Recent advances in the theoretical understanding of equilibria in property markets predict that the equilibrium hedonic price function will typically be highly nonlinear. Rather than adopting progressively more flexible econometric specifications to deal with this nonlinearity we adopt an alternative estimation strategy based on a further insight provided by the theoretical literature. That insight is that in equilibrium the market may not be characterised by a continuum of properties over attribute space. Rather the market may well be lumpy, being well-provided with properties exhibiting certain combinations of characteristics and sparsely provided elsewhere. We test the predictions of two different models; one that suggests that the market will be characterised by clusters of properties with similar physical attributes, one that the market will be characterised by clusters of neighbourhoods exhibiting similar socioeconomic compositions. We identify clusters by applying techniques of model-based clustering which allow the data to inform on the nature and the number of clusters. Our estimation strategy for handling nonlinearity, therefore, is to avoid estimating the hedonic price function over the entire attribute space. Rather, we fit separate price functions for the properties in each cluster thereby forming local approximations to the hedonic price surface over the attribute area spanned by the properties in each cluster. Finally we test to see which partitioning of the data, either according to the attributes of properties or the socioeconomics of neighbourhoods, is capable of explaining more of the variability in the data.