PROBLEMS WITH THE ESTIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS USING STRUCTURAL EQUATIONS MODELS.

The present paper deals with the identification and maximum likelihood estimation of systems of linear stochastic differential equations using panel data. So we only have a sample of discrete observations over time of the relevant variables for each individual. A popular approach in the social sciences advocates the estimation of the exact discrete model after a reparameterization with LISREL or similar programs for structural equations models. The exact discrete model corresponds to the continuous time model in the sense that observations at equidistant points in time that are generated by the latter system also satisfy the former. In the LISREL approach the reparameterized discrete time model is estimated first without taking into account the nonlinear mapping from the continuous to the discrete time parameters. In a second step, using the inverse mapping, the fundamental system parameters of the continuous time system in which we are interested, are inferred. However, some severe problems arise with this indirect approach. First, an identification problem may arise in multiple equation systems, since the matrix exponential function defining some of the new parameters is in general not one-to-one, and hence the inverse mapping mentioned above does not exist. Second, usually some sort of approximation of the time paths of the exogenous variables is necessary before the structural parameters of the system can be estimated with discrete data. Two simple approximation methods are discussed. In both approximation methods the resulting new discrete time parameters are connected in a complicated way. So estimating the reparameterized discrete model by OLS without restrictions does not yield maximum likelihood estimates of the desired continuous time parameters as claimed by some authors. [ABSTRACT FROM AUTHOR]