The author deals with the location-dependent AR process \(\{X_t(u):u\in[0,1]^2\}\), where \(X_t(u)\) has the representation \[ X_t(u)=\sum_{j=1}^p a_j(u)X_{t-j}(u)+\sigma(u)\xi_t(u),\;t=l,\dots,T, \] where \(u=(x,y)\in[0,1]^2\), \(\{a(u);j=l,\dots,p\}\) and \(\sigma(u)\) are nonparametric functions, the innovations \(\{\xi_t(u):u\in[0,1]^2\}\) are independent in time spatially stationary processes with \(E[\xi_t(u)]=0\) and \(\text{var}[\xi_t(u)]=1\). The results of this article do not rely on any distributional assumptions on \(\xi_t(u)\). Note that if the \(\{a_j(u)\}\) are not constant over space, then \(\{X_t(u)\}\) is a spatially nonstationary process. The problem of estimation of the value of the spatio-temporal process \(X_t(u_0)\) at an arbitrary unobserved location \(u_0\) using known neighbouring observations \(\{X_1(u_s),\dots,X_T(u_s),\;s=1,\dots,m\}\) is investigated. The estimate depends on the parameters \(\{a_j(u_0)\}\) which are unknown and can not be estimated using standard methods since observations \(\{X_1(u_0),\dots,X_T(u_0)\}\) at location \(u_0\) are not given.
The author proposes two methods for estimating the \(\{a_j(u_0)\}\) under the assumption that the AR functions \(\{a_j(u)\}\) are continuous in the space. Both methods are based on a localized least squares criterion. The first estimator is a localized least squares estimator with constant regressors, whereas the second estimator is a local linear least squares estimator. The sampling properties of both estimators are considered in two cases, where: (i) the number of locations are fixed and time \(T\to\infty\); and (ii) both the number of locations and \(T\to\infty\). In the case in which the number of locations is fixed, it is shown that both estimators are asymptotically normal but biased (in probability). In the case in which the number of locations also grows, the estimators are asymptotically consistent. A test for spatial stationarity is developed, which is based on testing for homogeneity. The limiting distributions of the test statistic under the null and alternative hypotheses of spatial stationarity and nonstationarity are evaluated. The methods and the test for spatial stationarity are illustrated by simulations.