This project develops a new framework of statistical spatial analysis of time series based on the concept of local time. Local time for a general stochastic process measures the sojourn time that the underlying process spends in a neighborhood of each spatial point. This project uses the concept of local time and its variants (integrated local time, expected integrated local time, functionals of local time) to empirically analyze time series in economics and finance. The goal is to develop a general spatial analysis for economic and financial time series. The methodology developed in the project applies to nonstationary as well as stationary time series, for both of which the local time provides the spatial distribution. The local time, however, reduces essentially to the density of the time invariant distribution if the underlying time series is stationary. The newly developed methodology may therefore be regarded as an extension to the nonstationary time series of the usual distributional analysis for the stationary time series. The approach used in the project is nonparametric, and imposes very weak and minimal conditions on the underlying time series. In particular, the observations are allowed to be generated from a wide class of stochastic processes with stationary and mixing increments, or general markov processes including all diffusion models that are commonly used in practice.
The concept of local time has long been established, and has also been refined and extended recently by many experts in stochastic processes including, among others, Borodin, Jacod and Sorensen. The local time, however, has rarely been used for practical applications. It has a clear potential to be useful for the time series analysis in the fields of economics and finance: The (expected) local time of a stochastic process determines the sum of the (expected) utilities that it generates over time, and therefore, the analysis of local time and expected local time is critically important for economic decision making over time. The proposed research explores this potential, and extends many of the static concepts used frequently in empirical economics and finance to accommodate general dynamic environments. For instance, a popular financial risk measure such as VaR, may now be measured, evaluated and predicted over a certain period of time using an estimate of the expected local time for the underlying asset price. Moreover, such a concept as stochastic dominance, which has been widely used to compare investment strategies, income distributions and various social programs, can be extended so that it allows us to order the prospects based on their performances over a time period rather than at a fixed time.
The Broader Impact of the Research This project will have significant and far-reaching impacts on both theoretical and practical analyses of economic and financial data. The researchers in economics, finance and other related fields may use the newly developed tools to perform more precise inferences on a wide class of models. In particular, the tools allow them to analyze a broad range of the economic and financial problems in a more realistic and dynamic context. The practitioners working in the applied fields of economics and finance will also benefit much from this project. The new tools developed in the project may be readily applicable to improve the current practices of the applied economic research and financial engineering, in relation to the forecasting and analyzing economic and financial markets and the financial engineering for risk management, derivative pricing and portfolio selections. The research outcomes will be presented at various conferences in the fields of economics, finance and applied probability and statistics, and the computer programs will be prepared and distributed to make them more easily accessible to the practitioners.
