Keeping Variables within Bounds: Using Information between Observations

This research develops an econometric framework to analyze time series processes with bounds. The framework is general enough that it can incorporate several different kinds of bounding information that constrain continuous-time stochastic processes between discretely-sampled observations. It applies to situations in which the process is known to remain within an interval between observations, by way of either a known constraint or through the observation of extreme realizations of the process. The main statistical technique employs the theory of maximum likelihood estimation. This approach leads to the development of the asymptotic distribution theory for the estimation of the parameters in bounded diffusion models. The results of this analysis present several implications for empirical research. The advantages are realized in the form of efficiency gains, bias reduction and in the flexibility of model specification. A bias arises in the presence of bounding information that is ignored, while it is mitigated within this framework. An efficiency gain arises, in the sense that the statistical methods make use of conditioning information, as revealed by the bounds. Further, the specification of an econometric model can be uncoupled from the restriction to the bounds, leaving the researcher free to model the process near the bound in a way that avoids bias from misspecification. One byproduct of the improvements in model specification is that the more precise model estimation exposes other sources of misspecification. Some processes reveal themselves to be unlikely candidates for a given diffusion model, once the observations are analyzed in combination with the bounding information. A closer inspection of the theoretical foundation behind diffusion models leads to a more general specification of the model. This approach is used to produce a set of algorithms to make the model computationally feasible and more widely applicable. Finally, the modeling framework is applied to a series of interest rates, which, for several years, have been constrained by the lower bound of zero. The estimates from a series of diffusion models suggest a substantial difference in estimation results between models that ignore bounds and the framework that takes bounding information into consideration.

THE TRANSPORT OF HEAT BY FLOWING GROUNDWATER IN A DISCRETE FRACTURE

In typical theoretical or experimental studies of heat migration in discrete fractures, conduction and thermal dispersion are commonly neglected from the fracture heat transport equation, assuming heat conduction into the matrix is predominant. In this study analytical and numerical models are used to investigate the significance of conduction and thermal dispersion in the plane of the fracture for a point and line sources geometries. The analytical models account for advective, conductive and dispersive heat transport in both the longitudinal and transverse directions in the fracture. The heat transport in the fracture is coupled with a matrix equation in which heat is conducted in the direction perpendicular to the fracture. In the numerical model, the governing heat transport processes are the same as the analytical models; however, the matrix conduction is considered in both longitudinal and transverse directions. Firstly, we demonstrate that longitudinal conduction and dispersion are critical processes that affect heat transport in fractured rock environments, especially for small apertures (eg. 100 μm or less), high flow rate conditions (eg. velocity greater than 50 m/day) and early time (eg. less than 10 days). Secondly, transverse thermal dispersion in the fracture plane is also observed to be an important transport process leading to retardation of the migrating heat front particularly at late time (eg. after 40 days of hot water injection). Solutions which neglect dispersion in the transverse direction underestimate the locations of heat fronts at late time. Finally, this study also suggests that the geometry of the heat sources has significant effects on the heat transport in the system. For example, the effects of dispersion in the fracture are observed to decrease when the width of the heat source expands.