The Spatial Analysis of Time Series

In this paper, we propose a method of analyzing time series, called the spatial analysis. The analysis consists mainly of the statistical inference on the distribution given by the expected local time, which we define to be the spatial distribution, of a given time series. The spatial distribution is introduced primarily for the analysis of nonstationary time series whose distributions change over time. However, it is well defined for both stationary and nonstationary time series, and reduces to the time invariant stationary distribution if the underlying time series is indeed stationary. The spatial analysis may therefore be regarded as an extension of the usual inference on the distribution of a stationary time series to accommodate for nonstationary time series. In fact, we show that the concept of the spatial distribution allows us to extend many notions and ideas built upon the presumption of stationarity and make them applicable also for the analysis of nonstationary data. Our approach is nonparametric, and imposes very mild conditions on the underlying time series. In particular, we allow for the observations generated from a wide class of stochastic processes with stationary and mixing increments, or general markov processes including virtually all diffusion models used in practice. For illustration, we provide some empirical applications of our methodology to various topics such as the risk management, distributional dominance and option pricing.