Methodological Issues in the Analysis of Residential Preferences,Residential Mobility,and Neighborhood Change

2.1. Stated Preferences

An example of stated preference data involves measures of residential race-ethnic preferences from the 1992–1994 Multi-City Study of Urban Inequality (MCSUI) (Bobo et al. 2000, appendix D). The MCSUI presented survey respondents with cards depicting five neighborhood vignettes of 14 houses that vary in their race-ethnic composition. The respondent’s house is located in the center of the neighborhood. Although the study as a whole examined four groups (whites, blacks, Asians, and Hispanics), each card shows only two groups, the respondent’s group and one other group. Figure 1 shows the cards shown to black respondents concerning white neighbors. The survey used a split-ballot design in Boston and Los Angeles, such that each respondent had a one-third probability of being shown a particular vignette out-group. The data include three measures of racial residential preferences. First, for each of the five neighborhood vignettes, each respondent is asked whether he or she would move into that neighborhood. (Whites were asked if they would move out of the neighborhood.) The data consist of five binary responses, each one corresponding to a different proportion own-group and featured out-group. Second, respondents were asked to rank the five vignettes in order of attractiveness. Finally, respondents were given another card with the same configuration of 14 empty houses, but they were asked to assign each house to one of the four race-ethnic groups according to his or her “ideal” neighborhood composition. Exact wording of these three types of questions is shown in Appendix A. The second of these three types of response provide a full ranking of alternatives. The binary responses to the “would you move in/out” question provide a partial ranking of the five neighborhoods. The neighborhoods that the respondent would move into are ranked higher than the ones that the respondent would not move into, but the relative desirability beyond this dichotomy is unknown to the analyst. The ideal neighborhood ethnic configuration response indicates that the chosen configuration is preferred to all other possible configurations, but the relative desirability of the configurations that were not chosen is unknown to the analyst.

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Figure 1.

Neighborhood vignettes shown to black respondents asked about white neighbors.

These data have been analyzed using a variety of approaches, including descriptive statistics, OLS regression, and categorical response models of various types (e.g., Farley 1978; Farley et al. 1978, 1993, 1994; Charles 2000, 2005; Krysan and Farley 2002). Although these analyses have been informative, they typically do not make full use of information available in the data. In contrast to these approaches, the discrete choice models proposed in this paper make full use of the quantitative information about race-ethnic composition in these data, allow for full examination of complex interactions among race-ethnic groups, generalize to data that include more dimensions of neighborhood variation than just race-ethnic makeup, and provide a natural comparison to analyses of actual residential choices.

The MCSUI vignettes contain information only on neighborhood racial composition; all other neighborhood characteristics are ignored. Thus, it is difficult to know whether to interpret these data as representing the degree of an individual’s true ethnic “tolerance” or a response to other neighborhood characteristics (e.g., crime, school quality, and housing costs) associated with race (Quillian 1995; Harris 1999). Emerson, Chai, and Yancey (2001) use vignette neighborhoods that vary along a number of dimensions: school quality, ethnic composition, property values, and crime rate. They find that, after controlling for non-race/ethnic neighborhood characteristics, whites’ aversion to a predominantly Hispanic or Asian neighborhood is no longer statistically significant in their sample, but whites’ apparent aversion to black neighborhoods remains. Krysan et al. (2009) construct video vignettes that vary the race of actors portraying neighborhood residents, but they hold constant key visual indicators of the socioeconomic composition of neighborhoods (e.g., the upkeep of yards and the types of cars in driveways). Multidimensional vignette data in principle allow the analyst to “control for” any potential confounding neighborhood characteristics. However, it is hard to represent multidimensional neighborhoods using pictures, and complex verbal descriptions are difficult for respondents to understand. A more straightforward way of exploring how multiple factors affect residential choice is to use data on actual moves.

2.2. Mobility Histories

Residential choices and preferences may also be observed in actual mobility behavior. Information about mobility and neighborhood choice may be obtained from cross section data, such as the U.S. Decennial Census, which documents both current neighborhood of residence and also year moved into current unit (to identify recent movers). Alternatively, mobility data may come from retrospective survey questions that ask individuals to recall their previous addresses over some specified time period. For example, wave 1 of the Los Angeles Family and Neighborhood Survey (LA FANS) asked individuals to report all moves and addresses lived in over the past two years, and wave 2 asked for a residential history between wave 1 and wave 2 (Sastry et al. 2006). Residential mobility data may also be prospective, identifying respondents at the beginning of a time period and tracking their subsequent moves.

For example, the Panel Study of Income Dynamics (PSID) records where a respondent lives at the time of each interview. The population represented by a set of mobility data, of course, depends on the survey instrument. For example, the data may be nationally representative data, as in the Census or the PSID, or data focused on a particular metropolitan area, as in the LA FANS.

Several studies have used the PSID panel data to examine neighborhood mobility. Some treat the decision to move out of one’s current neighborhood (e.g., analyses of “white flight”) as a binary outcome variable (e.g., South and Crowder 1997; Rosenbaum and Friedman 2001), whereas others use the demographic (typically race-ethnic) composition of the destination neighborhood as a polytomous or quantitive outcome variable (Crowder, South, and Chavez 2006; Crowder and South 2008). The outcome is often characterized by its racial composition (e.g., its percentage of white, black, or Hispanic). Typically the outcome is modeled using a binary logit (did or did not move out) or multinomial logit (with destinations categorized into types). The goal of these analyses is to predict choice of destination conditional on individual and/or household characteristics, characteristics of the current residential census tract, and characteristics of the metropolitan area as a whole.

Although these studies usefully describe mobility among neighborhood types and covariates of this mobility, they are ill-suited to the study of residential decision-making by individuals and the impact of these decisions on segregation or other aspects of population distribution. Whereas analyses of mobility rates among neighborhoods with varying percentages of a given ethnic group examine only a single dimension of destination neighborhoods, households evaluate potential destination neighborhoods on several dimensions—for example, racial composition, economic level, housing price, and school quality—when making residential decisions. Any single dimension, when considered by itself, may be confounded with other distinct but correlated dimensions. Additionally, these studies only allow respondents’ own characteristics, characteristics of their current neighborhood, and the racial composition of the chosen tract to affect destinations, omitting the possible effects of the comparative characteristics of potential destinations on mobility decisions. As we show below, a fruitful alternative approach is to adapt models for discrete choice to the analysis of residential decision making. This approach incorporates the effects of both neighborhood and individual characteristics on residential location choice, a multidimensional approach to measuring neighborhood attractiveness, and a natural way to extrapolate to aggregate neighborhood change. Additionally, it allows us to examine both stated preferences and actual mobility decisions within a common analytic framework.

2.3. Stated Preferences Versus Mobility Histories

Stated preference (vignette) and mobility history data have several complementary strengths and weaknesses. The most important advantage of stated preference data is that the hypothetical characteristics of neighborhoods are under the control of the investigator. Thus, it is possible to assign descriptions of neighborhoods that vary along one or more dimensions to different individuals or to administer to the same individual an array of possible neighborhood configurations. Randomization combined with observations of repeated choices can control for unmeasured differences among individuals. This is a relatively low-cost means of data collection inasmuch as it does not require the collection of residential mobility histories or large samples of individuals, only a fraction of whom have moved in the recent past. It also allows for the specification of relatively rare types of neighborhoods that would otherwise require an extremely large sample of actual moves. Furthermore, stated preference designs elicit individuals’ preferences; in theory these preferences are unconstrained by affordability constraints, housing supply, discrimination, and other factors that affect actual moves.

The weaknesses of neighborhood vignettes arise because they are administered in interviews, which poorly approximate the contexts in which actual choices are made. First, preference for neighborhoods that vary in their racial makeup is potentially a sensitive subject and thus respondents may express socially desirable preferences. Second, vignettes are typically administered to individuals, but mobility decisions may be made collectively by multiple household members. Third, it is usually impractical to vary more than two or three dimensions of neighborhood desirability in vignette studies (e.g., racial makeup, poverty rate, age of housing), precluding the investigation of complex interactions among determinants of housing desirability (Harris 1999). Fourth, because neighborhood vignettes are hypothetical, stated preferences abstract from the virtually limitless array of alternatives that people may have in a real choice situation, as well as their substantial proclivity not to move (that is, to choose their current residence) as a result of the search and moving costs. Finally, as discussed further in Section 7, stated preferences may be sensitive to how interview questions are phrased.

