Chapter 20 Continuous time stochastic models and issues of aggregation over time

This chapter describes statistical methods that are applicable to a class of continuous time stochastic models and discusses the theoretical foundations of these methods. An important feature of the class of models is that such models allow for the incorporation of a priori restrictions, such as those derived from economic theory, through the structural parameters of the continuous time system. They are used to represent a dynamic system of causal relations in which each variable adjusts continuously in response to the stimulus provided by other variables, and the adjustment relations involve the basic structural parameters in some optimization theory. These structural parameters are estimated from a sample comprising a sequence of discrete observations of the variables that are a mixture of stock variables and flow variables. In this way, it is possible to take advantage of the a priori restrictions derived from economic theory without making the unrealistic assumption that the economy moves in discrete jumps among successive positions of temporary equilibrium. On the theoretical side an important and relatively unexplored field of research is in the development of methods of estimation for systems of nonlinear stochastic differential equations. In some cases, it is possible to derive the exact likelihood function in terms of the discrete observations generated by a system of nonlinear stochastic differential equations. But, more generally, approximate methods are relied on, possibly involving the use of numerical solutions to the nonlinear differential equations system.