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Multivariate Markov-switching score-driven models: an application to the global crude oil market

  • Szabolcs Blazsek , Alvaro Escribano EMAIL logo and Adrian Licht


A new class of multivariate nonlinear quasi-vector autoregressive (QVAR) models is introduced. It is a Markov switching score-driven model with stochastic seasonality for the multivariate t-distribution (MS-Seasonal-t-QVAR). As an extension, we allow for the possibility of having common-trends and nonlinear co-integration. Score-driven nonlinear updates of local level and seasonality are used, which are robust to outliers within each regime. We show that VAR integrated moving average (VARIMA) type filters are special cases of QVAR filters. Using exclusion, sign, and elasticity identification restrictions in MS-Seasonal-t-QVAR with common-trends, we provide short-run and long-run impulse response functions for the global crude oil market.

Corresponding author: Alvaro Escribano, Universidad Carlos III de Madrid, Department of Economics, Madrid 126, Getafe 28903, Madrid, Spain, E-mail:

Funding source: Community of Madrid, Agencia Estatal de Investigación, Ministry of Economics of Spain, Universidad Francisco Marroquín

Award Identifier / Grant number: MadEco-CM S2015/HUM-3444

Award Identifier / Grant number: 2019/00419/001

Award Identifier / Grant number: ECO2016/00105/001MDM 2014-0431

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

  2. Research funding: Szabolcs Blazsek and Adrian Licht acknowledge funding from the School of Business of Universidad Francisco Marroquin. Alvaro Escribano acknowledges funding from the Ministry of Economics (ECO2016-00105-001, MDM 2014-0431), the Community of Madrid (MadEco-CM S2015/HUM-3444), and Agencia Estatal de Investigacion (2019/00419/001) of Spain.

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


Abramson, A., and I. Cohen. 2007. “On the Stationarity of Markov-Switching GARCH Processes.” Econometric Theory 23 (3): 485–500. in Google Scholar

Baumeister, C., and J. D. Hamilton 2019. “Structural Interpretation of Vector Autoregressions with Incomplete Identification: Setting the Record Straight.” Unpublished Manusript, in Google Scholar

Baumeister, C., and G. Peersman. 2013a. “The Role of Time-Varying Price Elasticities in Accounting for Volatility Changes in the Crude Oil Market.” Journal of Applied Econometrics 28 (7): 1087–109. in Google Scholar

Baumeister, C., and G. Peersman. 2013b. “Time-Varying Effects of Oil Supply Shocks on the US Economy.” American Economic Journal: Macroeconomics 5 (4): 1–28. in Google Scholar

Baxter, M., and R. G. King. 1999. “Measuring Business Cycles: Approximate Band-Pass Filter for Economic Time Series.” The Review of Economics and Statistics 81 (4): 575–93. in Google Scholar

Bjørnland, H. 2019. “Supply Flexibility in the Shale Patch: Facts, No Fiction.” In Working Paper No. 08/2019 Centre for Applied Macroeconomics and Commodity Prices (CAMP), BI Norwegian Business School, 11250/2629918.10.2139/ssrn.3434198Search in Google Scholar

Bjørnland, H., F. Nordvik, and M. Rohrer. 2017. “Supply Flexibility in the Shale Patch: Evidence from North Dakota.” In Working Paper No. 2/2017 Centre for Applied Macroeconomics and Commodity Prices (CAMP), BI Norwegian Business School in Google Scholar

Bjørnland, H. C., and J. Zhulanova 2019. “The Shale Oil Boom and the US Economy: Spillovers and Time-Varying Effects.” In CAMA Working Paper No. 59/2019, 14979/shale-oil-boom-and-us-economy-spillovers-and-time.10.2139/ssrn.3436499Search 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

Blasques, F., S. J. Koopman, and A. Lucas. 2017. “Maximum Likelihood Estimation for Score-Driven Models.” In TI 2014-029/III Tinbergen Institute Discussion Paper in Google Scholar

Blazsek, S., A. Escribano, and A. Licht. 2017. “Score-Driven Nonlinear Multivariate Dynamic Location Models.” In Working Paper 17-14: University Carlos III of Madrid, Department of Economics in Google Scholar

