A general framework for spatial GARCH models

Otto, P., Schmid, W. (2022): A general framework for spatial GARCH models. Statistical Papers (DOI)

Abstract

In time-series analysis, particularly in finance, generalized autoregressive conditional heteroscedasticity (GARCH) models are widely applied statistical tools for modelling volatility clusters (i.e., periods of increased or decreased risk). In contrast, it has not been considered to be of critical importance until now to model spatial dependence in the conditional second moments. Only a few models have been proposed for modelling local clusters of increased risks. In this paper, we introduce a novel spatial GARCH process in a unified spatial and spatiotemporal GARCH framework, which also covers all previously proposed spatial ARCH models, exponential spatial GARCH, and time-series GARCH models. In contrast to previous spatiotemporal and time series models, this spatial GARCH allows for instantaneous spill-overs across all spatial units. For this common modelling framework, estimators are derived based on a non-linear least-squares approach. Eventually, the use of the model is demonstrated by a Monte Carlo simulation study and by an empirical example that focuses on real estate prices from 1995 to 2014 across the postal code areas of Berlin. A spatial autoregressive model is applied to the data to illustrate how locally varying model uncertainties (e.g., due to latent regressors) can be captured by the spatial GARCH-type models.

Background

The paper develops a unified framework for spatial and spatiotemporal GARCH-type models that allows instantaneous variance spillovers across locations. The framework nests earlier spatial ARCH and spatial log-GARCH specifications as well as classical time-series GARCH, and provides a consistent nonlinear least-squares (NLSE) estimator. An empirical illustration uses Berlin condominium price changes.

Key takeaways

  • Unified model class: one setup covers spatial ARCH, spatial GARCH, spatial log-GARCH, and (as a special case) time-series GARCH.
  • Instantaneous spatial spillovers: conditional variances can influence each other within the same index set (no temporal lag needed).
  • Estimation: parameters estimated via NLSE; consistency is shown under regularity conditions; practical implementations available (R package spGARCH).
  • Use either as error model or direct process: model residual uncertainty in spatial regressions, or model observed volatility itself (e.g., real-estate, labour, networks).
  • Flag to watch for: significant Moran’s I of squared residuals in a mean model suggests missing spatial volatility structure.

Intuition

Time-series GARCH lets today’s variance depend on past shocks. Spatially, it is natural that a location’s variance depends on its neighbours: uncertainty itself clusters in space (Tobler’s law applied to variance). The framework formalises this by letting local conditional variances co-move via spatial weight matrices.

Minimal maths

Core representation:

\[ Y = \operatorname{diag}(h)^{1/2}\,\varepsilon,\qquad F = \alpha + W_1\,\gamma\!\left(Y^{(2)}\right) + W_2\,F, \] where \(h=(h(s_i))_{i=1}^n\), \(Y^{(2)}=(Y(s_i)^2)_{i=1}^n\), and \(F=(f(h(s_i)))_{i=1}^n\). Choosing \(f,\gamma,W_1,W_2\) yields different models. Existence/uniqueness follow from a contraction (random fixed-point) argument under Lipschitz-type conditions.

Nested examples.

  • Spatial ARCH (spARCH): \(f(x)=x\), \(\gamma_i(x)=x_i\), \(W_2=0\) → \(h=\alpha+W_1 Y^{(2)}\).
  • Spatial GARCH (spGARCH): \(f(x)=x\), \(\gamma_i(x)=x_i\) → \(h=\alpha+W_1 Y^{(2)}+W_2 h\) so \(h=(I-W_2-W_1E)^{-1}\alpha\) (when the inverse exists; \(E=\mathrm{diag}(\varepsilon^2)\)).
  • Spatial log-GARCH: \(f(x)=\log x\) → \(\log h=\alpha+W_1 \log\!\left(Y^{(2)}\right)+W_2 \log h\) with simpler existence condition \(\|W_1+W_2\|\le 1 \).

Estimation (NLSE) and weights

In practice one often sets \(W_1=\rho W_1^*,\,W_2=\lambda W_2^*,\,\alpha=\alpha\mathbf{1}\) and estimates \((\rho,\lambda,\alpha)\) by nonlinear least squares using a log-squared transformation; consistency and identification conditions are provided. Choice of \(W_k^*\) is flexible (contiguity, distance, anisotropy, network).

Practical advice

  • Diagnostic: after fitting your mean model (SAR/SEM/GLM), compute Moran’s I on \(\xi^2\) (squared residuals). If significant, add a spatial GARCH-type variance model.
  • Weights sensitivity: compare plausible \(W\) (contiguity vs. distance; directional for flows); check stability of \((\hat\rho,\hat\lambda)\) and fit criteria.
  • Interpretation: large \(\hat h(s)\) indicates locally elevated uncertainty; \(\hat\rho,\hat\lambda\) quantify variance spillovers and persistence in space.
  • Software: an implementation is available in R: CRAN package spGARCH.
  • Instantaneous vs. lagged spillovers: Unlike many spatiotemporal GARCH variants that embed spatial effects with a time lag, this framework permits instantaneous spatial/network interactions (due to simultaneity of investors' decisions) and also covers purely spatial settings.

Application: Berlin real-estate prices

Using long-term logarithmic price changes by postcode, the paper fits SAR mean models and then spatial GARCH to residuals. Squared residuals show spatial autocorrelation that the spGARCH captures; resulting uncertainty maps highlight districts with higher local risk. This matters for real-estate pricing and investment: ignoring volatility clustering can lead to biased valuations, underestimation of risks, and poorly informed policy or lending decisions. By explicitly modelling spatial dependence in volatility, the framework offers a way to measure local market risk — a key concern for banks, investors, and urban policy makers.

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