Continuous Time ARMA Processes: Discrete Time Representation and Likelihood Evaluation

Michael Alan Thornton, Marcus Chambers

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)48-65
Number of pages18
JournalJournal of Economic Dynamics and Control
Early online date31 Mar 2017
Publication statusPublished - 1 Jun 2017

Bibliographical note

© Elsevier, 2017. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy.


  • continuous time
  • ARMA process
  • state space
  • discrete time representation
  • mixed frequency
  • Discrete time representation
  • Mixed frequency
  • Continuous time
  • State space

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