ANALYZING COMOVEMENTS IN HOUSING PRICES USING VINE COPULAS

I. INTRODUCTION When housing prices were rapidly appreciating from 1999 to 2006, investment firms created structured securities by combining mortgages from houses located in different parts of the country, and then traders bought and sold those securities, and also pieces of those securities called "tranches," in secondary markets. The most familiar of those structured securities was the collateralized debt obligation (CDO). The main appeal of CDOs rested in the belief that, due to the localized nature of housing markets, houses in separate geographic markets would be unlikely to simultaneously experience large decreases in prices. Credit rating agencies offered support to this thinking by awarding many CDOs the highest possible safety rating. However, it soon became clear that CDOs offered less diversified protection than originally thought when, starting in 2006, housing prices in different geographic areas, even those located far apart, simultaneously plummeted in value. As a consequence, CDOs lost much of their value, with the global CDO market shrinking from $482 billion globally in 2007 to only $8 billion in 2010. (1) In the wake of the housing crisis, financial analysts and policy makers have questioned why, compared to pre-crisis expectations, housing prices showed strong correlations across different geographic areas. In addition, a branch of research has emerged that attempts to identify sources of, and quantify magnitudes of, housing price comovements (Apergis and Payne 2012; Barros, Gil-Alana, and Payne 2012). The popular press quickly assigned blame to the statistical method used to analyze linkages between housing markets: the Gaussian copula (Li 2000). Notably, the March 2009 issue of the technology magazine Wired featured an article on the Gaussian copula entitled "Recipe for Disaster: The Formula that Killed Wall Street." Similar ideas have reached the general public through the works of Nassim Taleb (Taleb 2007). The Gaussian copula became popular due, in part, to its link to the familiar multivariate normal distribution. But the multivariate normal distribution has asymptotic independence, such that events, regardless of the strength of their correlation, become independent if one pushes far enough into the tails (Embrechts, McNeil, and Straumann 2002). Thus, in the midst of the housing crisis, which might be thought of as a lower tail event, the Gaussian copula predicted near independence in price movements across different areas, when in fact, prices plummeted simultaneously throughout most of the United States. But the Gaussian copula's link to the normal distribution was not its only appeal. Perhaps a bigger reason for its popularity was that the Gaussian copula, much like the related normal distribution, readily extends to higher dimensions. Certainly, credit rating agencies were not considering simple bivariate movements between two locations, but rather multivariate movements across many locations. Recent studies argue that alternative specifications, especially copulas that depart from normality, more accurately reflect correlations in housing price movements during extreme market swings (Ho, Huynh, and JachoChavez 2014; Zimmer 2012). However, those improved fits have been achieved only in bivariate models that compare housing price movements between two locations. It remains an open question whether non-Gaussian copulas can accurately reflect housing price movements in higher dimensional settings. Unfortunately, copulas other than the Gaussian do not readily extend to higher-than-bivariate dimensions (Nelsen 2006, 105). Attempts to develop higher dimensional copulas, some of which are discussed below, either impose unrealistic restrictions or present difficulties when applied to data. As an alternative, this article develops multivariate models of housing price comovements based on vine copulas. The approach requires marginal distributions, which financial analysts should know with some certainty, and bivariate copulas, which have well-understood statistical properties. …