## 1 Introduction

Continuous time models are natural, as time is a continuously flowing quantity without steps. On the other hand, empirical data in the social sciences and economics are mostly available only at certain time points, e.g. daily, weekly, quarterly or at arbitrary times. Only physical quantities such as voltages, pressures, levels of rivers, etc. may be measured on a continuous basis. Therefore, there has been a tendency to formulate dynamical models in discrete time (times series and panel analysis). Thus, the causal relations are specified between the arbitrary discrete measurement times. Bartlett (1946) argues as follows:

It will have been apparent that the discrete nature of our observations in many economic and other time series does not reflect any lack of continuity in the underlying series. Thus theoretically it should often prove more fundamental to eliminate this imposed artificiality.

An unemployment index does not cease to exist between readings, nor does Yule's pendulum cease to swing. (emphasis H.S.)

Indeed there are many disadvantages of discrete time models. One of the most basic defects is that the dynamics are modeled between the (arbitrarily sampled) measurements and not between the dynamically relevant system states. For example, a physical system like a pendulum (cf. the citation above) fulfils a simple linear relation (Newton's equation for small amplitudes) between the state and its velocity change (acceleration), whereas the relationship between sampled measurements (e.g. daily) is very complicated and nonlinearly dependent on the parameters (mass, length of the pendulum, etc.) and the sampling interval. Moreover, the velocity cannot be measured with discrete time data (latent variable).

Discrete time studies with different sampling intervals cannot be compared, because the causal parameters relate to the chosen interval. Moreover, if the same dataset is analyzed with different intervals (select a weekly or monthly dataset from daily measurements), one gets estimates corresponding to these intervals which can be in contradiction.

Nevertheless, the *continuous–discrete state-space model* is able to combine both points of view:

- 1a continuous time dynamical model;
- 2discrete time (sampled) measurements.

This hybrid model first appeared in engineering (Jazwinski, 1970), but is now well known in econometrics, sociology, and psychology. One can estimate the parameters of the continuous time model from time series or panel measurements. This is achieved by computing the conditional probability density between the measurement times. In the linear Gaussian case, only the time-dependent conditional mean and autocovariance are needed. More generally, in the presence of latent states and errors of measurement, a measurement model can be defined, mapping the continuous time state to observable discrete time data.