Goodness of Fit for Lattice Processes

The paper discusses tests for the correct speci…cation of a model when data is observed in a d-dimensional lattice, extending previous work when the data is collected in the real line. As it happens with the latter type of data, the asymptotic distribution of the tests are functionals of a Gaussian sheet process, say B ( ), 2 [0; ]. Because it is not easy to …nd a time transformation h ( ) such that B (h ( )) becomes the standard Brownian sheet, a consequence is that the critical values are di¢ cult, if at all possible, to obtain. So, to overcome the problem of its implementation, we propose to employ a bootstrap approach, showing its validity in our context. JEL Classi…cation: C21, C23. 1. INTRODUCTION The paper is concerned with testing the goodness of …t of a parametric family of models for data collected in a lattice. More speci…cally, we are concerned with the correct speci…cation (or model selection) of the dynamic structure with time series and/or spatial stationary processes fx (t)gt2Z de…ned on a d-dimensional lattice. The key idea of the test is to compare how close is the parametric and nonparametric …ts of the data to provide support for the null hypothesis. In the paper, we shall speci…cally consider data for which d 3. The motivation to focus on the case d 3 lies in the fact that the most often type of data available in economics is when d = 2, say with agricultural or environmental data, or when d = 3. An important example of the latter is the spatial-temporal data sets, that is data collected in a lattice during a number of periods. However, we ought to mention that extensions to higher index lattice processes can be adapted under suitable modi…cations. All throughout the paper we will assume that the (spatial) process fx (t)gt2Zd can be represented by the multilateral model (1.1) x (t) = X j2Zd (j) " (t j) ; X j2Zd 2 (j) <1 (0) = 1, for some sequence f" (t)gt2Zd satisfying E (" (t)) = 0 and E (" (0) " (t)) = " if t = 0; and = 0 for all t 6= 0. Notice that because our model is multilateral, the sequence f" (t)gt2Zd loses its interpretation as the “prediction” error or that they can be regarded as innovations. Under (1:1), the spectral density function of fx (t)gt2Zd can be factorized as f ( ) = " (2 ) d j ( )j , 2 , where = ( ; ] and with (1.2) ( ) = X j2Zd (j) exp ( ij ) . Date : 1 February 2008. Key words and phrases. Goodness of …t tests. Spatial linear processes. Spectral domain. Bootstrap tests. 1