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The Estimation of Higher-Order Continuous Time Autoregressive Models


  • Harvey, A. C.
  • Stock, James H.


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(01), pages 97-117, April.
  • Handle: RePEc:cup:etheor:v:1:y:1985:i:01:p:97-117_01

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    1. repec:eee:dyncon:v:79:y:2017:i:c:p:48-65 is not listed on IDEAS
    2. 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.
    3. Stock, James H., 1987. "Measuring Business Cycle Time," Scholarly Articles 3425950, Harvard University Department of Economics.
    4. Lawrence J. Christiano & Martin Eichenbaum, 1985. "A continuous time, general equilibrium, inventory-sales model," Working Papers 361, Federal Reserve Bank of Minneapolis.
    5. 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.
    6. 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.
    7. Lo, Andrew W., 1988. "Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data," Econometric Theory, Cambridge University Press, vol. 4(02), pages 231-247, August.
    8. 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.
    9. 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.).
    10. Lars Peter Hansen & Thomas J. Sargent, 1993. "Recursive linear models of dynamic economies," Proceedings, Federal Reserve Bank of San Francisco, issue Mar.
    11. Michael A. Thornton & Marcus J. Chambers, 2013. "Temporal aggregation in macroeconomics," Chapters,in: Handbook of Research Methods and Applications in Empirical Macroeconomics, chapter 13, pages 289-310 Edward Elgar Publishing.
    12. 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.
    13. Comte, F., 1998. "Discrete and continuous time cointegration," Journal of Econometrics, Elsevier, vol. 88(2), pages 207-226, November.
    14. 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.
    15. 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.
    16. 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.

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