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Continuous Time ARMA Processes: Discrete Time Representation and Likelihood Evaluation

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  • Michael Thornton
  • Marcus Chambers

Abstract

This paper explores the representation and estimation of mixed continuous time ARMA (autoregressive moving average) systems of orders p, q. Taking the general case of mixed stock and flow variables, we discuss new state space and exact discrete time representations and demonstrate that the discrete time ARMA representations widely used in empirical work, based on differencing stock variables, are members of a class of observationally equivalent discrete time ARMA(p + 1, p) representations, which includes a more natural ARMA(p, p) representation. We compare and contrast two approaches to likelihood evaluation and computation, namely one based on an exact discrete time representation and another utilising astate space representation and the Kalman-Bucy filter.

Suggested Citation

  • Michael Thornton & Marcus Chambers, 2016. "Continuous Time ARMA Processes: Discrete Time Representation and Likelihood Evaluation," Discussion Papers 16/10, Department of Economics, University of York.
  • Handle: RePEc:yor:yorken:16/10
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    References listed on IDEAS

    as
    1. Chambers, Marcus J., 2009. "Discrete Time Representations Of Cointegrated Continuous Time Models With Mixed Sample Data," Econometric Theory, Cambridge University Press, vol. 25(4), pages 1030-1049, August.
    2. Zadrozny, Peter A., 2016. "Extended Yule–Walker identification of VARMA models with single- or mixed-frequency data," Journal of Econometrics, Elsevier, vol. 193(2), pages 438-446.
    3. Thornton, Michael A. & Chambers, Marcus J., 2016. "The exact discretisation of CARMA models with applications in finance," Journal of Empirical Finance, Elsevier, vol. 38(PB), pages 739-761.
    4. Harvey, A. C. & Stock, James H., 1989. "Estimating integrated higher-order continuous time autoregressions with an application to money-income causality," Journal of Econometrics, Elsevier, vol. 42(3), pages 319-336, November.
    5. Harvey, A. C. & Stock, James H., 1988. "Continuous time autoregressive models with common stochastic trends," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 365-384.
    6. Bergstrom, A. R., 1986. "The Estimation of Open Higher-Order Continuous Time Dynamic Models with Mixed Stock and Flow Data," Econometric Theory, Cambridge University Press, vol. 2(3), pages 350-373, December.
    7. Bergstrom, A.R., 1997. "Gaussian Estimation of Mixed-Order Continuous-Time Dynamic Models with Unobservable Stochastic Trends from Mixed Stock and Flow Data," Econometric Theory, Cambridge University Press, vol. 13(4), pages 467-505, February.
    8. Marcus J. Chambers, 2011. "Cointegration and sampling frequency," Econometrics Journal, Royal Economic Society, vol. 14(2), pages 156-185, July.
    9. Harvey, A. C. & Stock, James H., 1985. "The Estimation of Higher-Order Continuous Time Autoregressive Models," Econometric Theory, Cambridge University Press, vol. 1(1), pages 97-117, April.
    10. Bergstrom,Albert Rex & Nowman,Khalid Ben, 2012. "A Continuous Time Econometric Model of the United Kingdom with Stochastic Trends," Cambridge Books, Cambridge University Press, number 9781107411234, April.
    11. Bergstrom, A. R., 1985. "The Estimation of Parameters in Nonstationary Higher Order Continuous-Time Dynamic Models," Econometric Theory, Cambridge University Press, vol. 1(3), pages 369-385, December.
    12. Phillips, P C B, 1991. "Error Correction and Long-Run Equilibrium in Continuous Time," Econometrica, Econometric Society, vol. 59(4), pages 967-980, July.
    13. Chambers, Marcus J., 2016. "The estimation of continuous time models with mixed frequency data," Journal of Econometrics, Elsevier, vol. 193(2), pages 390-404.
    14. Chambers, Marcus J. & Thornton, Michael A., 2012. "Discrete Time Representation Of Continuous Time Arma Processes," Econometric Theory, Cambridge University Press, vol. 28(01), pages 219-238, February.
    15. Marcellino, Massimiliano, 1999. "Some Consequences of Temporal Aggregation in Empirical Analysis," Journal of Business & Economic Statistics, American Statistical Association, vol. 17(1), pages 129-136, January.
    16. Chambers, Marcus J., 1999. "Discrete time representation of stationary and non-stationary continuous time systems," Journal of Economic Dynamics and Control, Elsevier, vol. 23(4), pages 619-639, February.
    17. Zadrozny, Peter, 1988. "Gaussian Likelihood of Continuous-Time ARMAX Models When Data Are Stocks and Flows at Different Frequencies," Econometric Theory, Cambridge University Press, vol. 4(1), pages 108-124, April.
    18. Zadrozny, Peter A., 1998. "An eigenvalue method of undetermined coefficients for solving linear rational expectations models," Journal of Economic Dynamics and Control, Elsevier, vol. 22(8-9), pages 1353-1373, August.
    19. Bergstrom, Albert Rex, 1983. "Gaussian Estimation of Structural Parameters in Higher Order Continuous Time Dynamic Models," Econometrica, Econometric Society, vol. 51(1), pages 117-152, January.
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    Cited by:

    1. Chambers, MJ & McCrorie, JR & Thornton, MA, 2017. "Continuous Time Modelling Based on an Exact Discrete Time Representation," Economics Discussion Papers 20497, University of Essex, Department of Economics.
    2. Sikora, Grzegorz & Michalak, Anna & Bielak, Łukasz & Miśta, Paweł & Wyłomańska, Agnieszka, 2019. "Stochastic modeling of currency exchange rates with novel validation techniques," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1202-1215.
    3. Antoine GODIN & Sakir-Devrim YILMAZ, 2020. "Modelling Small Open Developing Economies in a Financialized World: A Stock-Flow Consistent Prototype Growth Model," Working Paper 5eb7e0e8-560f-4ce6-91a5-5, Agence française de développement.
    4. Thornton, Michael A. & Chambers, Marcus J., 2016. "The exact discretisation of CARMA models with applications in finance," Journal of Empirical Finance, Elsevier, vol. 38(PB), pages 739-761.

    More about this item

    Keywords

    Continuous time; ARMA process; state space; discrete time representation.;

    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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