https://orcid.org/0000-0003-4859-5536
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ABSTRACT
Existing volatility models normally emphasize the behaviour of prices in a temporal sense and comparatively few studies have explicitly analysed the spatial variation of volatility. This paper proposes a flexible spatial volatility model for squared returns using a Box–Cox transformation that includes the linear and log-linear forms as special cases, thus providing a unified framework for simultaneously testing space-varying volatility and its functional form. The use of the model is illustrated by a substantive application to housing price data in the US city of Chicago. The estimation results suggest that housing returns in Chicago show that the volatility exhibits strong spatial dependence and the log-linear functional form is appropriate. In the final log-linear model, a new practical indicator, called neighbourhood elasticity, is proposed that determines how volatility in one neighbourhood is linked to that in surrounding neighbourhoods.
ACKNOWLEDGEMENTS
This paper is based on the first chapter of my PhD dissertation, ‘Three essays on housing economics’, submitted to the University of Illinois at Urbana-Champaign (UIUC) in June 2021. I am extremely grateful to my PhD supervisor, Anil K. Bera, for his invaluable guidance and support. I also appreciate my committee members, Geoffrey J. D. Hewings, Daniel McMillen and JiHyung Lee, for their insightful conversations and feedback. I also thank the editor for his kind consideration and the anonymous referees for many pertinent comments and suggestions which improved the paper. The views expressed here are those of the author and do not reflect the position of the Federal Reserve Bank of Richmond or the Federal Reserve System. Of course, I retain the responsibility for any remaining errors.
DISCLOSURE STATEMENT
No potential conflict of interest was reported by the author.
Notes
1 According to the Federal Reserve’s 2016 Survey of Consumer Finances (SCF), at US$24.2 trillion, the primary residence accounted for about one quarter of all assets held by households in 2016. The survey also reveals that the value of the primary mortgage debt was the largest liability faced by the homeowners (https://www.federalreserve.gov/econres/scfindex.htm).
2 This pattern is also detected when using an alternative measure of volatility, absolute returns, suggesting that the dependence behaviour is not sensitive to the choice of a particular measure of volatility. Robust analysis using this alternative is addressed in Section C in the Appendix in the supplemental data online.
3 As indicated by Bollerslev et al. (Citation1992), squared returns of not only exchange rate data but also all speculative price series typically exhibit volatility clustering and ARCH-type models are appropriate for volatility estimations.
4 Lee (Citation2009) examines the determinants of housing price volatility for eight capital cities in Australia by using consumer price index, income, unemployment rate and mortgage rate. Miller and Peng (Citation2006) also investigate interactions between housing volatilities and economic conditions in 277 MSAs. They demonstrate that the volatility series is Granger-caused by the home appreciation rate and the growth in gross metropolitan product (GMP).
5 Although Bera and Simlai (Citation2005) use data from 506 census tracts in the Boston metropolitan area, the data are taken from the 1970 census, which does not reflect recent housing cycles.
6 Although Le Gallo et al. (Citation2020) and the present paper have similarities in a sense that both try to detect spatial clusters of observations, they focus on model residuals rather than on the model itself. Therefore, an important difference from Le Gallo et al. is that rather than assuming a certain type of model specification such as SAR and then investigating their residuals, I allow for added flexibility in determining the appropriate degree of non-linearity by developing a new type of model.
7 An editor suggested using the quadratic terms of the explanatory variables to explain the volatility of prices in different locations. However, I found the corresponding parameter estimates to be insignificant.
8 The MLS is a private real estate-listing service where real estate property information is listed and searched by participating members (e.g., real estate agents).
9 Census tracts are small, relatively stable spatial units with population ranging between 2500 and 8000, with an average of approximately 4000, and designed to be homogeneous with respect to population characteristics, economic status and living conditions.
10 The community areas are well-defined, stable geographical regions designated by the Social Science Research Committee at the University of Chicago and officially recognized by the city of Chicago.
11 Hayunga et al. (Citation2019) argue that homeowner characteristics influence their maintenance and home-improvement behaviours, which in turn affect home values and thus the observed pecuniary return. This analysis requires mortgage information data at initial purchase, which are not available from the MLS records; the current study thus treats observed housing returns as actual returns despite the possibility of overstatement.
12 In the finance literature, several papers have documented that an autocorrelation of stock returns exists, particularly during the postcrisis period (Islam et al., Citation2007; Sarwar & Khan, Citation2017).
13 Kelejian and Prucha (Citation2001) introduce a central limit theorem (CLT) that can be used to establish the asymptotic distribution of the Moran’s I test under certain regularity conditions. See Kelejian and Prucha’s equations (2.4) and (4.1) for the test statistic and asymptotic distribution, respectively.
14 The LM test in is based on spatial lag model. Other LM test statistics are consistent and available from the author upon request.
15 The fact that the log-linear model is less misspecified than the linear model can also be illustrated by several test statistics based on the information matrix as described in Section B in the Appendix in the supplemental data online.
16 Millo and Piras (Citation2012) develop the splm package in R to estimate the fixed effect spatial panel model that accounts for both spatial lag and spatial error effects, but it does not take account of temporal heterogeneity.
17 Most existing spatial panel estimation methods are designed for balanced panel data, and are not effective for unbalanced panels because of the computational burden associated with inverting a large spatial weighting matrix.