Implementation of Multivariate Continuous-Time ARMA Models

  • Helgi TómassonEmail author


The multivariate continuous-time ARMA model is a tool to capture the relationship between multivariate time series. In this chapter, a particular computational implementation of a stationary normal multivariate CARMA model is illustrated. A review of a parametric setup is shown. Data are assumed to be observed at irregular non-synchronous discrete time points. The computational approach for calculating the likelihood is based on a state-space form and the Kalman filter. Interpretation of the CARMA models is discussed. The computational algorithms have been implemented in R packages. Examples of a simulated and real data are shown.


  1. Belcher, J., Hampton, J., & Tunnicliffe Wilson, G. (1994). Parameterization of continuous autoregressive models for irregularly sampled time series data. Journal of the Royal Statistical Association, Series B, 56(1), 141–155.MathSciNetzbMATHGoogle Scholar
  2. Björk, T. (1998). Arbitrage theory in continuous time. Oxford: Oxford University Press.
  3. Box, G. E. P., & Jenkins, G. M. (1976). Time series analysis: Forecasting and control. San Fransisco: Holden Day.zbMATHGoogle Scholar
  4. Brockwell, P. J. (2014). Recent results in the theory and applications of carma processes. Annals of the Institute of Statistical Mathematics, 66(4), 647–685. MathSciNetCrossRefGoogle Scholar
  5. Brockwell, P. J., & Davis, R. A. (1991). Time series: Theory and methods. New York: Springer. CrossRefGoogle Scholar
  6. Brockwell, P. J., & Marquardt, T. (2005). Levy-driven and fractionally integrated ARMA processes with continuous time parameter. Statistica Sinica, 15, 477–494.MathSciNetzbMATHGoogle Scholar
  7. Fasen, V., & Fuchs, F. (2013). Spectral estimates for high-frequency sampled continuous-time autoregressive moving average processes. Journal of Time Series Analysis, 34(5), 532–551. MathSciNetCrossRefGoogle Scholar
  8. Garcia, I., Klüppelberg, C., & Müller, G. (2011). Estimation of stable CARMA models with an application to electricity spot prices. Statistical Modelling, 11(5), 447–470. MathSciNetCrossRefGoogle Scholar
  9. Greene, W. H. (2012). Econometric analysis (7th ed.). Upper Saddle River, NJ: Pearson.Google Scholar
  10. Harvey, A. C. (1989). Forecasting, structural time series models and the Kalman filter. Cambridge: Cambridge University Press.
  11. Iacus, S. M., & Mercuri, L. (2015). Implementation of Lévy Carma model in Yuima package. Computational Statistics, 30(4), 1111–1141. MathSciNetCrossRefGoogle Scholar
  12. Jouzel, J., et al. (2007). EPICA Dome C ice core 800kyr deuterium data and temperature estimates. IGBP PAGES/World Data Center for Paleoclimatology Data Contribution Series # 2007-091. NOAA/NCDC Paleoclimatology Program, Boulder CO, USA.Google Scholar
  13. Kawai, R. (2015). Sample path generation of Lévy-driven continuous-time autoregressive moving average processes. Methodology and Computing in Applied Probability, 19(1), 175–211. CrossRefGoogle Scholar
  14. Kreyszig, E. (1999). Advanced engineering mathematics (8th ed.). London: Wiley (Linear fractional transformation, p. 692).Google Scholar
  15. Luthi, D., Floch, M. L., Bereiter, B., Blunier, T., Barnola, J.-M., Siegenthaler, U., et al. (2008). High-resolution carbon dioxide concentration record 650,000-800,000 years before present. Nature, 453, 379–382. CrossRefGoogle Scholar
  16. Lütkepohl, H. (1991). Introduction to multiple time series analysis. Berlin: Springer. CrossRefGoogle Scholar
  17. Øksendal, B. (1998). Stochastic differential equations: An introduction with applications (5th ed.). Berlin: Springer. CrossRefGoogle Scholar
  18. Oud, J. H. L., & Voelkle, M. C. (2014). Do missing values exist? incomplete data handling in cross-national longitudinal studies by means of continuous time modeling. Quality & Quantity, 48(6), 3271–3288. CrossRefGoogle Scholar
  19. Schlemm, E., & Stelzer, R. (2012). Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes. Bernoulli, 18(1), 46–64. MathSciNetCrossRefGoogle Scholar
  20. Singer, H. (1992). Continuous-time dynamical systems with sampled data, errors of measurment and unobserved components. Journal of Time Series, 14(5), 527–544. CrossRefGoogle Scholar
  21. Singer, H. (1995). Analytical score function for irregularly sampled continuous time stochastic processes with control variables and missing values. Econometric Theory, 11(4), 721–735. MathSciNetCrossRefGoogle Scholar
  22. Tómasson, H. (2015). Some computational aspects of Gaussian CARMA modelling. Statistics and Computing, 25(2), 375–387. MathSciNetCrossRefGoogle Scholar
  23. Tsai, H., & Chan, K. (2000). A note on the covariance structure of a continuous-time ARMA process. Statistica Sinica, 10, 989–998.MathSciNetzbMATHGoogle Scholar
  24. Zadrozny, P. (1988). Gaussian likelihood of continuous-time ARMAX models when data are stocks and flows at different frequencies. Econometric Theory, 4(1), 108–124. CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of EconomicsUniversity of IcelandReykjavíkIceland

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