Bayesian inference in spatial GARCH models: an application to US house price returns

ABSTRACT

In this paper we consider a high-order spatial generalized autoregressive conditional heteroskedasticity (GARCH) model to account for the volatility clustering patterns observed over space. The model consists of a log-volatility equation that includes the high-order spatial lags of the log-volatility term and the squared outcome variable. We use a transformation approach to turn the model into a mixture of normals model, and then introduce a Bayesian Markov chain Monte Carlo (MCMC) estimation approach coupled with a data-augmentation technique. Our simulation results show that the Bayesian estimator has good finite sample properties. We apply a first-order version of the spatial GARCH model to US house price returns at the metropolitan statistical area level over the period 2006Q1–2013Q4 and show that there is significant variation in the log-volatility estimates over space in each period.

KEYWORDS:

• spatial generalized autoregressive conditional heteroskedasticity (SGARCH)
• volatility
• spatial autoregressive model
• spatial dependence
• Bayesian inference
• Markov chain Monte Carlo (MCMC)
• house price returns

JEL:

• C11
• C21
• C22

ACKNOWLEDGEMENTS

The authors thank the editor, three anonymous referees and Philipp Otto for their constructive comments that substantially improved an earlier draft.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the authors.

Notes

1 In Figure 1, the returns are classified into positive and negative categories, and the squared returns are categorized according to their sample average. For details of the dataset, see section 5.

2 The parameters in Table 1 are chosen by Omori et al. (2007) by matching the first four moments of the 10-component Gaussian mixture distribution with that of $p({\epsilon }_i^{\ast })$ given in (2.4).

3 In our simulation, we set the tuning parameter to $c=0.1059$ and then adjust according to $c^{\ast }=c/1.05$ if the acceptance rate is less than 40%, and to $c^{\ast }=c\times 1.05$ if the acceptance rate is more than 50%.

4 The empirical mean is the average of the estimated posterior means over 200 resamples.

5 The house price data are taken from the Freddie Mac House Price Index (FMHPI) at the MSA level. For details of this dataset, see Aquaro et al. (2021) and Yang (2021).

6 As in Aquaro et al. (2021), we use the ordinary least squares (OLS) residuals obtained from the regressions of $\mathbf{y}$, ${\mathbf{z}}_1$ and ${\mathbf{z}}_2$ on (1) an intercept; (2) three quarterly dummy variables; and (3) national and regional cross-sectional averages of $\mathbf{y}$, ${\mathbf{z}}_1$ and ${\mathbf{z}}_2$.

7 For colored versions, see the supplemental data online.