Cointegration Between Two Intrinsically Stationary Spatial Processes

The concept of the intrinsic processes proposed by Matheron (1973) provides an elegant mathematical framework for modeling nonstationary spatial phenomena. It can be viewed as a direct analogue of taking differences of nonstationary time series to achieve stationarity. But it is applicable to spatial data observed on irregular grids. The goal of this paper is to establish the inference methods and the relevant theory for identifying the cointegration between two simple intrinsic processes. We apply the least squares estimation, like Engle and Granger (1987). However, the asymptotic property of the estimation is much more complex, depending on the underlying processes as well as the way the observations were taken. We propose some bootstrap approximations for the asymptotic distribution of the estimators. It turns out that the wild bootstrap procedure is adaptive automatically to varying convergence rates under the different schemes of taking the observations. Therefore, it paves the way for constructing practically feasible confidence intervals for cointegration coefficients. A new and easy-to-use statistical test is constructed for testing the cointegration. The proposed methods, as well as the associated asymptotic results under various settings, are illustrated in simulation. The application to a real data example is also reported.

Qiwei Yao is a Professor of Statistics at London School of Economics. His main research interest includes High-dimensional time series, factor models, dynamic network, spatio-temporal processes, non-stational processes and cointegration, and nonlinear processes. He is Fellow of both ASA and IMS.

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