The estimation of continuous-time systems using discrete data

Abstract

This thesis is concerned with the estimation of parameters in continuous-time systems, when the available data consist of time series sampled at regular intervals of time. Chapter 1 begins with a discussion of the circumstances in which the work may be relevant. We also describe useful results in matrix theory and in the spectral theory of continuous stationary processes. In Chapter 2 the general continuous-time model is specified by a sequence of detailed assumptions. It may be regarded as the solution of a system that is linear in the endogenous and exogenous variables, the parameters possibly occuring in a non-linear fashion and satisfying non-linear constraints. The basic method of estimation involves the Fourier transformation of the model and the insertion of the discrete Fourier transforms to produce an approximate model that is of regression type and is thus relatively easy to handle, although the estimation must generally rely on numerical methods. We establish the strong consistency of the estimates and the asymptotic normality of the normed estimates with respect to the true model. We do not assume independence or normality for our processes but under our, much weaker, conditions the Fourier transformed residuals have these properties in large samples and it is then possible to choose estimates which are, in a sense, efficient. The topic of Chapter 3 is the estimation of a regression matrix of less than full rank, a problem related to canonical correlation analysis. Asymptotic theory follows from the previous chapter since a non-linear regression approach is employed but the model is unlagged and much of this work was carried out before the author thought of the general model of Chapter 2. Possible computational procedures are suggested and the related problem of regression on an unobservable variable is considered .