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IABFU Distribution of Returns Economy Question

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A Flexible Model for Spatial Volatility with an Application to the Chicago Housing Market * Jiyoung Chae † November 9, 2019 JOB MARKET PAPER Abstract Existing volatility models normally emphasize the behavior of prices in a temporal sense and comparatively few studies have explicitly analyzed 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 maximum likelihood method is used to estimate the model and Monte Carlo simulations are conducted to investigate the finite sample performance of the maximum likelihood estimator. The use of the model is also illustrated by a substantive application to housing price data in the city of Chicago. The estimation results suggest that housing returns in Chicago show 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 neighborhood elasticity, is proposed that determines how volatility in one neighborhood is linked to that in surrounding neighborhoods. From a practical point of view, this indicator provides a tool to help policy-makers avoid volatility transmission and the risk of contagion in the housing market. Keywords: Spatial volatility clustering, Spatial dependence, Maximum Likelihood, Monte Carlo simulation JEL Classification: C13, C21, C49, C55, R39 * I am extremely grateful to my advisor 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. This paper has also benefited from comments and suggestions by Amit Batabyal and Pavel Krivenko at the Mid-Continent Regional Science Association (MCRSA 2019) Conference in Madison, Wisconsin, and my mentor, Xu Lin, at the Annual Meeting of the Midwest Econometrics Group (MEG 2019) in Columbus, Ohio. I gratefully acknowledge the financial support provided by Illinois REALTORS® to the Regional Economics Applications Laboratory (REAL) that provided research funding. All errors are mine. † Department of Economics, University of Illinois at Urbana-Champaign, U.S., email: 1 1 Introduction The volatility of housing prices has important implications for household behavior and welfare as well as for the aggregate U.S. economy. At the household level, it can be easily argued that housing is the most important asset for many households. It is usually both the largest asset they own and the most readily available source of collateral against which they can borrow.1 Higher housing price volatility thus has the potential to pose substantial risk to household welfare. For example, it could distort a household’s housing choices, lead to a higher likelihood of mortgage foreclosure, and also affect home building and intergenerational equity (Miller and Peng, 2006; Oxley and Haffner, 2010; Stephens, 2011). From a macroeconomic perspective, the impact of housing price volatility is similarly damaging as the housing sector is vital to the national economy. Recent experience has made painfully clear the importance of the housing market in the U.S. A catastrophic and systematic collapse of the U.S. housing market triggered an economic recession, the so-called Great Recession, that rippled throughout the global economy. The level of housing price volatility is more important today than ever, as more U.S. households are headed by renters (and therefore housing investors) than at any point since at least 1965. According to a Pew Research Center analysis of Census Bureau housing data in 2017, the number of households renting their homes increased significantly, from 31.2% in 2006 to 36.6% in 2016, near the high of 37% in 1965.2 The analysis also states that, in 2016, 65% of the nation’s households headed by people under age 35 are rental households. This indicates that higher housing price volatility levels may discourage newly formed households from committing to homeownership as they view housing as a risky investment vehicle, although previously housing was perceived as a stable investment and a reliable inflation hedge (Stevenson, 1999; Anari and Kolari, 2002). Research into housing price volatility has received increased attention in the housing literature in recent years. Much of this research has focused primarily on investigating whether housing price volatility is time-varying, i.e., housing prices exhibit volatility clustering or autoregressive conditional heteroskedasticity (ARCH) effects (e.g. Dolde and Tirtiroglu, 1997; Crawford and Fratantoni, 2003; Miller and Peng, 2006; Miles, 2008; Barros et al., 2015). While it is widely recognized that conditional heteroskedasticity is pervasive in studies of housing price volatility, there has been little research addressing such heteroskedasticity in the context of spatial volatility. Unlike other capital markets, the housing market is characterized by factors such as differences in spatial dynamics, demographic characteristics, and inherent locational features that may be linked interregionally. 