A Hierarchical Linear Model Approach for Assessing the Effects of House and Neighborhood Characteristics on Housing Prices

Journal of Real Estate Practice and Education, 2004 by Brown, Kenneth H, Uyar, Bulent

Abstract. This pedagogical paper illustrates how a hierarchical linear model (Raudenbush and Bryk, 2002) can be used by researchers and practitioners to estimate housing prices as a function of both house and neighborhood characteristics. While traditional hedonic regression models allow researchers to include both house and neighborhood characteristics in the study of housing prices, hedonic regression does not account for the inherent hierarchy in the housing purchase decision, namely that houses reside in neighborhoods, which in turn exist within cities and states. On the other hand, multinomial logit models incorporate such hierarchy but focus primarily on the housing purchase decision and not the housing price. Thus, the hierarchical linear model gives researchers a more flexible tool to model housing prices.

Introduction

A hedonic regression model has traditionally been used to model housing prices. The hedonic approach regresses the price of the house against various characteristics of the house to determine the importance of specific characteristics (such as the number of bathrooms) to explain housing prices. For selected examples and extensions of the hedonic approach, see Downes and Zabel (2002), Cheshire and Sheppard (1998), Palmquist (1984), Paterson and Boyle (2002) and Yinger, Bloom, Borsch-Supan and Ladd (1988). Most hedonic regression models include characteristics of the community where a house is located, as well as the house's characteristics. Since houses are located within neighborhoods, neighborhoods within cities, and so forth, residential location decisions are inherently hierarchical, proceeding in stages. In other words, and as pointed out by Quigley (1985), households begin the search process by choosing a town to live in, followed by a neighborhood to live in given the town, and finally a house to live in given the neighborhood and the town. The hedonic regression model, however, does not take advantage of this hierarchy when modeling housing prices.

An alternative approach used in the housing literature is the multinomial logit model (McFadden, 1978). This is a discrete choice approach that attempts to identify the importance of various housing characteristics on the housing purchase decision. This model does account for the inherent hierarchy in the data (Chattopadhyay, 2000; Nechyba and Strauss, 1998; and Quigley, 1985). Unlike the hedonic model, however, this approach does not attempt to explain housing prices, but instead examines the variables that determine whether a household buys a dwelling or not.

A relatively new approach to modeling hierarchical data is the hierarchical linear model (HLM). Raudenbush and Bryk (2002) outline the various applications and statistical techniques associated with the model. HLM has been widely used, particularly in the education literature (see Aitkin, Anderson and Hinde, 1981; Aitkin and Longford, 1986; Goldstein, 1987; Rumberger and Thomas, 1993). Its use in economics has been limited, though Kahane (2001) employed the HLM to determine the importance of team and player attributes on National Hockey League player salaries. His approach shows that the HLM can separate the effects that different levels of the hierarchy have on an outcome variable. Thus, the HLM can be employed to model housing prices, as the hedonic model does, while at the same time incorporating the hierarchical nature of the data into the analysis, as the multinomial logit does.

The objective in this paper is to demonstrate in a simple way, for the benefit of researchers and practitioners alike, how HLM can be used to develop models that account for the inherent hierarchy in determining housing prices. This paper does not attempt to build a complete model of housing prices using HLM. Therefore, a two-level model is considered that specifies housing prices as a function of dwelling and neighborhood characteristics, and includes only one variable for each level. While this limits the scope of the study, it does permit a discussion of the key ideas of the HLM model and the statistical results obtained from such models in a pedagogical framework. Future research should, of course, expand the list of explanatory variables at each level to develop a more complete model.

An HLM Model of Housing Prices

The goal in this paper is to demonstrate how HLM can be used to separate the variation in housing prices into that portion that depends on house-specific characteristics and that portion that depends on neighborhood-specific characteristics. The HLM model is detailed in Raudenbush and Bryk (2002) and summarized in Kahane (2001).

Raudenbush and Bryk (2002) refer to Equation (5) as the "combined" model. As they point out, while Equation (5) is linear, it cannot be estimated using traditional ordinary least squares (OLS) methods. Standard OLS methods require the random errors to be independent and to have constant variance. The error structure in Equation (5) does not meet these requirements for two reasons. First, the errors are dependent within each neighborhood since u^sub 0j^ and u^sub 1j^ are common to every house in neighborhood j. Second, the errors have unequal variances since u^sub 0j^ u^sub 1j^(X^sub ij^ - X^sub j^) depend on u^sub 0j^ and u^sub 1j^, which vary across neighborhoods and on (X^sub ij^ - X^sub j^), which varies across houses. In this instance, OLS estimates will be both biased and inconsistent. For a derivation of the optimal estimator for this model, see Raudenbush and Bryk (2002).