ZISAR: A Zero-Inflated Spatial Autoregressive Model for Count Data with Random Effects

Spatial count data with excess zeros are common in regional science, epidemiology and urban economics, but no frequentist model combines a structural spatial autoregressive mechanism with zero-inflation, overdispersion and unobserved heterogeneity. We introduce the Zero-Inflated Spatial Autoregressive (ZISAR) model, which places a simultaneous autoregressive (SAR) process on the latent log-mean of a zero-inflated negative binomial model, with unit-level Gaussian random effects. A general version adds a spatial field to the zero-inflation predictor, represented on a lattice of knots. Parameters are estimated by maximising the Laplace approximation to the marginal likelihood with Template Model Builder, using automatic differentiation and a sparse Cholesky log-determinant, so that tens of thousands of units are fitted in minutes on a laptop. We establish consistency and asymptotic normality and derive direct, indirect and total effects that account for both the mixture and the spatial multiplier. A Monte Carlo study shows that ignoring either zero-inflation or spatial dependence biases the autoregressive and regression coefficients, whereas ZISAR recovers them, and that Moran tests on residuals miss dependence that the likelihood detects. In the adoption of domestic solar photovoltaic systems across the 32,322 small areas of England during the Feed-in Tariff boom, adoption is strongly dependent in space (estimated ρ = 0.63), nearly half of the zeros are structural and follow the share of flats, the effects of tenure and property value found by aspatial models shrink by an order of magnitude, and indirect effects account for about 60% of every total effect. The estimator is available in the R package zisar (https://www.niccolosalvini.me/packages/zisar/).