. Tian, and S. van Norden. 2015. “Trend in Cycle or Cycle in Trend? New Structural Identifications for Unobserved-Components Models of US Real GDP.” Macroeconomic Dynamics 19 (4): 776–90. in Google Scholar

Durbin, J., and S. J. Koopman. 1997. “Monte Carlo Maximum Likelihood Estimation for Non-Gaussian State Space Models.” Biometrika 84 (3): 669–84. in Google Scholar

Durbin, J., and S. J. Koopman. 2002. “A Simple and Efficient Simulation Smoother for State Space Time Series Analysis.” Biometrika 89 (3): 603–16. in Google Scholar

Durbin, J., and S. J. Koopman. 2012. Time Series Analysis by State Space Methods, Vol. 38. Oxford, UK: Oxford University Press.10.1093/acprof:oso/9780199641178.001.0001Search in Google Scholar

Ejrnæs, M., and M. Browning. 2014. “The Persistent–Transitory Representation for Earnings Processes.” Quantitative Economics 5 (3): 555–81.10.3982/QE239Search in Google Scholar

Fisher, L. A., H.-S. Huh, and A. R. Pagan. 2016. “Econometric Methods for Modelling Systems with a Mixture of I(1) and I(0) Variables.” Journal of Applied Econometrics 31 (5): 892–911. in Google Scholar

Frühwirth-Schnatter, S., and H. Wagner. 2010. “Stochastic Model Specification Search for Gaussian and Partial Non-Gaussian State Space Models.” Journal of Econometrics 154 (1): 85–100. in Google Scholar

Galí, J. 2015. “Hysteresis and the European Unemployment Problem Revisited.” In National Bureau of Economic Research Working Paper.10.3386/w21430Search in Google Scholar

Galí, J. 2016. “Insider-outsider Labor Markets, Hysteresis and Monetary Policy.” In Universitat Pompeu Fabra, Department of Economics Working Papers (1506).10.3386/w27385Search in Google Scholar

Gonzalo, J., and S. Ng. 2001. “A Systematic Framework for Analyzing the Dynamic Effects of Permanent and Transitory Shocks.” Journal of Economic Dynamics and Control 25 (10): 1527–46. in Google Scholar

Harvey, A. 2011. “Modelling the Phillips Curve with Unobserved Components.” Applied Financial Economics 21 (1-2): 7–17. in Google Scholar

Harvey, A. C., and N. Shephard. 1993. “Structural Time Series Models.” In Handbook of Statistics, Vol. 11, edited by G.S. Maddala, C. R.Rao and H.D. Vinod, 261–302. Amsterdam: North Holland.10.1016/S0169-7161(05)80045-8Search in Google Scholar

Herwartz, H., and H. Lütkepohl. 2014. “Structural Vector Autoregressions with Markov Switching: Combining Conventional with Statistical Identification of Shocks.” Journal of Econometrics 183 (1): 104–16. in Google Scholar

Holston, K., T. Laubach, and J. C. Williams. 2017. “Measuring the Natural Rate of Interest: International Trends and Determinants.” Journal of International Economics 108: S59–75. in Google Scholar

Hosszejni, D., and G. Kastner. 2018. “Approaches toward the Bayesian Estimation of the Stochastic Volatility Model with Leverage.” In International Conference on Bayesian Statistics in Action, 75–83. Springer.10.1007/978-3-030-30611-3_8Search in Google Scholar

Hwu, S.-T., and C.-J. Kim. 2019. “Estimating Trend Inflation Based on Unobserved Components Model: Is it Correlated with the Inflation Gap?” Journal of Money, Credit, and Banking 51 (8): 2305–19.10.1111/jmcb.12600Search in Google Scholar

Justiniano, A., and G. E. Primiceri. 2008. “The Time-Varying Volatility of Macroeconomic Fluctuations.” The American Economic Review 98 (3): 604–41. in Google Scholar

Kass, R. E., and A. E. Raftery. 1995. “Bayes Factors.” Journal of the American Statistical Association 90 (430): 773–95. in Google Scholar

Keating, J. W. 2013a. “Interpreting Permanent Shocks to Output when Aggregate Demand May Not Be Neutral in the Long Run.” Journal of Money, Credit, and Banking 45 (4): 747–56. in Google Scholar

Keating, J. W. 2013b. “What Do We Learn from Blanchard and Quah Decompositions of Output if Aggregate Demand May Not Be Long-Run Neutral?” Journal of Macroeconomics 38: 203–17. in Google Scholar

Lanne, M., H. Lütkepohl, and K. Maciejowska. 2010. “Structural Vector Autoregressions with Markov Switching.” Journal of Economic Dynamics and Control 34 (2): 121–31. in Google Scholar

Laubach, T., and J. C. Williams. 2003. “Measuring the Natural Rate of Interest.” The Review of Economics and Statistics 85 (4): 1063–70. in Google Scholar

Lütkepohl, H. 1984. “Linear Transformations of Vector ARMA Processes.” Journal of Econometrics 26 (3): 283–93. in Google Scholar

