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The Estimation of Higher-Order Continuous Time Autoregressive Models

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  • Harvey, A. C.
  • Stock, James H.

Abstract

A method is presented for computing maximum likelihood, or Gaussian, estimators of the structural parameters in a continuous time system of higherorder stochastic differential equations. It is argued that it is computationally efficient in the standard case of exact observations made at equally spaced intervals. Furthermore it can be applied in situations where the observations are at unequally spaced intervals, some observations are missing and/or the endogenous variables are subject to measurement error. The method is based on a state space representation and the use of the Kalman–Bucy filter. It is shown how the Kalman-Bucy filter can be modified to deal with flows as well as stocks.

Suggested Citation

  • 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.
    • Handle: RePEc:cup:etheor:v:1:y:1985:i:01:p:97-117_01
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    Cited by:

    1. 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.
    2. 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.
    3. D. Stephen G. Pollock, 2020. "Linear Stochastic Models in Discrete and Continuous Time," Econometrics, MDPI, vol. 8(3), pages 1-22, September.
    4. 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.
    5. 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.
    6. Andrew Filardo & Marco Jacopo Lombardi & Marek Raczko, 2018. "Measuring financial cycle time," BIS Working Papers 755, Bank for International Settlements.
    7. J. Roderick McCrorie, 2000. "The Likelihood of a Continuous-time Vector Autoregressive Model," Working Papers 419, Queen Mary University of London, School of Economics and Finance.
    8. Stock, James H., 1987. "Measuring Business Cycle Time," Scholarly Articles 3425950, Harvard University Department of Economics.
    9. Lo, Andrew W., 1988. "Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data," Econometric Theory, Cambridge University Press, vol. 4(2), pages 231-247, August.
    10. Hermann Singer, 2003. "Simulated Maximum Likelihood in Nonlinear Continuous-Discrete State Space Models: Importance Sampling by Approximate Smoothing," Computational Statistics, Springer, vol. 18(1), pages 79-106, March.
    11. D.S.G. Pollock, "undated". "Linear Stochastic Models in Discrete and Continuous Time," Discussion Papers in Economics 19/10, Division of Economics, School of Business, University of Leicester.
    12. Ghysels, E. & Jasiak, J., 1994. "Stochastic Volatility and time Deformation: An Application of trading Volume and Leverage Effects," Cahiers de recherche 9403, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    13. Lawrence J. Christiano & Martin S. Eichenbaum, 1985. "A continuous time, general equilibrium, inventory-sales model," Working Papers 361, Federal Reserve Bank of Minneapolis.
    14. Parra-Alvarez, Juan Carlos & Polattimur, Hamza & Posch, Olaf, 2021. "Risk matters: Breaking certainty equivalence in linear approximations," Journal of Economic Dynamics and Control, Elsevier, vol. 133(C).
    15. Lars Peter Hansen & Thomas J. Sargent, 1993. "Recursive linear models of dynamic economies," Proceedings, Federal Reserve Bank of San Francisco, issue Mar.
    16. Michael A. Thornton & Marcus J. Chambers, 2013. "Temporal aggregation in macroeconomics," Chapters, in: Nigar Hashimzade & Michael A. Thornton (ed.), Handbook of Research Methods and Applications in Empirical Macroeconomics, chapter 13, pages 289-310, Edward Elgar Publishing.
    17. Hermann Singer, 2011. "Continuous-discrete state-space modeling of panel data with nonlinear filter algorithms," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 95(4), pages 375-413, December.
    18. Comte, F., 1998. "Discrete and continuous time cointegration," Journal of Econometrics, Elsevier, vol. 88(2), pages 207-226, November.
    19. Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," Econometrica, Econometric Society, vol. 63(4), pages 767-804, July.
    20. Roderick McCrorie, J., 2001. "Interpolating exogenous variables in continuous time dynamic models," Journal of Economic Dynamics and Control, Elsevier, vol. 25(9), pages 1399-1427, September.
    21. 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.
    22. Michael A. Thornton & Marcus J. Chambers, 2013. "Continuous-time autoregressive moving average processes in discrete time: representation and embeddability," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(5), pages 552-561, September.
    23. Milena Hoyos, 2020. "Mixed First‐ and Second‐Order Cointegrated Continuous Time Models with Mixed Stock and Flow Data," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(2), pages 249-267, March.
    24. 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.
    25. Tucker S. McElroy & Thomas M. Trimbur, 2007. "Continuous time extraction of a nonstationary signal with illustrations in continuous low-pass and band-pass filtering," Finance and Economics Discussion Series 2007-68, Board of Governors of the Federal Reserve System (U.S.).

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