Modeling irregularly spaced residual series as a continuous stochastic process

In this paper, the background and functioning of a simple but effective continuous time approach for modeling irregularly spaced residual series is presented. The basic equations were published earlier by von Asmuth et al. (2002), who used them as part of a continuous time transfer function noise model. It is shown that the methods behind the model are build on two principles: The first is the fact that the equations of a Kalman filter degenerate to a form that is equivalent to “conventional” autoregressive moving average (ARMA) models when the modeled data are considered to be free of measurement errors. This assumption, in comparison to the “full” Kalman filter, also yields a better prediction efficiency (Ahsan and O'Connor, 1994). The second is the mathematical equivalence between discrete time AR parameters and continuous exponentials and the point that continuous time models provide an elegant solution for modeling irregularly spaced observations (e.g., Harvey, 1989). Because simple least squares methods do not apply in case of modeling irregular data, a sum of weighted squared innovations (SWSI) criterion is introduced and derived from the likelihood function of the innovations. In an example application it is shown that the estimates of the SWSI criterion converge to maximum likelihood estimates for larger sample sizes. Finally, we propose to use the so-called innovation variance function as an additional diagnostic check, next to the well-known autocorrelation and cross-correlation functions.