1 According to the Federal Reserve’s 2016 Survey of Consumer Finances (SCF), at $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. 2 Cilluffo et al. (2017). 2 Pioneer work by Meen (1999) uses a theoretical model to explain how real estate markets in separate locations interact with each other and describes changes in housing prices between two regions under normal market conditions. The study provides four possible explanations that may cause spillover effects among regional housing markets — migration, equity transfer, spatial arbitrage, and spatial patterns in the determinants of housing prices.3 Beginning with Meen (1999), a number of studies have investigated regional housing price diffusion in the U.S. housing market (e.g. Clapp et al., 1991; Pollakowski and Ray, 1997; Gupta and Miller, 2012). In contrast, research on volatility spillovers in the U.S. regional housing market has only recently received attention from the literature (e.g. Miao et al., 2011; Zhu et al., 2013). Also, most studies adopt primarily Case-Siller home price indices and examine regional housing markets based on a state or metropolitan statistical area (MSA) data, subject to availability. Piazzesi et al. (2007) argue that half of the volatility in individual housing prices reflects city-level variation, while one-quarter of the individual volatility is aggregate house price variation. This illustrates the importance of understanding the variation in the housing prices in narrow locational contexts. The initial motivation of the present study arises from compelling graphical evidence for the presence of space-varying volatility in housing returns at the census-tract level in the Chicago housing market, presented in Figure 1. Construction of the return data is described in Section 5. Considering the representative spatial plot for the returns and squared returns as a volatility proxy, it is evident that, while returns appear to be randomly distributed over space, squared returns are spatially correlated, with distinct clusters of high and low volatility readily identifiable, e.g., high volatility clusters are detected in Chicago’s west-side neighborhoods (Austin (23); West Garfield Park (26) and East Garfield Park (27); North Lawndale (29) and South Lawndale (30)) and south side neighborhoods (South Chicago (46); West Englewood (67) and Englewood (68); Greater Grand Crossing (69); Auburn Gresham (71)).4 Shifting the 2D view to the 3D map view given in Figure 2, it can be easily seen that clusters of high volatility are more pronounced in the southern part of the Chicago. This clustering pattern may occur in any housing market, as the nature of change in the U.S. housing market has opened significant gaps between regions in recent decades. Motivated by the visual implications of Figure 1 and Figure 2, this paper develops a spatial ARCH-type 3 The migration of households from higher-priced regions to lower priced regions to take advantage of regional housing price differentials as well as the process of equity transfer may lead to an increase in housing prices in the relocation regions, providing an avenue for linking regional housing markets. Alternatively, households may engage in spatial arbitrage whereby financial capital moves from higher-priced regional housing markets to acquire houses in lower-priced regions in anticipation of higher future returns in the lower-priced regions. Finally, if the underlying determinants of housing prices across regions experience correlated movement, then regional housing prices may exhibit the same correlated movement. 4 This pattern is also detected when using an alternative measure of volatility, absolute returns, suggesting that the dependence behavior is not sensitive to the choice of a particular measure of volatility. Robust analysis using this alternative is addressed in Section 6. 3 (a) Returns (b) Squared returns FIGURE 1 Spatial distribution of (a) returns and (b) squared returns, 2017 Note: The above figure shows the spatial distribution of returns and squared returns by census tract in the city of Chicago for 2017. The listed numbers refer to the codes for Chicago community areas (see Table 13 in the Appendix for details). model of squared returns to examine housing price volatility in the context of a spatial regression framework.5 Specifically, the proposed model incorporates space-varying volatility (or spatial volatility clustering), which is the spatial equivalent of time-varying volatility in a time-series context. When analyzing economic data that take only positive values, the use of transformations is very common and may be helpful when the usual assumptions are not satisfied in one’s original model. In the finance literature, the logarithmic transformation of realized volatility (the sum of squared intraday returns) is often used in empirical applications owing to its superior finite sample properties. Barndorff-Nielsen