Dynamic Spatiotemporal ARCH Models

TL;DR

This paper introduces a dynamic spatiotemporal ARCH model for geo-referenced data, capturing spatial and temporal dependencies in volatility, with a GMM estimator proven to be consistent and effective in simulations and real data analysis.

Contribution

It develops a novel dynamic spatiotemporal ARCH model with a GMM estimation approach, accounting for fixed effects and heteroscedasticity in geo-referenced data.

Findings

Significant spatial, temporal, and spatiotemporal effects on volatility in Berlin condominium prices.

GMM estimator shows good finite-sample properties in simulations.

Empirical application confirms the model's ability to capture volatility dynamics.

Abstract

Geo-referenced data are characterized by an inherent spatial dependence due to the geographical proximity. In this paper, we introduce a dynamic spatiotemporal autoregressive conditional heteroscedasticity (ARCH) process to describe the effects of (i) the log-squared time-lagged outcome variable, i.e., the temporal effect, (ii) the spatial lag of the log-squared outcome variable, i.e., the spatial effect, and (iii) the spatial lag of the log-squared time-lagged outcome variable, i.e., the spatiotemporal effect, on the volatility of an outcome variable. Furthermore, our suggested process allows for the fixed effects over time and space to account for the unobserved heterogeneity. For this dynamic spatiotemporal ARCH model, we derive a generalized method of moments (GMM) estimator based on the linear and quadratic moment conditions of a specific transformation. We show the consistency and asymptotic normality of the GMM estimator, and determine the best set of moment functions. We investigate the finite-sample properties of the proposed GMM estimator in a series of Monte-Carlo simulations with different model specifications and error distributions. Our simulation results show that our suggested GMM estimator has good finite sample properties. In an empirical application, we use monthly log-returns of the average condominium prices of each postcode of Berlin from 1995 to 2015 (190 spatial units, 240 time points) to demonstrate the use of our suggested model. Our estimation results show that the temporal, spatial and spatiotemporal lags of the log-squared returns have statistically significant effects on the volatility of the log-returns.