Actual mobility histories also have their own advantages and disadvantages. On the one hand, they provide true measures of real mobility decisions, albeit subject to constraints. Additionally, because they measure choices made by heterogeneous individuals for neighborhoods that vary in a wide range of attributes, they allow the analyst to represent mobility using a rich set of individual and neighborhood covariates. Finally, probability samples of individuals and households include both movers and nonmovers and, in individual mobility histories, periods of stable residence as well as episodes of mobility. This enables the analyst to examine differences in how decision makers evaluate their own locations relative to other potential destinations, and to thus explore how origins as well as destinations affect choice.

On the other hand, actual moves are not pure measures of residential preferences. Rather, they result from preferences about desired locations in the context of constraints on residential options. If the analyst can specify the true choice set for each individual, this will reduce the extent to which constraints dominate the choice process. In practice, however, we seldom know an individual’s true range of alternatives. Additionally, mobility histories are comparatively expensive to collect. Because recent mobility is usually a relatively rare event, large amounts of data must be collected, whether through lengthy retrospective mobility histories, long prospective panels, or shorter residential histories obtained from large samples of individuals. The need for large numbers of observations is exacerbated, moreover, when the analyst wishes to look at the selection of relatively rare neighborhoods.

In principle, we can combine the strengths of stated and revealed preference data by pooling them into one model. Louviere, Hensher, and Swait (2000) discuss this possibility for studying consumer choice. To our knowledge, this approach has not yet been taken in the study of residential choice.

3.1. Residential Mobility as a Market Process

In most of the models discussed below, we represent residential choice as a “demand-side” process whereby individuals or households select from an array of possible destinations. This is a partial view of residential mobility inasmuch as moves in fact result from interactions between buyers and sellers or landlords and renters who negotiate the exchange of housing units. Discrete choice models capture housing demand conditional on housing supply, but these models do not represent how the actions or motivations of housing suppliers (e.g., the steering decisions of real-estate agents, the lending decisions of banks, or the building decisions of developers) affect the number and type of available units. For the limited purpose of analyzing individual choice, it suffices to assume that housing vacancies and housing prices are given and a one-sided approach is sufficient. For studying the realistic aggregate dynamics of a housing market, it may be necessary to take the supply as well as the demand side of the market into account. In later sections, we discuss how to incorporate prices into models of individual residential choice and to use price equilibrium assumptions to assess the effects of changes in aggregate demand. (An alternative modeling strategy is to model explicitly the interactions between housing suppliers and housing seekers. Such models could rely on optimal matching of housing seekers and providers [e.g., Roth and Sotomayor 1990] and use extensions of available “two-sided” statistical models for joint decisions of actors on both the supply and demand sides of a market [Logan 1996, 1998; Logan, Hoff, and Newton 2008]. Specification and implementation of such a model for housing markets is beyond the scope of this paper.)

3.2. Outcome Variable and Data Structure

In discrete choice models, the outcome is either a single choice (representing the “best” possible outcome given available opportunities) or a set of ranked choices. Rankings contain more information on preferences than single choices, which reveal the top-ranked choice but not the relative desirability of the remaining options. In data on actual choices, we typically observe only a single choice (or a series of choices made over some period of time). In stated preferences, respondents may rank neighborhoods in order of desirability. The models used to estimate parameters based on these two outcomes are similar, except that the ranked outcome model includes additional elements to the likelihood function, one for each ranking given the current set of unranked items. We discuss this in more detail below.

Table 1 shows the data structure for estimating discrete choice models. Each of the I individuals has J lines of data, one for each of potential destination alternatives. We refer to each line of data as an “individual-alternative” and the set of J alternatives as the individual’s choice set. In the example shown in Table 1, J = 5 for all individuals, but in general it is possible for the size of choice set to vary across individuals. Individual characteristics (X _i ,) are constant within individuals, but features of neighborhood alternatives (Z _j ), such as neighborhood proportion own-race, vary across alternatives within individuals.

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Table 1.

Data Structure Used in Estimation of Discrete Choice Models

ID	Own Race	City	Neighborhood	Other Race	Rank	Move
1	black	Detroit	0	white	4	0
1	black	Detroit	29	white	2	1
1	black	Detroit	50	white	1	1
1	black	Detroit	86	white	3	1
1	black	Detroit	100	white	5	0
2	white	Detroit	0	black	.	1
2	white	Detroit	7	black	.	1
.	.	.	.	.	.	.
.	.	.	.	.	.	.

Note: The variables are defined as own-race (respondent’s own race), city (survey city), neighborhood (proportion other-race), other-race (other racial group in vignette neighborhood), rank (attractiveness ranking assigned to vignette neighborhood), and move (whether or not the respondent would move into this vignette neighborhood).

3.3. Conditional Logit Model

Let Y _ij be an indicator variable denoting which neighborhood (indexed by j) is chosen by the ith individual (i = 1, . . ., I; j = 1, . . ., J). Let U _ij denote the (latent) utility or attractiveness that the ith individual attaches to the jth neighborhood. Let p _ij denote the probability that the ith individual chooses the jth neighborhood. The utility of a neighborhood for an individual depends on neighborhood characteristics, possibly interacting with characteristics of individuals. These characteristics may or may not be known by the researcher, but they are known to the individuals to whom they apply. Let Z ¯ i be a vector of observed (to the analyst) characteristics of the jth neighborhood (e.g., the race-ethnic makeup of the neighborhood). Let X ¯ i denote a vector of observed characteristics of the ith individual or household. These characteristics include fixed demographic characteristics such as race and sex, and time-varying characteristics such as income, employment status, housing roster, and residential history. Let φ _ij represent the contribution of unobserved attributes of individuals and potential neighborhoods to utility. The attractiveness of neighborhoods is represented as

U ij = F ( Z ¯ j , X ¯ i , φ ij ) .

If F is a linear random utility model, then, for example, for a single observed neighborhood and personal characteristic (Z and X respectively), the model is

U i j = β Z j + γ Z j X i j + φ j ,

where β and γ are parameters to be estimated. When individuals choose where to live, they implicitly compare neighborhoods in their choice set—that is, neighborhoods that they know about and where they may move with a nonzero probability. The difference in utility between the jth and the kth neighborhood is

U ij − U ik = β ( Z j − Z k ) + γ ( Z j − Z k ) X i + ( φ ij − φ ik ) .

Utility differences among neighborhoods for a given individual are thus a function of differences in observed and unobserved characteristics of neighborhoods and individuals. Because utility comparisons take place within individuals, their characteristics X _i do not affect the utility comparison additively. These characteristics, however, may interact with neighborhood characteristics. For example, the effect of differences in the proportion of persons in a neighborhood in a given ethnic group on the relative attractiveness of the neighborhoods may differ between individuals who are members of that ethnic group and those who are not. Unmeasured characteristics of individuals may also modify the effects of neighborhood characteristics, as we show below. These unmeasured characteristics can induce random variation in the effects of measured neighborhood characteristics β. For example, the effect of the proportion of persons in the neighborhood who are ethnic minorities may depend on an individual’s level of tolerance, which is unobserved to the analyst.

Given data on the characteristics of individuals and neighborhoods and the behaviors or stated preferences of individuals for neighborhoods and an assumed probability distribution of the unobserved characteristics of individuals and neighborhoods, it is possible to estimate the parameters of the discrete choice model. If the φ _ij follow a type I extreme value (Gumbel) distribution, we obtain the conditional logit model ²

p ij ( Z j , X i , C ( i ) ) = exp ( β Z j + γ Z j X i ) ∑ k ∈ C ( i ) exp ( β Z k + γ Z k X i )

where C _(i) denotes the choice set for the ith individual, which may be restricted to incorporate discrimination, prices, or information constraints (McFadden 1978). ³ For example, the choice set may be restricted to units within a given radius of a person’s current home, to units in neighborhoods that are at least 10% own-race, or to units where monthly rent or mortgage payments would be less than some fraction of individuals’ incomes. Typically these models are estimated using maximum likelihood, where the likelihood is

L = Π i = 1 N Π j = 1 J y ij exp ( β Z j + γ Z j X i ) ∑ k ∈ C ( i ) exp ( β Z k + γ Z k X i ) .

