The Spatial Analysis of Time Series

In this paper, we propose a method of analyzing time series in the spatial domain. The analysis is based on the inference on the local time and its expectation. Both for the stationary and nonstationary time series, the spatial distributions are provided by the local time, and some of their important characteristics can be examined through the investigation of the expected local time. The methodology developed in the paper for the analysis of the expected local time applies to both stationary and nonstationary time series. The expected local time, however, reduces to the density of the time invariant distribution if the underlying time series is stationary. Our analysis may therefore be regarded as an extension to the nonstationary time series of the usual distributional analysis for the stationary time series. Our approach is nonparametric, and imposes very weak and minimal conditions on the underlying time series. In particular, we allow for 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. Proposed are several interesting applications of our methodology, such as forecast of spatial distribution, test of structural break in spatial domain, specification test in spatial domain, test of equality in spatial distribution and test of spatial dominance