This dissertation is devoted to the study and empirical implementation of new methods in the econometrics of continuous-time finance. The first chapter is concerned with the nonparametric estimation of the drift and diffusion function of general continuous-time homogeneous stochastic differential equations. Minimal requirements are placed on the data generating mechanism, allowing for both stationary and nonstationary systems, and the available data is assumed to be a set of discrete sample observations. Econometric estimation proceeds by constructing refined sample analogues of unknown drift and diffusion function. Cross-restrictions on the functional forms are not imposed, nor is the existence of a time-invariant marginal data density and, in consequence, the new approach is robust against deviations from stationarity. We prove consistency of the point estimates and pointwise weak convergence to mixtures of normal laws, where the mixture variates depend on the chronological local time of the underlying semi-martingale, that is on the amount of time spent by the process in the spatial vicinity of each point.
The second chapter focuses on the application of this new method to a well-known problem in empirical finance, namely the estimation of the short-term interest rate dynamics in a continuous-time framework. The approach to data analysis is twofold. First, a descriptive analysis of the time series is conducted using econometric estimates of the local time, which is treated as a spatial density function, along lines pioneered in Phillips (1998). Spatial densities (and various functionals of them, such as spatial hazard rates) are newly developed descriptive tools for data analysis that are applicable when the series is nonstationary or, more strictly, when stationarity cannot be guaranteed [c.f. Phillips (1998) and Phillips and Park (1998)]. Second, nonparametric estimates of the drift and diffusion function and associated confidence intervals are obtained for the interest rate process.
The third chapter of this thesis discusses the finite sample performance of fully nonparametric estimators of the drift and diffusion function of general, potentially nonlinear and homogeneous stochastic differential equations. We compare the estimators in the first chapter to those suggested in recent papers by Jiang and Knight (1997) and Stanton (1997). Theoretical justification for the different functional approaches is based on specific assumptions on the limit theory and the underlying process. The stringency of these assumptions in finite sample is investigated by evaluating the performance of the estimators in the presence of various simulated underlying processes.
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