Bayesian Inference in Spatial Stochastic Volatility Models: An Application to House Price Returns in Chicago*
| Published date | 01 October 2021 |
| Author | Süleyman Taşpınar,Osman DoĞan,Jiyoung Chae,Anil K. Bera |
| Date | 01 October 2021 |
| DOI | http://doi.org/10.1111/obes.12425 |
Bayesian Inference in Spatial Stochastic Volatility
Models: An Application to House Price Returns in
Chicago*
SÜLEYMAN TAŞPINAR†,OSMAN DO˘
GAN‡,JIYOUNG CHAE‡and
ANIL K. BERA‡
†Department of Economics, Queens College, The City University of New York, New York, NY,
USA (e-mail: staspinar@qc.cuny.edu)
‡Department of Economics, University of Illinois at Urbana-Champaign, Urbana, IL, USA
(e-mail: odogan@illinois.edu; jchae3@illinois.edu; abera@illinois.edu)
Abstract
In this study, we propose a spatial stochastic volatility model in which the latent log-
volatility terms follow a spatial autoregressive process. Though there is no spatial
correlation in the outcome equation (the mean equation), the spatial autoregressive
process defined for the log-volatility terms introduces spatial dependence in the
outcome equation. To introduce a Bayesian Markov chain Monte Carlo (MCMC)
estimation algorithm, we transform the model so that the outcome equation takes the
form of log-squared terms. We approximate the distribution of the log-squared error
terms of the outcome equation with a finite mixture of normal distributions so that the
transformed model turns into a linear Gaussian state-space model. Our simulation
results indicate that the Bayesian estimator has satisfactory finite sample properties. We
investigate the practical usefulness of our proposed model and estimation method by
using the price returns of residential properties in the broader Chicago Metropolitan
area.
I. Introduction
The stochastic volatility models and the autoregressive conditional heteroskedasticity
(ARCH)-type models are designed to capture volatility clustering phenomenon
observed in financial time series. Unlike the ARCH-type models, the standard
stochastic volatility model consists of separate independent error processes for the
conditional mean and the conditional variance. The process for the conditional variance
is specified as a log-normal autoregressive process with independent innovations. There
JEL Classification numbers:C11. C21. C22.
*
We thank the editor, Debopam Bhattacharya, and two anonymous referees for their constructive comments and
suggestions on the earlier versions of this paper. We also benefitted from insightful comments by the
econometric seminar participants at University of Illinois at Urbana-Champaign in March 2018.
1243
©2021 The Department of Economics, University of Oxford and John Wiley & Sons Ltd
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 83, 5 (2021) 0305-9049
doi: 10.1111/obes.12425
is evidence that the stochastic volatility models can offer increased flexibility over the
ARCH-type models (Jacquier, Polson and Rossi, 1994, 2004; Fridman and Harris,
1998; Kim, Shephard and Chib, 1998). The purpose of this paper is to extend the
standard stochastic volatility model to spatial data. We suggest a parsimonious
specification in which we directly model the log-volatility terms through a first-order
spatial autoregressive process. The resulting spatial stochastic volatility model shares
similar properties with the standard stochastic volatility model in the time-series
literature, and it is designed to capture the volatility clustering observed in spatial data.
As in the standard stochastic volatility model, our specification consists of two
independent error terms for the outcome and the log-volatility equations, respectively.
Under the assumption that both error terms have the normal distribution, our spatial
process implies a leptokurtic symmetric distribution for spatial data. More importantly,
although our specification implies no spatial correlation in the outcome variable, the
spatial autoregressive process defined for the log-volatility introduces spatial correlation
in the higher moments of the outcome variable, implying spatial dependence in the
outcome variable. As such, a test based on the spatial autoregressive parameter in the
log-volatility equation can be formulated to test for the existence of spatial dependence.
To introduce an estimation approach for our specification, we transform the model
so that the outcome equation takes the form of log-squared terms. For a similar spatial
model, Robinson (2009) approximates the distribution of the log-squared error terms
with the normal distribution and establishes the asymptotic consistency and normality
of the resulting Gaussian pseudo-maximum likelihood estimator (PMLE). In the time-
series literature, the PMLEs obtained in this way for the stochastic volatility models
also attain the standard large sample properties, but they are sub-optimal in the sense
that they have poor finite sample properties (Jacquier et al., 1994; Shephard, 1994;
Kim et al., 1998; Sandmann and Koopman, 1998). Therefore, we propose
approximating the distribution of the log-squared error terms by a mixture of Gaussian
distributions (Kim et al., 1998; Shephard, 1994). The resulting system of equations
constitutes a Gaussian state-space model, where the log-volatility equation is the state
equation. We then introduce a Bayesian Markov chain Monte Carlo (MCMC)
estimation approach in which a data augmentation scheme is used to treat the latent
log-volatility terms as additional parameters which are estimated as a natural by-
product of the estimation process. In a Monte Carlo study, we investigate the finite
sample properties of our Bayesian estimator along with a (naive) Bayesian estimator
based on an algorithm in which the distribution of the log-squared error terms is
approximated by the normal distribution. Our results indicate that the Bayesian
estimator based on the finite mixture of normal distributions performs well, whereas
the naive estimator has poor finite sample properties. To test the presence of spatial
correlation in the log-volatility, i.e., to test the existence of spatial dependence in the
outcome variable, we suggest using the Savage–Dickey density ratio (SDDR) for the
calculation of the Bayes factor (Dickey, 1971; Verdinelli and Wasserman, 1995).
To investigate the practical usefulness of our methodology, we apply it to the price
returns calculated from the residential property sales in the broader Chicago
Metropolitan area in 2014 and 2015. We first use the Moran I test to check for the
presence of spatial correlations in the returns and the squared returns. We find a mild
©2021 The Department of Economics, University of Oxford and John Wiley & Sons Ltd
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