Blazsek, S., A. Escribano, and A. Licht. 2020a. “Identification of Seasonal Effects in Impulse Responses Using Score-Driven Multivariate Location Models.” Journal of Econometric Methods 10 (1): 53–66. in Google Scholar

Blazsek, S., A. Escribano, and A. Licht. 2020b. “Nonlinear Common Trends for the Global Crude Oil Market: Markov-Switching Score-Driven Models of the Multivariate t-Distribution.” In Working Paper 20-04: University Carlos III of Madrid, Department of Economic, in Google Scholar

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

Brandt, A. 1986. “The Stochastic Equation Yn+1 = AnYn + Bn with Stationary Coefficients.” Advances in Applied Probability 18 (1): 211–20. in Google Scholar

Caivano, M., A. C. Harvey, and A. Luati. 2016. “Robust Time Series Models with Trend and Seasonal Components.” SERIEs Journal of the Spanish Economic Association 7 (1): 99–120. in Google Scholar

Caldara, D., M. Cavallo, and M. Iacoviello. 2019. “Oil Price Elasticities and Oil Price Fluctuations.” Journal of Monetary Economics 103: 1–20. in Google Scholar

Clark, T. E., and S. J. Terry. 2010. “Time Variation in the Inflation Passthrough of Energy Prices.” Journal of Money, Credit, and Banking 42 (7): 1419–33. in Google Scholar

Creal, D., S. J. Koopman, and A. Lucas. 2008. “A General Framework for Observation Driven Time-Varying Parameter Models.” In Discussion Paper 08-108/4: Tinbergen Institute, framework-for-observation-driven-time-varying-parameter-modelsdiscussion.10.2139/ssrn.1297183Search in Google Scholar

Creal, D., S. J. Koopman, and A. Lucas. 2011. “A Dynamic Multivariate Heavy-Tailed Model for Time-Varying Volatilities and Correlations.” Journal of Business & Economic Statistics 29 (4): 552–63. in Google Scholar

Creal, D., S. J. Koopman, and A. Lucas. 2013. “Generalized Autoregressive Score Models with Applications.” Journal of Applied Econometrics 28 (5): 777–95. in Google Scholar

Creal, D., B. Schwaab, S. J. Koopman, and A. Lucas. 2014. “Observation-Driven Mixed-Measurement Dynamic Factor Models with an Application to Credit Risk.” The Review of Economics and Statistics 96 (5): 898–915. in Google Scholar

Elton, J. H. 1990. “A Multiplicative Ergodic Theorem for Lipschitz Maps.” Stochastic Processes and their Applications 34 (1): 39–47. in Google Scholar

Escanciano, J. C., and I. N. Lobato. 2009. “An Automatic Portmanteau Test for Serial Correlation.” Journal of Econometrics 151 (2): 140–9. in Google Scholar

Gray, S. 1996. “Modeling the Conditional Distribution of Interest Rates as a Regime-Switching Process.” Journal of Financial Economics 42 (1): 27–62. in Google Scholar

Haas, M., S. Mittink, and M. S. Paolella. 2004. “A New Approach to Markov-Switching GARCH Models.” Journal of Financial Econometrics 2 (4): 493–530. in Google Scholar

Hahn, E., and R. Mestre. 2011. “The Role of Oil Prices in the Euro Area Economy since the 1970s.” In Working Paper No 1356: European Central Bank, in Google Scholar

Hamilton, J. D. 1989. “A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle.” Econometrica 57 (2): 357–84. in Google Scholar

Hamilton, J. D. 1994. Time Series Analysis. Princeton: Princeton University Press.10.1515/9780691218632Search in Google Scholar

Harvey, A. C. 2013. Dynamic Models for Volatility and Heavy Tails. Cambridge: Cambridge University Press.10.1017/CBO9781139540933Search in Google Scholar

Harvey, A. C., and T. Chakravarty. 2008. “Beta-t-(E)GARCH.” In Cambridge Working Papers in Economics 0840: Faculty of Economics, University of Cambridge, in Google Scholar

Harvey, A. C., and A. Luati. 2014. “Filtering with Heavy Tails.” Journal of the American Statistical Association 109 (507): 1112–22. in Google Scholar

