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 denning 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. Third, a further limitation of estimating the reparameterized model with programs for structural equations models is that even simple restrictions on the original fundamental parameters of the continuous time system cannot be dealt with. This issue is also discussed in some detail. For these reasons the "indirect method" cannot be recommended. In many cases the approach leads to misleading inferences. We strongly advocate the direct estimation of the continuous time parameters. This approach is more involved, because the exact discrete model is nonlinear in the original parameters. A computer program by Hermann Singer that provides appropriate maximum likelihood estimates is described.

This paper deals with the identification and maximum likelihood estimation of the parameters of a stochastic differential equation from discrete time sampling. Score function and maximum likelihood equations are derived explicitly. The stochastic differential equation system is extended to allow for random effects and the analysis of panel data. In addition, we investigate the identifiability of the continuous time parameters, in particular the impact of the inclusion of exogenous variables.

Slow event-related potentials (ERP) were examined in healthy and aphasic subjects in two-stimulus designs comprising a word comprehension and a rhyming task. Aphasics, though selected to perform above chance level, made significantly more errors and responded more slowly than controls, although canonical correlations did not indicate a statistical relationship between performance measures and ERP amplitudes. A discriminant analysis of ERP amplitudes distinguished the groups for the slow wave (SW; 0.5-1.0 s post-S1 onset) in the word comprehension, for the SW and the initial contingent negative variation (iCNV; 1.0-2.0 s post-S1 onset) in the rhyming task. Similarly for both tasks, ERP topography showed left-anterior predominance of the negative SW and iCNV in controls, whereas participants with aphasia showed smaller anterior and larger left-posterior amplitudes. The centroparietal terminal CNV (tCNV; 1 s pre-S2) was smaller in participants with aphasia than in controls, but similar in topography. Results suggest left-anterior activation for those language processes that were presumably provoked in the present tasks, like lexical access, or phonological encoding. The pattern of participants with aphasia may indicate effects of language impairment and recovery, but also consequences of the brain damage.

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