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Continuous time ARMA processes: Discrete time representation and likelihood evaluation

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  • Thornton, Michael A.
  • Chambers, Marcus J.

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 a state space representation and the Kalman–Bucy filter. We demonstrate the value of our approach in two applications: a univariate study of the yield curve at different frequencies; and, a multivariate study of the relationship between US GDP and oil prices, taking account of the mixed frequencies with which these data are available.

Suggested Citation

  • Thornton, Michael A. & Chambers, Marcus J., 2017. "Continuous time ARMA processes: Discrete time representation and likelihood evaluation," Journal of Economic Dynamics and Control, Elsevier, vol. 79(C), pages 48-65.
    • Handle: RePEc:eee:dyncon:v:79:y:2017:i:c:p:48-65
    DOI: 10.1016/j.jedc.2017.03.012
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    16. 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.
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    18. 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.
    19. 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.
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    2. 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.
    3. Fasen-Hartmann, Vicky & Mayer, Celeste, 2023. "Empirical spectral processes for stationary state space models," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 319-354.
    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.
    5. Vicky Fasen-Hartmann & Celeste Mayer, 2022. "Whittle estimation for continuous-time stationary state space models with finite second moments," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(2), pages 233-270, April.
    6. Vicky Fasen‐Hartmann & Sebastian Kimmig, 2020. "Robust estimation of stationary continuous‐time arma models via indirect inference," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(5), pages 620-651, September.
    7. Ralf Korn & Bilgi Yilmaz, 2022. "House Prices as a Result of Trading Activities: A Patient Trader Model," Computational Economics, Springer;Society for Computational Economics, vol. 60(1), pages 281-303, June.
    8. 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.
    9. 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.
    10. Szarek, Dawid & Bielak, Łukasz & Wyłomańska, Agnieszka, 2020. "Long-term prediction of the metals’ prices using non-Gaussian time-inhomogeneous stochastic process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 555(C).

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    More about this item

    Keywords

    Continuous time; ARMA process; State space; Discrete time representation; Mixed frequency;
    All these keywords.

    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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