Early applications of the basic discrete choice model to residential mobility analysis include McFadden (1978) and Lerman (1975). Gabriel and Rosenthal (1989) use a multinomial logit model to examine how race and other traits of individuals affect residential mobility among five counties in the Washington, D.C., area. Sermons and Koppelman (2001) estimate a discrete choice model of residential choice that explores how men and women differ in their sensitivity to commuting time. ⁴

3.4. Independence from Irrelevant Alternatives Assumption

The conditional logit form of the discrete choice model assumes independence from irrelevant alternatives (IIA). It is a model for pairwise comparison and assumes that the odds of preferring an alternative in a pairwise comparison is unaffected by the other available alternatives. That is, after accounting for observable features of choices, the remaining (unobserved) features of choices are uncorrelated (that is, E [ φ ij , φ ik ] = 0 ). IIA is really an assumption about proper model specification, which implies that there is no omitted variable bias and also that the choice set is exhaustive and well defined (McFadden, Train, and Tye 1981). The IIA property implies that the ratio of probabilities for any two choices is unaffected by the utilities of all other alternatives implying that the ratio is not affected by the addition or exclusion of alternatives. The conditional probability of choosing the jth neighborhood given a choice between neighborhood j or k is

⪻ ( Y ij = 1 | Y ij ∈ { j , k } ) = ⪻ ( Y ij = 1 ) ⪻ ( Y ij = 1 ) + ⪻ ( Y ik = 1 ) = exp ( β Z j + δ Z j X i ) exp ( β Z j + δ Z j X i ) + exp ( β Z k + δ Z k X i )

This probability does not depend on the traits of neighborhoods other than j and k. If valid, this assumption makes it possible to estimate choice models on a subset of alternatives in the choice set. Additionally, we can make out-of-sample predictions because the parameter estimates from the model are invariant to the inclusion or exclusion of alternatives in individuals’ choice sets.

However, in practice the IIA assumption is often not met. We rarely observe all attributes of destinations that affect mobility behavior. Some neighborhoods have similar characteristics and, if one of them were omitted, individuals would disproportionately choose a similar neighborhood rather than distribute themselves proportionately across both similar and dissimilar neighborhoods. Unless the sources of similarity and dissimilarity among neighborhoods are controlled in the choice model, the model is likely to yield incorrect predictions about the effects of omitting one of the neighborhoods. The most common way of testing for IIA is through partitioning the choice set, and comparing estimates from a full model with those from a model estimated using a subset of the choice set (Hausman and McFadden 1984; Small and Hsiao 1985). ⁵

There are three ways of dealing with IIA violations. First, we can ignore violation of the IIA assumption but recognize that the estimated parameters are at best an approximation of choice behavior and are not appropriate for making inferences about substitution patterns. Second, we can in principle modify the discrete choice model by adding additional covariates that represent sources of neighborhood resemblance. However, usually we cannot capture all the unobserved correlation in choice behavior explicitly. Finally, if available data permit, we can use a mixed logit specification, preferably with panel data that permit identification of unobserved time invariant neighborhood heterogeneity. We discuss these models in more detail below.

3.5. Unmeasured Heterogeneity

Even neighborhoods that are identical on measured characteristics may vary in their desirability to individuals. For example, neighborhoods may vary in amenities that have not been measured (nearness to the ocean or availability of charming coffee shops). Additionally, even among individuals who have identical measured attributes, we may observe variation in their mobility behavior. Unaccounted for features of individuals or neighborhoods that affect choice behavior can lead to correlations in the disturbance φ _ij across alternatives. Another form of unobserved heterogeneity arises if we incorrectly assume that people select one neighborhood directly from a given choice set when in fact they decide sequentially, systematically narrowing down their options based on some criterion. For example, choosers may first select part of a city, then select a neighborhood within that part, and then a house within the neighborhood. In this case, all neighborhoods within the chosen region and all vacant houses within the chosen neighborhood have a higher than average probability of selection irrespective of their measured characteristics. When the number of alternatives is small, we can represent the average level of attractiveness of each residential choice by including alternative-specific constants, which enter as dichotomous variables in the choice model. However, when the choice set is large, when we seek to parameterize the effects of measured attributes of neighborhoods on choice probabilities, or when the concern is with unobserved attributes of individuals that influence choice behavior, it is more appropriate to estimate a model that allows for correlation in the attractiveness of observations within or among individuals. Several models are available to represent correlation of attractiveness across observations, including the nested and mixed logit models. We discuss these in turn.

3.5.2. Mixed Logit Model

Mixed logit models are a more general class of models that can accommodate both alternative- and individual-specific unmeasured heterogeneity, and they are useful if the analyst believes that the unobserved heterogeneity is correlated with observable characteristics of neighborhoods. The model is an extension of Equation (4). In particular, the error component φ _ij is broken out into two parts; that is,

U ij = β Z j + γ Z j X i + μ i W ij + ε ij ,

where μ _i is an individual-specific (alternative invariant) random vector with zero mean, the W _ij are one or more vectors of data related to the jth alternative, and φ ij = μ i W ij + ε ij . The W _ij represent characteristics of alternatives that may or may not include interactions with individual-level variables, and the ε _ij follow a type I extreme value distribution. The specification of the W _ij generates correlation in alternatives over the unobserved portion of utility because the covariance between any two alternatives is

E ( [ μ i W ij + ε ij ] [ μ i W ik + ε ik ] ) = W ′ ij V ( μ ) W ik ,

where V(µ) is the covariance matrix for µ. Given some value µ _i , the conditional choice probability follows the logistic distribution since the remaining error component ε _ik follows an extreme value distribution:

p ij ( μ i ) = exp ( β Z j + γ Z j X i + μ i W ij ) ∑ k ∈ c ( i ) exp ( β Z k + γ Z k X i + μ i W ik ) .

Because the µ _i are unobserved, the unconditional probability is the logit formula integrated over all possible values for µ _i , weighted by the density of µ.

p ij = ∫ p ij ( μ ) f ( μ | Ω ) d μ

where Ω denotes support for the distribution of µ. These models are referred to as “mixed logit” because their probabilities are heterogeneous with f as the mixing distribution (Train 2003). The mixing distribution is assumed by the analyst, and it can be normal, lognormal, or other shapes. Because choice probabilities do not have closed form solutions, they cannot be estimated directly. Instead, the probabilities can be simulated by drawing values of µ, from its assumed distribution, using a Gibbs sampler, EM algorithm, or some other form of iterative estimation (see Train 2003, Chs. 8–10). These models can be estimated using specialized software for discrete choice estimation, such as the NLOGIT package for LIMDEP.

The choice probabilities depend on parameters β, γ, and Ω. Different patterns of correlation are specified based on the choice of W _ij . For example, in the nested logit model with N nests, W _ij is a set of dummy variables, d j c , indicating whether the jth alternative belongs in the cth nest ( W ¯ = { d j 1 , d j 2 … d j N } ) . In this case, the µ _i are IID random deviates where V(µ) is a diagonal matrix with elements σ _n , n = 1, 2,. . ., N. The unobserved component is correlated within but not between nests, with covariances E ( [ μ i W ij + ε ij ] [ μ i W ik + ε ik ] ) = σ n if alternatives j and k are both in the nth nest, and equals zero otherwise.

If the pattern of unobserved heterogeneity across alternatives is unknown, the W _ij can be specified as error components that, along with ε _ij , make up the random component of utility. In the usual conditional logit model, W _ij are zero, which means there is no correlation in utility over alternatives after conditioning on observables. When W ij ≠ 0 , utility is correlated over alternatives, even when the error components are independent across observations such that V(µ) is a diagonal matrix.

Because this specification includes no measured neighborhood characteristics to identify the correlation across observations, it requires strong assumptions about the distribution of the W _ij random deviates.

