Continuous-time Gaussian autoregression

The problem of fitting continuous-time autoregressions (linear and nonlinear) to closely and regularly spaced data is considered. For the linear case Jones (1981) and Bergstrom (1985) used state-space representations to compute exact maximum likelihood estimators, and Phillips (1959) did so by fitting an appropriate discrete-time ARMA process to the data. In this paper we use exact conditional maximum likelihood estimators for the continuously-observed process to derive approximate maximum likelihood estimators based on the closely-spaced discrete observations. We do this for both linear and non-linear autoregressions, and indicate how the method can be modified also to deal with non-uniformly but closely-spaced data. Examples are given to indicate the accuracy of the procedure.