On the Identification of Trend and Correlation in Temporal and Spatial Regression

In longitudinal or spatial regression problems, estimation of temporal or spatial trends is often of primary interest, while correlation itself is of secondary interest or is regarded as a nuisance component. In other situations, the stochastic process inducing the correlation may be of interest in itself. In this paper, we investigate for some simple time series and spatial regression models, how well trend and correlation can be separated if both are modeled in a flexible manner. In this contribution we shed some further light on this puzzle from a Bayesian perspective.We focus on approaches with Bayesian smoothing priors for modeling trend functions, such as random walk models or extensions to Bayesian penalized (P-)splines. If the correlation-generating error process has similar stochastic structure as the smoothing prior it seems quite plausible that identi.ability problems can arise. In particular, it can become difficult to separate trend from correlation. We first exemplify this using a simple time series setting in Section 2. In Section 3 we move on to the corresponding spatial situation, which arises in geostatistics. Section 4 brie.y points out extensions to the general class of structured additive regression (STAR) models.