Mixed logit models can also represent heterogeneity in individual behavior by assuming that W_ij= Z (or Z_j X _i when the random coefficient refers to interaction between alternative- and individual-specific variables) such that U ij = β Z j + γ Z j X i + μ i Z j + ε ij . Under this circumstance, β i = β + μ i and thus the coefficients of beta vary over individuals, with mean β and deviations µ _i . Elements of Z_j that do not enter into W_ij have fixed parameters that do not vary over the population. Similarly, elements of W _ij that do not enter into Z_j are variables whose parameters vary within the population but have means of 0. This is analogous to the standard random coefficient framework for linear models. For example, if W_ij includes a variable that is the difference between the jth neighborhood’s median income and the ith individual’s household income, the estimated model would allow for individual variation in response to neighborhood median income, potentially reflecting unobserved differences in consumption patterns.

While mixed logit models are widely used in transportation and land-use research, there are only a few studies that apply them specifically to the analysis of residential choice. In their analysis of Dallas County households’ choices to live in a particular land-use zone, Bhat and Guo (2004) estimate a mixed spatially correlated logit that allows for both unobserved taste variation among movers and also spatial correlation among adjacent zones. More recently, Hoshino (2011) uses a mixed logit model to analyze stated preference data collected in Tokyo.

3.6. Functional Form

Discrete choice models allow the analyst to specify a variety of ways that people may respond to characteristics of neighborhoods. For example, in models of the relationship between neighborhood racial composition and the probability of entering or leaving a neighborhood, it is not just the average level of tolerance that matters but also the shape of the response curve. Schelling (1971, 1978) showed that a high level of segregation results when individuals have a threshold response to the proportion own-group in their neighborhood—that is, when people are indifferent to neighborhood characteristics within some interval and only care about whether a neighborhood characteristic is above or below the threshold. In a simple model where only neighborhood characteristic Z_j enters into the choice equation, the utility in a threshold specification is

U i j = { 1 if Z j > thresold 0 otherwise ,

where the threshold is a specific value of Z_j. An alternative behavioral response is that people have a continuous response to neighborhood composition; in other words they are sensitive to even small changes in composition regardless of the actual level of the compositional variable. That is, utility is a continuous specification of neighborhood composition—for example, U ij = β Z j . In addition, a number of intermediate functional form specifications allow for indifference over some intervals of neighborhood composition with a threshold response at key points. These functional form assumptions about how people respond to neighborhoods have implications for neighborhood turnover and segregation dynamics. Bruch and Mare (2006, 2009) show how the shape of choice functions affects segregation dynamics.

3.7. Models for Ranked Data

The discrete choice models discussed thus far assume that the analyst observes only the chosen alternative and has no information on the relative utilities of unchosen alternatives. Stated preference data, however, may provide information on full or partial ranking of alternatives, albeit for a hypothetical choice set (Allison and Christakis 1994). ⁹ Ties occur in the data when respondents assign multiple items the same rank, and incomplete rankings occur when respondents leave certain items un-ranked. In this case, we observe groups of items that are ranked together, providing a partial ranking. The rank-ordered logit accommodates tied rankings (Allison and Christakis 1994:206–8). The likelihood function is an extension of the simpler discrete choice likelihood Equation (5), except that Y _ij is a rank rather than a 0/1 indicator for the chosen alternative, and the model includes an additional term δ _ijk , which equals 1 if the ranking of the kth choice is greater than or equal to the ranking of the jth choice, and is zero otherwise. That is,

L = Π i = 1 N Π j = 1 J y ij exp ( β Z j + γ Z j X i ) ∑ k ∈ C ( i ) δ ijk exp ( β Z k + γ Z k X i ) .

In the case where one alternative is ranked “first” and all others are tied for “last,” the rank-ordered logit model simplifies to the discrete choice model for a single choice.

4.1. Aggregation of Alternatives

In actual residential choice, individuals select among housing units, apartments, or even rooms. Typically, however, we observe choices of aggregate units such as Census tracts. When the units that individuals actually choose are not the ones that we observe, it is necessary to modify the choice model to take into account the differential size and variability of the aggregate units (Ben-Akiva and Lerman 1985, Ch. 9). Denote by L the actual choice set (e.g., housing units). P _i (l) is the probability that the ith decisionmaker chooses the lth housing unit (where l ∈ L ). The L housing units are partitioned into J nonoverlapping aggregates (e.g., Census tracts denoted as C _j ) such that the total number of units in the jth aggregate, M j = ∑ l ∈ C j housing unit _l . The probability of choosing the jth tract is equal to the sum of the probabilities that the respondent chooses each of the tract’s constituent housing units. Thus, the probability that the chooser selects a housing unit located in the jth parcel is P i ( j ) = ∑ l ∈ C j P i ( l ) , and the utility associated with the jth aggregate is the average utility of all its housing units:

U ¯ ij = 1 M j ∑ l ∈ C j U il .

An implication of this result is that, all else being equal, aggregate utilities and choice probabilities vary with the size of the aggregate units. Census tracts with more housing units will, ceteris paribus, be chosen more often than those with fewer units. Further, within tracts, individual dwelling units may be heterogeneous in their desirability. Thus, the estimated effects of other measured characteristics of tracts may be distorted by their correlations with tract size and variability. To take these complications into account we modify the general choice model in Equation (4) as follows:

P j ( i ) = exp ( U ¯ ij + μ 1 ln M j + μ 2 ln B j ) ∑ k = 1 K exp ( U ¯ ik + μ 1 ln M k + μ 2 ln B k ) ,

where U ¯ ij is the average utility of the housing units within the jth Census tract, M_j is the number of housing units in the jth Census tract, B_j measures the variation in the utilities of housing units within the jth Census tract, and µ₁,µ₂ are positive scaling coefficients (Ben-Akiva and Lerman 1993). Estimates of the M_j are typically available from Census data and thus can be straightforwardly included as regressors in the discrete choice model. However, we rarely have complete descriptions of the distribution of utilities of individual housing units and thus do not know the B_j. ¹⁰

4.2. Large Number of Potential Destinations

When the residential choice set is all neighborhoods or housing units in a city or other large area, the number of observations can be very large in a discrete choice model, making it computationally burdensome to compute choice probabilities for every individual-alternative observation. For example, a discrete choice model for 1,000 individuals (and their location decisions) in a metropolitan area of 2,000 Census tracts has 1,000*2,000 = 2,000,000 individual-alternative combinations (if each tract is in the choice set of every sampled individual). Such a large data set makes computation difficult. However, we can obtain consistent estimates of the discrete choice model by sampling from the individual-destination observations within each respondent (McFadden 1978; Ben-Akiva and Lerman 1985). This procedure can be accomplished without significant loss of information, if we use all information on actually chosen alternatives and a random subsample of unchosen alternatives. This is analogous to the procedure of subsampling the risk sets in survival analysis (e.g., Breslow et al. 1983) or subsampling controls in case-control designs (Jewell 2004). If we subsample unchosen alternatives, it is possible to estimate a modified version of the model shown in Equation (4), which is

p ij ( Z j , X i , C ( i ) ) = exp ( β Z j + γ Z j X i − ln q ij ) ∑ k ∈ c ( i ) exp ( β Z k + γ Z k X i − ln q ik ) ,

where q _ij denotes the known probability of sampling thejth destination for the ith respondent. We sample according to the following rules:

For example, if we sample the unchosen alternatives with probability 0.05, this procedure yields a sample of 1,000 + (1,999 × 1,000) × 0.05 = 100,950, a more manageable number of alternative-individual observations. This model can be estimated using standard maximum likelihood approaches for the discrete choice model, subject to the constraint that the coefficient on q _ij is 1.0. In practice, there are no firm guidelines for selecting a value of q _ij . The value will depend on both the sample size and also the size of the choice set. However, the computational burden of estimating the choice model is linear in both the number of observations and the number of alternatives. Thus if we have sufficient observations, it is more fruitful to analyze a sample of many observations with a small number of sampled alternatives rather than fewer observations with a large number of alternatives (Ben-Akiva and Lerman 1985:263). In practice, we can do sensitivity analyses to determine how alternative subsampling probabilities affect the estimated coefficients and standard errors. For example, we can vary the subsampling fraction and pick the smallest fraction that does not result in marked loss of precision of estimates.

