Dynamic spatiotemporal ARCH models

Abstract

ABSTRACTGeo-referenced data are characterised by an inherent spatial dependence due to 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, the temporal effect, (ii) the spatial lag of the log-squared outcome variable, the spatial effect, and (iii) the spatiotemporal effect on the volatility of an outcome variable. We derive a generalised method of moments (GMM) estimator based on the linear and quadratic moment conditions. We show the consistency and asymptotic normality of the GMM estimator. After studying the finite-sample performance in simulations, the model is demonstrated by analysing monthly log-returns of condominium prices in Berlin from 1995 to 2015, for which we found significant volatility spillovers.Preprint: This paper is based on the preprint arXiv:2202.13856KEYWORDS: Spatial ARCHGMMvolatility clusteringvolatilityhouse price returnslocal real-estate marketJEL: C13C23P25R31 DISCLOSURE STATEMENTNo potential conflict of interest was reported by the author(s).Notes1 Note that the matrix equation ABC=D, where D, A, B, and C are suitable matrices, can be expressed as vec(D)=(C′⊗A)vec(B), where vec(B) denotes the vectorisation of the matrix B (Abadir & Magnus, Citation2005, p. 282). This property can be applied to (U1∗,U2∗,…,UT−1∗)=(U1,U2,…,UT)FT,T−1 by setting D=(U1∗,U2∗,…,UT−1∗), C=FT,T−1, B=(U1,U2,…,UT) and A=In.2 In applying Lemma 1 in the Appendix in the supplemental data online, we use the fact that tr(A′B)=vec′(A)vec(B)=vec′(B)vec(A), where A and B are any two N×N matrices.3 The explicit forms of D1N and D2N are given in Section C of the Appendix.4 Note that when t=1, we may simply use H1=c1((In−1T−1∑h=1T−1Ah)Y0∗−1T−1∑r=1T−1(∑h=0T−r−1Ah)S−1(Xrβ0+αr,01n)).5 Note that when T is large, μ~0=(μ0+μϵ1n) can be estimated by μ~ˆN=1T∑t=1T(ϑˆt−1n1n′ϑˆt1n).