Herwartz, H., and H. Lütkepohl. 2000. “Multivariate Volatility Analysis of VW Stock Prices.” International Journal of Intelligent Systems in Accounting, Finance & Management 9 (1): 35–54.<35::aid-isaf176>;2-v.10.1002/(SICI)1099-1174(200003)9:1<35::AID-ISAF176>3.0.CO;2-VSearch in Google Scholar

Hindrayanto, I., J. P. A. M. Jacobs, D. R. Osborn, and J. Tian. 2018. “Trend-Cycle-Seasonal Interactions: Identification and Estimation.” Macroeconomic Dynamics 23 (8): 3163–88. in Google Scholar

Johansen, S. 1995. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press.10.1093/0198774508.001.0001Search in Google Scholar

Kilian, L. 2008. “The Economic Effects of Energy Price Shocks.” Journal of Economic Literature 46 (4): 871–909. in Google Scholar

Kilian, L. 2009. “Not All Oil Price Shocks Are Alike: Disentangling Demand and Supply Shocks in the Crude Oil Market.” The American Economic Review 99 (3): 1053–69. in Google Scholar

Kilian, L., and D. P. Murphy. 2012. “Why Agnostic Sign Restrictions Are Not Enough: Understanding the Dynamics of Oil Market VAR Models.” Journal of the European Economic Association 10 (5): 1166–88. in Google Scholar

Kilian, L., and D. P. Murphy. 2014. “The Role of Inventories and Speculative Trading in the Global Market for Crude Oil.” Journal of Applied Econometrics 29 (3): 454–78. in Google Scholar

Kim, C. J., and C. R. Nelson. 1999. State-Space Models with Regime Switching. Cambridge: The MIT Press.Search in Google Scholar

King, R. G., and M. W. Watson. 1997. “Testing Long-Run Neutrality.” Federal Reserve Bank of Richmond Economic Quarterly 83 (3): 69–101.10.3386/w4156Search in Google Scholar

Klaassen, F. 2002. “Improving GARCH Volatility Forecasts with Regime-Switching GARCH.” Empirical Economics 27 (2): 363–94. in Google Scholar

Krolzig, H.-M. 1997. Markov-Switching Vector Autoregressions: Modelling, Statistical Inference, and Application to Business Cycle Analysis. Berlin: Springer-Verlag.10.1007/978-3-642-51684-9Search in Google Scholar

Li, W. K. 2004. Diagnostic Checks in Time Series. Boca Raton: Chapman & Hall.10.1201/9780203485606Search in Google Scholar

Lütkepohl, H. 2005. New Introduction to Multivariate Time Series Analysis. Berlin: Springer-Verlag.10.1007/978-3-540-27752-1Search in Google Scholar

Millard, S. and T. Shakir. 2013. “Oil Shocks and the UK Economy: The Changing Nature of Shocks and Impact over Time.” In Working Paper No. 476: Bank of England, oil-shocks-and-the-uk-economy-the-changing-nature-of-shocks-and-impact-over-time.pdf.10.2139/ssrn.2313021Search in Google Scholar

Newey, K., and K. D. West. 1987. “A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica 55 (3): 703–8. in Google Scholar

Riggi, M., and F. Venditti. 2015. “The Time Varying Effect of Oil Price Shocks on Euro-Area Exports.” Journal of Economic Dynamics and Control 59: 75–94. in Google Scholar

Rubio-Ramirez, J. E., D. Waggoner, and T. Zha. 2010. “Structural Vector Autoregressions: Theory for Identification and Algorithms for Inference.” The Review of Economic Studies 77 (2): 665–96. in Google Scholar

Stock, J. H., and M. W. Watson. 2001. “Vector Autoregressions.” The Journal of Economic Perspectives 15 (4): 101–15. in Google Scholar

Straumann, D., and T. Mikosch. 2006. “Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach.” Annals of Statistics 34 (1): 2449–95. in Google Scholar

Valcarcel, V., and M. Wohar. 2013. “Changes in the Oil Price-Inflation Pass-Through.” Journal of Economics and Business 68: 24–42. in Google Scholar

Van Robays, I. 2012. “Macroeconomic Uncertainty and the Impact of Oil Shocks.” Working Paper No 1479: European Central Bank, in Google Scholar

Supplementary Material

The online version of this article offers supplementary material (

Received: 2020-08-25
Revised: 2021-03-21
Accepted: 2021-04-04
Published Online: 2021-04-28

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