4.3. Choice-based Sampling

Many surveys employ a form of stratified sampling that overrepresents some kinds of neighborhoods and underrepresents others. For example, surveys may oversample poor neighborhoods within a city or be drawn from schools or school districts with atypical minority or socioeconomic representation. Whereas this stratification scheme may be exogenous for some analytic purposes, it results in endogenous stratification for the study of neighborhood choice. Neighborhood stratified samples, therefore, are choice-based (Manski and Lerman 1977), in that the sampling procedure is confounded with the residential choices of the respondents. Without correction for sample design, estimates of parameters in discrete choice models are not, in general, consistent. If choice-based sampling probabilities are known, however, we can obtain consistent estimates of the model parameters using sampling weights. Manski and Lerman (1977) introduce an estimator in which each observed residential choice is weighted by its representation in the population as a whole. We define a function for each respondent,

W T i = V i H i ,

where V_i denotes the population shares and H _i denotes the sample shares for that respondent’s type. These weights enter the likelihood function for the model as

L = Π i = 1 N Π j = 1 J WT i × y ij exp ( β Z j + γ Z j X i ) ∑ k ∈ C ( i ) exp ( β Z k + γ Z k X i ) .

In practice, the correction weights for choice-based sampling can be estimated using the “importance weights” option in statistical estimation packages. For example, consider a sample of households where the proportion of respondents in high-poverty neighborhoods (≥30% of households below the poverty line) and low-poverty neighborhoods (<30% of households below the poverty line) are each 0.5, whereas the population proportions of households in high- and low-poverty neighborhoods are 0.3 and 0.7, respectively. In this case, the Manski-Lerman weights are 0.3/0.5 for respondents in high poverty tracts and 0.7/0.5 for respondents in low poverty tracts.

4.4. Nuances of Behavior

6.1. Housing Markets and Housing Prices

Although housing prices affect choice behavior, the estimated effects of prices may be contaminated by factors omitted from the model that affect neighborhood desirability and thus also affect demand for housing in an area and housing prices. Estimating discrete choice models that include housing costs without taking into account this problem of unmeasured sources of desirability will result in inconsistent parameter estimates. In linear models, a possible solution is to use instrumental variables to eliminate correlation between the error term and covariates. However, discrete choice models are more complicated because of the nonlinearity of the model and possible interactions between the characteristics of individuals and the characteristics of their potential choices. To address these problems, Berry (1994) and Berry, Levinsohn, and Pakes (1995) estimate a series of alternative-specific constants that capture average demand for different alternatives (based on both observed and unobserved characteristics) and incorporate them into a conditional logit or mixed logit model. When applied to neighborhood choice data, the alternative-specific constants absorb the unobserved component of neighborhood desirability. This removes the simultaneity problem that arises out of correlation between prices and unobserved features of neighborhoods in models of individual choice.

This approach decomposes unobserved determinants of neighborhood choice into (1) the average utility that individuals derive from unobserved neighborhood characteristics ( ξ j ) and (2) random individual deviations in the utility (ε _ij ). The utility function can be written

U ij = β Z j + γ Z j X i − α p j + ε ij + ξ j ,

where p_j denotes the average house price in the jth neighborhood. The negative coefficient indicates that neighborhood utility varies inversely with housing prices, all else being equal. The endogeneity problem is that prices depend on both observed and unobserved attributes of neighborhoods that affect desirability and thus demand. In other words, prices are a function of ξ _j . The solution is to introduce a constant for each neighborhood that captures its average utility (based on both observed and unobserved characteristics). This moves ξ _j out of the error term and into this alternative specific constant. Rearranging terms in (18), we have

U ij = [ β Z j − α p j + ξ j ] + γ Z j X i + ε ij ,

where the term in brackets does not vary over individuals. If we denote the alternative specific constants as δ j = β Z j − α p j + ξ j then

U ij = δ j + γ Z j X i + ε ij .

This choice model no longer has an endogeneity problem because ξ _j are subsumed into the alternative specific constants, which can be estimated along with the other parameters of the model. (We present this solution for the standard conditional logit model, but this strategy can also be applied to other models, including the mixed logit model). This model provides estimates of the alternative specific coefficient and the remaining parameters for choice behavior. However, the parameters associated with the utility for a given neighborhood that is common to all individuals remain subsumed in the δ _j . Fortunately, because these parameters enter the definition of the alternative specific constants linearly, they can be treated as outcomes in a regression model where the dependent variable is the alternative specific constant and the explanatory variables are characteristics of the neighborhood, including price. Here ξ _j is endogenous, but there are well-developed IV procedures for handling endogeneity in a linear model. The practical problem with this approach is that when the number of alternatives is large it is not feasible to estimate the alternative specific constants. Berry et al. (1995) provide an algorithm for estimating these parameters when there is a large number of alternatives.

Bayer and colleagues (Bayer and McMillan, 2008; Bayer, McMillan, and Rueben 2004) use this method in their analyses of residential choice and segregation dynamics. To obtain consistent estimates of the relationship between housing costs and mobility behavior, they divide their discrete choice utility function into a house-specific fixed effect, δ _j , and individual-specific interaction component, λ _ij , such that U ij = δ j + λ ij + ε ij . They estimate model parameters using an iterative two-step procedure. In step 1, they estimate the parameters in λ _ij and the average utilities δ _j using a discrete choice model. In step 2, they instrument for prices to recover the parameters in δ _j . They then use a measure of the relative scarcity of a given housing unit or neighborhood in the housing market as the instrument. Neighborhoods that are unique or occur less frequently—for example, a perfectly racially mixed area that contains new housing stock—command higher prices assuming there is some demand.

7.1. Stated Residential Preferences in MCSUI Data

We illustrate the analysis of stated preference data using the MCSUI data for Los Angeles. For illustrative purposes, we analyze only the “ranked attractiveness” and “would move in” data. The ranked-attractiveness data were collected only for nonwhite respondents. Table 2 shows the percentage of neighborhoods that were ranked first or second by black, Asian, and Hispanic respondents who were asked about neighbors of different race-ethnicities. Among black respondents asked about white, Asian, or Hispanic neighbors, the most attractive neighborhoods were those with a minority of other-group neighbors. However, a nontrivial proportion of black respondents identified the entirely other-group neighborhood (e.g., 100% white) as the most attractive neighborhood. Asian respondents were also most likely to rank neighborhoods with a minority of other-group neighbors as most attractive, although they found Hispanic and black neighbors less attractive than white neighbors. Similarly, Hispanic respondents found white neighbors more attractive than black or Asian neighbors but were most likely to rank neighbors with a strong Hispanic presence most attractive.

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Table 2.

Percentage of Neighborhoods Ranked First or Second, by Respondent’s Ethnicity and Percentage Other in Neighborhood

	Other Group in Neighborhood
	Black	White	Asian	Hispanic
Black Respondents
Proportion Other-Group
0	—	22.1	25.3	30.6
29	—	30.5	33.8	32.9
50	—	18.0	13.6	14.2
86	—	18.3	16.8	14.8
100	—	11.1	10.5	7.4
N=	—	356	374	380
Asian Respondents
Proportion Other-Group
0	47.0	14.8	—	42.0
29	43.6	31.6	—	38.6
50	7.0	15.8	—	11.3
86	1.9	23.6	—	6.7
100	0.6	14.3	—	1.5
N=	356	343	—	344
Hispanic Respondents
Proportion Other-Group
0	41.9	21.1	32.6	—
29	41.5	29.1	35.2	—
50	7.7	17.4	12.9	—
86	5.8	18.6	11.9	—
100	3.1	14.0	7.4	—
N=	307	341	338

Table 3 shows the percentage of white, black, Hispanic, and Asian respondents willing to move into a neighborhood based on its neighborhood proportion other (where the other-group may be white, black, Asian, or Hispanic). The first column of the table, which shows how white, Asian, and Hispanic respondents evaluate black neighbors, indicates that all groups avoid majority black neighborhoods. These descriptive tables show the distribution of responses over categories of neighborhood proportion other, but they do not provide a succinct way of showing the relationship between neighborhood preferences and neighborhood characteristics.

