Multilevel modeling using spatial processes: Application to the Singapore housing market

Abstract

Customary spatial modeling with point-referenced data introduces a modeling specification that includes a mean term, a spatial error or random effects term and a pure error term. The spatial random effects are usually modeled through a mean zero spatial process. If the mean term includes an intercept, then the spatial random effects can be interpreted as local spatial adjustments to the intercept. If the mean term is a familiar linear regression then it makes sense to ask whether the regression coefficients are constant or whether they might vary spatially, analogous to the intercept. This has been previously considered and the benefits of the increased flexibility have been demonstrated. The situation with replicates available at spatial locations is considered. This enables the building of the spatial analog of a multilevel model—replicate level covariates to explain the replicate level responses and location level covariates to explain the location level coefficients. The particular motivation for this modeling effort is a data set on condominium sales in Singapore. In this case, the replicates are the sales of condominiums within a building. Unit level features are available to explain the selling price of the unit and building level attributes to explain the coefficients. Anticipating dependence between coefficients, a multivariate spatial process specification is provided. This process is specified through kernel convolutions due to the computational challenges associated with fitting such models to a fairly large data set. There is flexibility in this kernel modeling necessitating model comparison. In particular, roughly 68,000 transactions across 1374 buildings (locations) are analyzed and the results and interpretation for the selected model are presented.