and Shephard (2005) demonstrate that the finite sample distribution of the log transformation of realized volatility is closer to the asymptotic standard normal distribution than to the raw version of realized volatility. Most empirical studies have also focused on the log transformation in volatility estimation and forecasting (Andersen et al., 2001; Corsi, 2009; Hansen et al., 2012; Koopman and Scharth, 2012). Recently, however, a more flexible power transformation suggested by Box and Cox (1964) has been considered in this context (Gonçalves and Meddahi, 2011; Nugroho and Morimoto, 2016; Weigand, 2014; Zheng and 5 As indicated by Bollerslev et al. (1992), squared returns of not only exchange rate data but of all speculative price series typically exhibit volatility clustering and ARCH/GARCH-type models are appropriate for volatility estimations. 4 (a) Returns (b) Squared returns FIGURE 2 Spatial distribution of returns and squared returns, 2017 5 Song, 2014; Taylor, 2017). The Box-Cox transformation indexed by the transformation parameter defines a general class of functional forms that includes the linear and log-linear forms as special cases. This feature allows the data added flexibility in model specification and provides a unified structure for statistically distinguishing between alternative functional specifications. Although the Box-Cox transformation has long been applied in finance, including to return-based volatility models, there has been almost no attempt to introduce the transformation in the spatial context. Studies conducted by Baltagi and Li (2004) and Li and Le (2010) are the most similar to the present paper, in that both explicitly consider functional form and spatial dependence simultaneously by applying the Box-Cox transformations in the context of the spatial regression model. In the former study, Baltagi and Li (2004) derive Lagrangian multiplier (LM) tests. In the latter study, the framework builds on double-length regression (DLR). Both of these studies indicate that choosing the correct functional form is beneficial in the presence of spatial dependence. In light of the foregoing, this paper generalizes the spatial ARCH model to a more general version with the Box-Cox transformation to derive the most appropriate functional form of spatial volatility, where the linear and log-linear models are special cases (hereafter referred to as the BC-SARCH model). The advantages of the BC-SARCH model are that both functional form and spatial dependence can be considered simultaneously, and that the validity of linear or log-linear specifications can be addressed. The proposed model would also be useful in another respect, in that the transformation might affect the distributional shape of the variable favorably. It is well known that the unconditional distribution of data for which typical linear models are used is frequently skewed and leptokurtic. Given that appropriate transformation may induce symmetry to the distribution, it seems probable that the proposed model would yield, apart from capturing the presence of spatial dependence, approximate symmetric and mesokurtic properties (i.e., normality) for the unconditional distribution. To estimate the BC-SARCH model, a maximum likelihood estimation (MLE) procedure is applied to both simulated and empirical data. The finite sample properties of the estimator is examined in a Monte Carlo experiment and the result suggests that the MLEs perform well in estimating parameters of interest. This study also carries out a detailed empirical analysis using housing sales price data within the city of Chicago in a period running from 2009 through 2018. It uses squared returns of observed housing sales prices that serve as a proxy for latent volatility and ensures that the study period covers the latest dynamics in the Chicago housing market. In this empirical application of the BC-SARCH model, the results suggest that volatility exhibits substantial spatial dependence and its functional form is close to the log-linear model. Judging by the associated model diagnostics and specification tests, the log-linear model is found to be superior to the linear model. Furthermore, the resulting spatial dependence parameter in the log-linear model serves as neighborhood elasticity, which measures how volatility in one neighborhood is linked to that in surrounding neighborhoods. For 2009 through 6 2018, the average value of the cross-sectional neighborhood elasticity is observed to be around 0.4, indicating that a 1% increase in volatility determines an increase of 0.4% in the volatility of neighboring locations. This analysis is further extended to the spatial panel data models to test temporal heterogeneity in neighborhood elasticity, i.e., whether neighborhood elasticity stays the same over time. Applying the method introduced by Xu and Yang (2019), this paper
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