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Table 3.

Percentage Willing to Move into Neighborhood, by Respondent’s Ethnicity and Percentage Other in Neighborhood

	Other Group in Neighborhood
	Black	White	Asian	Hispanic
White Respondents
Proportion Other-Group
0	82.4	—	89.6	86.5
29	78.9	—	88.0	87.5
50	72.4	—	85.7	82.3
86	52.1	—	77.2	68.1
100	37.9	—	64.2	54.9
N=	261	—	307	288
Black Respondents
Proportion Other-Group
0	—	77.4	81.7	81.6
29	—	99.2	98.9	99.2
50	—	98.9	98.4	98.7
86	—	87.6	88.8	88.2
100	—	37.9	42.6	38.8
N=	—	354	376	374
Asian Respondents
Proportion Other-Group
0	97.5	85.3	—	95.3
29	96.3	100.0	—	99.2
50	88.1	100.0	—	98.7
86	74.3	97.70	—	88.2
100	23.7	72.1	—	38.8
N=	354	341	—	374
Hispanic Respondents
Proportion Other-Group
0	91.2	76.3	85.2	—
29	97.1	98.0	97.9	—
50	87.0	98.0	95.9	—
86	66.8	88.3	82.0	—
100	21.8	53.4	44.4	—
N=	307	341	338

7.1.1. Models

We analyze the “ranked attractiveness” data by treating the five responses (one for each vignette neighborhood) as a full ranking of the alternatives. In contrast, we treat the five responses to the “would you move in/out” question as a partial ranking of the alternative vignette neighborhoods, and use these rankings to estimate rank-ordered logit models with ties. In Table 1 each respondent has five lines of data, one for each neighborhood ethnic composition vignette and the respondent’s rank of the vignette. The vignette rank is the dependent variable and is modeled as a function of the percent other-group in the neighborhood. ¹¹ Separate parameters are estimated for each combination of a respondent’s own race and the race of the other group in the vignette neighborhood. The nonlinear continuous model adequately describes residential preferences for these simple data. The coefficients from these models are shown in Table 4.

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Table 4.

Coefficients for Neighborhood Proportion Other Group on Residential Preferences, Neighborhood Attractiveness and Would Move In Questions, Nonlinear Continuous Functions

Preferences of Blacks
	β	SE(β)	|Z(β)|	β	SE(β)	|Z(β)|
	Attractiveness Rankings			Would Move In
neigh	1.202	0.190	6.33	2.412	0.400	6.02
neigh2	−3.312	0.199	16.63	−2.962	0.452	6.55
I(Hispanic other) × neigh	−0.739	0.189	3.91	−0.100	0.447	0.22
I(Hispanic other) × neigh2	0.290	0.298	0.97	0.142	0.485	0.29
I(Asian other) × neigh	−0.528	0.313	1.69	−0.030	0.404	0.08
I(Asian other) × neigh2	0.492	0.251	1.96	−0.011	0.459	0.02
Log-likelihood		−11997			−13375
N		15298			15556
Preferences of Hispanics
	β	SE(β)	|Z(β)|	β	SE(β)	|Z(β)|
	Attractiveness Rankings			Would Move In
neigh	0.891	0.329	2.71	2.550	0.460	5.54
neigh2	−2.211	0.538	4.11	−2.752	0.358	7.68
I(black other) × neigh	−1.420	1.219	1.16	−0.704	0.151	4.66
I(black other) × neigh2	−1.268	0.249	5.16	−0.169	0.295	0.57
I(Asian other) × neigh	−0.738	0.572	1.29	−0.577	0.122	4.72
I(Asian other) × neigh2	−0.377	0.239	1.57	0.257	0.143
Log-likelihood		−6230			−6949
N		8197			8282
Preferences of Asians
	β	SE(β)	|Z(β)|	β	SE(β)	|Z(β)|
	Attractiveness Rankings			Would Move In
neigh	0.745	0.146	5.11	1.426	0.069	20.62
neigh2	−2.123	0.171	12.40	−1.543	0.070	22.09
I(Hispanic other) × neigh	−0.373	0.238	1.57	−0.103	0.056	1.85
I(Hispanic other) × neigh2	−3.810	0.255	14.95	−0.722	0.030	24.03
I(black other) × neigh	−1.306	0.137	9.52	−0.077	0.072	1.08
I(black other) × neigh2	−1.306	0.838	7.30	−1.047	0.078	13.49
Log-likelihood		−2999			−4555
N		5285			5355
Preferences of Whites
	β	SE(β)	|Z(β)|	P	SE(β)	|Z(β)|
	Attractiveness Rankings			Would Move In
neigh				−0.250	0.226	1.10
neigh2				−1.800	0.159	12.05
I(Asian other) × neigh				0.270	0.150	1.80
I(Asian other) × neigh2				−0.127	0.549	0.23
I(black other) × neigh				−1.000	0.388	2.57
I(black other) × neigh2				−0.265	0.297	0.89
Log-likelihood					−10285
N					14414

The predicted probabilities from the models for two of the ethnic groups, blacks and Hispanics, are presented in Figures 2 and 3. Figure 2(a) shows the probability that black respondents rank a vignette neighborhood most attractive. Separate panels are shown for black-white, black-Hispanic, and black-Asian neighborhoods. Black respondents tend to rank as most attractive those neighborhoods where their own ethnic group is heavily represented most. However, when asked which neighborhoods they would be willing to move into, blacks display a strong preference for integrated neighborhoods. Blacks are also slightly more partial to white neighbors than Hispanic or black neighbors; they respond to all three groups in a similar way for both the neighborhood attractiveness and “would move in” questions. Figure 3 shows the corresponding response profiles for Hispanics. Like blacks, Hispanics tend to find neighborhoods where their own group is heavily represented more attractive. However, unlike blacks, Hispanics tend to respond to mixed neighborhoods differently depending on the ethnicity of the other group. Hispanics find black neighbors least attractive. Hispanics are most likely to move into diverse neighborhoods.

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Figure 2.

Predicted probabilities for black respondents, nonparametric (dummy variable) and nonlinear continuous models.

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Figure 3.

Predicted probabilities for Hispanics, nonparametric (dummy variable) and nonlinear continuous models.

7.1.2. Unobserved Heterogeneity

Within race-ethnic groups, individuals vary in their residential preferences and their expressed tolerance of other groups. To allow for unobserved heterogeneity within race-ethnic groups, we estimate a set of latent class models allowing for a distribution of responses to neighborhood composition within each ethnic group. This is a specific instance of the mixed logit model discussed above, where W ij = Z j and the mixing distribution f ( μ ) is discrete, with μ taking a finite set of values μ m ∈ { μ 1 , … , μ M } each with probability π _m . Here the mixed logit becomes a latent class model where subscript m denotes a particular class. The choice probability is then

p ij = ∑ m = 1 M π m exp ( β Z j + μ m Z j ) ∑ k ∈ C ( i ) exp ( β Z k + μ m Z k ) .

In our example below, we use the ranked-attractiveness data to estimate separate models by respondents’ race and by the race of their vignette neighbors. We estimate a nonparametric model with dummy variables for each vignette neighborhood (omitted category is the 100% own-group neighborhood). Here Z _j is a set of dummy variables that identify vignette neighborhoods, so that Z ¯ j = { Z 0 , Z 29 , Z 50 , Z 86 } and μ _m is the vector μ m = { μ m 0 , μ m 29 , μ m 50 , μ m 86 } . The utility for a member of the mth latent class is U im = ∑ j β j Z j + μ m j Z j , where j ∈ { 0 , 29 , 50 , 86 } and the estimated effect of each individual Z _j for group m = β j Z j + μ m j Z j . Separate coefficients are estimated for each own-race/other-race combination. ¹²

The results from estimating these models are shown for blacks and Hispanics in Figures 4 and 5, respectively. There is a clear pattern of response. For most respondents, the attractiveness of the neighborhood declines with the proportional representation of one’s own race/ethnic group. However, among Hispanic and black respondents who were asked about white neighbors, roughly one quarter indicate that the most attractive neighborhood is the one that is 100% white. Similarly, among blacks and Hispanics who were asked about living among Asians, 19% of Hispanics and 21% of blacks in the sample identify the all-Asian neighborhood as most attractive. These results are consistent with those reported by other analysts of the same data (e.g., Charles 2000).