Lütkepohl, H. 2005. New Introduction to Multiple Time Series Analysis. Berlin, Germany: Springer Science & Business Media.10.1007/978-3-540-27752-1Search in Google Scholar

Lütkepohl, H., and A. Netšunajev. 2017. “Structural Vector Autoregressions with Heteroskedasticity: A Review of Different Volatility Models.” Econometrics and Statistics 1: 2–18. in Google Scholar

Lütkepohl, H., and T. Wozniak. 2017. “Bayesian Inference for Structural Vector Autoregressions Identified by Markov-Switching Heteroskedasticity.” In DIW Berlin German Institute for Economic Research Discussion Paper 1707.10.1016/j.jedc.2020.103862Search in Google Scholar

McCracken, M. W., and S. Ng. 2016. “FRED-MD: A Monthly Database for Macroeconomic Research.” Journal of Business & Economic Statistics 34 (4): 574–89. in Google Scholar

Milunovich, G., and M. Yang. 2013. “On Identifying Structural Var Models via Arch Effects.” Journal of Time Series Econometrics 5 (2): 117–31. in Google Scholar

Mitra, S., and T. M. Sinclair. 2012. “Output Fluctuations in the G-7: An Unobserved Components Approach.” Macroeconomic Dynamics 16 (3): 396–422. in Google Scholar

Morley, J. C. 2007. “The Slow Adjustment of Aggregate Consumption to Permanent Income.” Journal of Money, Credit, and Banking 39 (2-3): 615–38. in Google Scholar

Morley, J. C., C. R. Nelson, and E. Zivot. 2003. “Why Are the Beveridge-Nelson and Unobserved-Components Decompositions of GDP So Different?” The Review of Economics and Statistics 85 (2): 235–43. in Google Scholar

Netsunajev, A. 2013. “Reaction to Technology Shocks in Markov-Switching Structural VARs: Identification via Heteroskedasticity.” Journal of Macroeconomics 36: 51–62. in Google Scholar

Oh, K. H., E. Zivot, and D. Creal. 2008. “The Relationship between the Beveridge-Nelson Decomposition and Unobserved Components Models with Correlated Shocks.” Journal of Econometrics 146: 207–19. in Google Scholar

Perron, P., and T. Wada. 2009. “Let’s Take a Break: Trends and Cycles in US Real GDP.” Journal of Monetary Economics 56 (6): 749–65. in Google Scholar

Shephard, N. 2015. “Martingale Unobserved Component Models.” In Unobserved Components and Time Series Econometrics, edited by S. J. Koopman, and N. Shephard, Chapter 10. Oxford, UK: Oxford University Press.10.1093/acprof:oso/9780199683666.003.0010Search in Google Scholar

Sinclair, T. M. 2009. “The Relationships between Permanent and Transitory Movements in US Output and the Unemployment Rate.” Journal of Money, Credit, and Banking 41 (2-3): 529–42. in Google Scholar

Stock, J. H., and M. W. Watson. 1998. “Median Unbiased Estimation of Coefficient Variance in a Time-Varying Parameter Model.” Journal of the American Statistical Association 93 (441): 349–58. in Google Scholar

Stock, J. H., and M. W. Watson. 2005. “Implications of Dynamic Factor Models for VAR Analysis.” In National Bureau of Economic Research Working Paper.10.3386/w11467Search in Google Scholar

Stock, J. H., and M. W. Watson. 2007. “Why Has Us Inflation Become Harder to Forecast?” Journal of Money, Credit, and Banking 39: 3–33. in Google Scholar

Trenkler, C., and E. Weber. 2016. “On the Identification of Multivariate Correlated Unobserved Components Models.” Economics Letters 138 (1): 15–8. in Google Scholar

Velinov, A., and W. Chen. 2015. “Do stock Prices Reflect Their Fundamentals? New Evidence in the Aftermath of the Financial Crisis.” Journal of Economics and Business 80: 1–20. in Google Scholar

Waggoner, D. F., and T. Zha. 2003. “A Gibbs Sampler for Structural Vector Autoregressions.” Journal of Economic Dynamics and Control 28 (2): 349–66. in Google Scholar

Waggoner, D. F., H. Wu, and T. Zha. 2016. “Striated Metropolis–Hastings Sampler for High-Dimensional Models.” Journal of Econometrics 192 (2): 406–20. in Google Scholar

Weber, E. 2011. “Analyzing U.S. Output and the Great Moderation by Simultaneous Unobserved Components.” Journal of Money, Credit, and Banking 43 (8): 1579–97. in Google Scholar

Wozniak, T., and M. Droumaguetb. 2015. “Assessing Monetary Policy Models: Bayesian Inference for Heteroskedastic Structural VARs.” Technical report.Search in Google Scholar

Supplementary Material

The online version of this article offers supplementary material (

Received: 2020-02-27
Revised: 2021-05-02
Accepted: 2021-05-18
Published Online: 2021-06-03

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 31.3.2023 from
Scroll to top button