Unit Root and Cointegration Tests in Spatial Cross-Section Data

Although nonstationarity arises in time series data, it may also arise in spatial cross-section data. A definition of spatial nonstationarity is provided in which spatial impulse responses do not weaken with distance. Hence, remote shocks have the same effect on outcomes in a location, as do shocks in its immediate neighbors. This definition implies that the SAR coefficients tends to unity. Analytical methods are used to derive spatial impulse responses when space is lateral (east–west), and simulation methods are used when space is bilateral (east–west, north–south). A lattice based spatial unit root test is proposed to test for spatial nonstationarity, and its critical values are calculated using Monte Carlo simulation methods. We show that these critical values depend on the specification of W. Since space is finite, the asymptotic properties of spatial unit root tests are undefined. Specifically, because finite space has edges, unit roots that would arise if space was infinite, do not arise when space is finite, because locations on the edge of space are less spatially connected than their counterparts in the epicenter of space. We use simulation methods to show how edge effects weaken spatial unit roots. If spatial cross-section data are nonstationary, spatial cross-section regression parameter estimates may be spurious or nonsense. A spatial cointegration test is proposed in which, under the null, the variables concerned are not spatially cointegrated, in which event the parameter estimates are genuine. They are spatially cointegrated if the residual errors are spatially stationary. If they are spatially cointegrated, their parameter estimates are super-consistent. Monte Carlo methods are used to calculate critical values to test for spatial cointegration in cross-section data.