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Figure 4.

Predicted probabilities for blacks, unmeasured heterogeneity models, 2 groups

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Figure 5.

Predicted probabilities for Hispanics, unmeasured heterogeneity models.

7.2. Actual Mobility Histories in the LA FANS Data

We illustrate how to analyze actual move data using the LA FANS Wave 1 data, which is a stratified sample of approximately 2,700 households in 65 Census tracts in Los Angeles County. The residential mobility history for each respondent was collected via an event history calendar for the 24 months preceding the survey date. Seventy percent of LA FANS respondents did not move during the two-year period prior to the interview, whereas 20% moved exactly once. Previous addresses in Los Angeles County are geolinked to the correspondent Census tract. However, we omit the small percentage (6.5%) of moves that occurred outside of Los Angeles County. We measure mobility in terms of annual moves, and observe up to two moves per respondent. Figure 6 shows one hypothetical mobility history for an LA FANS respondent. Because we examine annual mobility, multiple moves that occur within a single year are counted as a single move. Table 5 summarizes the information available for the analysis of residential mobility using the LA FANS data. ¹³ The 2,332 respondents provide information on 4,508 annual residential mobility decisions. ¹⁴ As indicated by the comparison with the 2000 Census data for Los Angeles County, our data overrepresent Hispanics and underrepresent non-Hispanic whites and Asians. Despite the relatively large number of mobility decisions faced by LA FANS respondents, they report only 412 annual between-tract moves during the two-year mobility window, and 105 within-tract moves. For the purposes of this analysis, we consider moves to occur only if a respondent changes Census tracts during the annual mobility period.

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Figure 6.

Example of one mobility history from the LA FANS.

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Table 5.

Summary of Observations in LA FANS and Race-Ethnic and Income Composition of L.A. County

Census Tracts (1990 Census): 1627
Respondents in LA FANS Data: 2332
Mobility Decisions	Total	White	Black	Hispanic	Asian
Year 1	2,178	600	227	1,186	162
Year 2	2,330	630	239	1,279	179
Total	4,508	1,230	466	2,465	341
Race-Ethnic Composition
LA FANS	100.00	27.29	10.38	54.77	7.56
2000 Census	100.00	31.10	10.90	44.60	13.10
Moves Between Tracts
Year 1	174	46	30	89	9
Year 2	238	43	37	147	11
Total	412	89	67	236	20
Person-Year-Options (Total)
Year 1	3,538,825	976,300	369,329	1,929,622	263,574
Year 2	3,789,289	1,026,637	388,853	2,082,566	291,233
Total	7,328,114	2,002,937	758,182	4,012,188	554,807
Person-Year-Options (Sampling the Choice Set)
Year 1	37,149	10,246	3,889	20,251	2,763
Year 2	39,797	10,753	4,100	21,890	3,054
Total	76,946	20,999	7,989	42,141	5,817

7.2.1. Choice-based Sampling

The LA FANS is a stratified sample that overrepresents neighborhoods where at least 40 percent of households have incomes below the poverty line. For the purpose of estimating models of neighborhood choice, LA FANS is a choice-based sample. Our models include Manski-Lerman weights (see Equation 16) to correct for the differential representation neighborhoods in the data. A further complication is that the data come from retrospective mobility histories. Thus, whereas LA FANS is a choice-based sample at the time of the survey, prior to that respondents could live anywhere conditional on living in one of the sampled tracts when the data were collected. Thus the sample is purely choice-based at the time of the survey (Year 2, as shown in Figure 6), but influenced in a complex way by the choice-based sample in the periods prior to the survey date. Thus, we create two sets of Manski-Lerman weights: one using the distribution of choices at the time the LA FANS sample was drawn (in Year 2 of the mobility window), and one using the distribution of choices one year prior (Year 1 of the mobility window).

Table 6 illustrates the construction of Manski-Lerman weights in the LA FANS. The first column (H _i ) shows the distribution of respondents across the sampling stratum in each of the two years, whereas the second column (W _i ) shows the distribution of the population across sampling stratum. The LA FANS overrepresents high-poverty neighborhoods in both years. The chosen neighborhoods of respondents were 28% high-poverty in Year 1 and 30% high-poverty in Year 2 (when the data were collected). In contrast, only 9% of Los Angeles County neighborhoods were high-poverty during this period. The sample distribution more accurately represents the population one year prior to the survey date because individuals could, in principle, live in any Los Angeles neighborhood during this period rather than only in one of the 65 sampled neighborhoods. The Manski-Lerman weights, which are the ratio of the population fractions to the sampled fractions in each stratum, are shown in column 3. The weights correct for over- and underrepresentativeness of sampled neighborhoods. The weights enter our discrete choice models using the “importance weights” option in Stata.

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Table 6.

Adjustment for Choice-based Sample

Sampling Stratum	Sample Fraction (Hi)	Population Fraction (Wi)	Manski-Lerman Weight
Year 1
40+% poverty	0.281	0.090	0.320
20–39% poverty	0.319	0.301	0.944
0–19% poverty	0.401	0.600	1.495
Year 2
40+% poverty	0.296	0.090	0.304
20–39% poverty	0.307	0.301	0.980
0–19% poverty	0.397	0.600	1.510

7.2.4 Models of Residential Choice

We estimate conditional logit models that incorporate the effects of individuals’ personal characteristics and the characteristics of neighborhoods to which they might move, assuming that the choice set of each individual is all Census tracts in Los Angeles County. We allow for the possibility that respondents evaluate their current location differently from other potential destinations, by including a dummy variable D_ij, which equals 1 when destination j is the neighborhood currently occupied by respondent i, and 0 otherwise. The model, which can be written as

p ij ( Z j , X i ) = exp ( β Z j + γ Z j X i + υ D ij + ϕ Z j D ij − ln q ij + ln M j ) ∑ k ∈ C ( i ) exp ( β Z k + γ Z k X i + υ D ik + ϕ Z k D ik − ln q ik + ln M k ) ,

incorporates terms for sampling the choice set, − ln q ij , for the number of households in a Census tract, M _j , for the “cost of moving” from one’s current location, and for the possibility that respondents evaluate their own neighborhood’s quality differently than they evaluate others. The model can be used to explore a number of possible behavioral aspects of residential choice. For example, an interaction between neighborhood proportion black and neighborhood proportion Hispanic could represent the idea that Hispanics provide a “buffer” between blacks and whites. Table 7 presents coefficient estimates for a somewhat simpler specification in which each ethnic group responds uniquely to its own group and individuals evaluate their own neighborhoods differently from other potential destinations. The marginal probabilities from the full model (1.3) are shown in Figure 7.

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Table 7.

Effects of Respondent and Tract Characteristics on Residential Choice, LA FANS data

Variable	Beta	|z(B)|	Beta	|z(B)|	Beta	|z(B)|
	Model 1.1 (Race Effects Only)		Model 1.2 (Mover-Stayer Only)		Model 1.3 (Race and Mover-Stayer Effects)
D ij			17.738	12.530	16.772	11.62
%black	0.469	0.62			−0.154	0.11
%black2	−20.396	11.75			−13.215	5.19
black × %black	13.607	9.76			3.301	1.29
black × %black2	−9.986	3.48			−0.250	0.06
%Hispanic	−14.224	18.97			−12.631	7.67
%Hispanic2	5.434	9.34			6.071	4.83
Hispanic × %Hispanic	5.950	8.90			4.416	3.07
Hispanic × %Hispanic2	−1.587	2.65			−1.994	1.35
%Asian	−4.338	6.00			−2.343	1.42
%Asian2	−7.526	7.65			−9.022	3.54
Asian × %Asian	17.529	11.62			3.986	1.28
Asian × %Asian2	−24.803	9.05			−3.267	0.52
%white2	−10.323	19.16			−7.755	7.63
white × %white2	4.159	17.17			2.050	4.58
ln(#Number of	1.172	30.62	1.021	9.080	0.952	8.29
households in tract)
Dij × ln(# Number of			−0.920	5.720	−0.834	5.09
households in tract)
N	76746		76746		76746
Log-likelihood	−30238		−4031		−3873

Note: Models include Manski-Lerman weights, and the offset term (q_ij ) for sampling the choice set

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Figure 7.

Predicted probabilities for whites, blacks, Asians, and Hispanics, LA FANS movers, by ethnic proportions

8.1. Interactive Markov Models

Markov models link a set of individual- or group-specific residential mobility probabilities to expected patterns of neighborhood turnover. A Markov model has a finite set of J states, S = { s 1 , s 2 , … , s j } . The states can be specific neighborhoods (for example, Census tracts in a city) or neighborhood types (for example, poor versus nonpoor neighborhoods). The expected distribution of the population across the J states at time t is

m [ t ] = [ m 1 1 ( t ) , … , m j 1 ( t ) , m 1 2 ( t ) , … , m j 2 ( t ) , … , m 1 G ( t ) , … , m j G ( t ) ] ,

where superscript g = 1 , 2 , … , G indexes group membership (e.g., race-ethnic groups). We also specify a GJ by GJ matrix P of conditional probabilities that a member of group g moves to state j at time t + 1 conditional on being in state i at time t. Markov models assume that the distribution of the population at time t + 1 depends only on characteristics and locations of the population at time t (and no prior time periods). The population distribution at time t + 1 is then

m [ t + 1 ] = Pm [ t ] .

This is equivalent to the operation of summing over transition probabilities within destinations

m [ t + 1 ] = ∑ j ∑ g m [ t ] gj × P t gj

where m [ t ] gj denotes the size of population group g in state j. Markov models usually assume time-invariant probabilities (P) of moving between states. However, if individuals both react to and transform features of their neighborhoods through their mobility behavior, then their behavior follows an interactive Markov model (IM) (Conlisk 1976), where the elements of P depend on the population distribution at time t:

m [ t + 1 ] = P ( m [ t ] ) m [ t ] .

Here m[t] represents the distribution of blacks and whites across neighborhoods, and the probability of moving into a given neighborhood is a function of its ethnic composition. In this model, preferences for neighborhood characteristics are fixed, but the attractiveness of specific neighborhoods changes as a result of their changing characteristics.

To illustrate the interactive Markov model, we consider a simple city with two neighborhoods and a population of 10 blacks and 10 whites. At time 0 the population is completely segregated; all blacks are in one state, and all whites are in the other. Next, we compute the population trajectory for whites and blacks using their respective preference functions. For example, if people evaluate their neighborhoods according to a simplified version of Equation (4), where the probability that the ith person selects the jth neighborhood is e Z j ∑ k e Z k where Z _j is neighborhood proportion own-group, then

m [ 1 ] = P 0 × m [ 0 ] = [ e m [ 0 ] [ 1 ] e m [ 0 ] [ 1 ] + e m [ 0 ] [ 2 ] e m [ 0 ] [ 1 ] e m [ 0 ] [ 1 ] + e m [ 0 ] [ 2 ] 0 0 e m [ 0 ] [ 2 ] e m [ 0 ] [ 1 ] + e m [ 0 ] [ 2 ] e m [ 0 ] [ 2 ] e m [ 0 ] [ 1 ] + e m [ 0 ] [ 2 ] 0 0 0 0 e m [ 0 ] [ 3 ] e m [ 0 ] [ 1 ] + e m [ 0 ] [ 4 ] e m [ 0 ] [ 3 ] e m [ 0 ] [ 1 ] + e m [ 0 ] [ 4 ] 0 0 e m [ 0 ] [ 3 ] e m [ 0 ] [ 1 ] + e m [ 0 ] [ 4 ] e m [ 0 ] [ 3 ] e m [ 0 ] [ 1 ] + e m [ 0 ] [ 4 ] ] × [ 10 0 0 10 ] = [ e 1 e 1 + e 0 e 1 e 1 + e 0 0 0 e 0 e 1 + e 0 e 0 e 1 + e 0 0 0 0 0 e 0 e 1 + e 0 e 0 e 1 + e 0 0 0 e 1 e 1 + e 0 e 1 e 1 + e 0 ] × [ 10 0 0 10 ] = [ 7 . 31 2 . 69 2 . 69 7 . 31 ]

At step 2,

m [ 2 ] = p 1 × m [ 1 ] [ e 0 . 731 e 0 . 731 + e 0 . 269 e 0 . 731 e 0 . 731 + e 0 . 269 0 0 e 0 . 731 e 0 . 731 + e 0 . 269 e 0 . 731 e 0 . 731 + e 0 . 269 0 0 0 0 e 0 . 269 e 0 . 731 + e 0 . 269 e 0 . 269 e 0 . 731 + e 0 . 269 0 0 e 0 . 731 e 0 . 731 + e 0 . 269 e 0 . 731 e 0 . 731 + e 0 . 269 ] × [ 7 . 31 2 . 69 2 . 69 7 . 31 ] = [ 0 . 613 0 . 613 0 0 0 . 387 0 . 387 0 0 0 0 0 . 387 0 . 387 0 0 0 . 613 0 . 613 ] × [ 0 . 731 0 . 269 0 . 269 0 . 731 ] = [ 6 . 14 3 . 87 3 . 87 6 . 14 ]

The process can continue until the neighborhoods reach equilibrium—that is, where Π m [ t + 1 ] = P t ( m [ t ] ) and m [ ∞ ] = P ∞ = Π P t m [ t ] . Given an estimated discrete choice function that can generate the P _t , it is possible to compute the expected pattern of residential segregation under the mobility regime summarized in mobility matrices P _t using the standard measures of residential segregation (Mare and Bruch 2003). Tuljapurkar, Bruch, and Mare (2010) provide a mathematical analysis of Markov models for segregation and neighborhood change. In principle, an interactive Markov models for mobility between individual neighborhoods can be represented as a fixed rate Markov model of mobility between neighborhood types (e.g., Hermanns 2002).

8.2. General Equilibrium Models with Price Effects

Another strategy for studying neighborhood dynamics is using general equilibrium (GE) models with price effects. Bayer and colleagues (Bayer and McMillan 2008; Bayer, McMillan, and Rueben 2004) use GE models to examine the relationship between residential choice behavior and neighborhood outcomes. The analysis consists of two parts: (1) estimating a discrete choice model and (2) simulating the expected distribution of individuals in each neighborhood implied by the choice model. GE models assume that observed neighborhoods are in equilibrium, such that each individual had made an optimal choice given the choices of all other individuals. The models can be used to show how a new equilibrium distribution of neighborhoods results from some change in initial conditions or behavior (e.g., assuming that people are indifferent to the racial composition of their neighborhoods or assigning all ethnic groups equal income distributions). The first step is assuming or estimating a discrete choice model for the effects of housing prices, neighborhood race-ethnic composition, and other factors (as discussed in Section 6). Given this model, it is possible to simulate the impact of counterfactual conditions. For example, choice-model coefficients associated with neighborhood race-ethnic composition may be set to zero, to represent a city in which people make race-blind residential decisions and, using this modified choice model, it is possible to compute a new equilibrium distribution of neighborhoods.

In the first stage, predicted probabilities are computed representing the likelihood that an individual with a given demographic profile chooses a neighborhood of a given demographic composition. These probabilities are summed over neighborhoods to generate the demographic composition of neighborhoods in the next time period. Residential choice probabilities are recomputed to take into account changing neighborhoods, and the procedure repeats. More formally, the demographic composition of neighborhoods at time t + 1 is Z j t + 1 = ∑ i P ij , where P _ij is the probability that the ith individual chooses the jth neighborhood. The process continues until a new equilibrium is reached, where Z j t + 1 = Z j t ∀ j . As the composition of neighborhoods changes, their desirability, reflected in housing prices, changes as well. The establishment of a new equilibrium requires an update of housing prices so that the market clears. Market clearing prices are set such that, given the valuation of neighborhood characteristics by different types of individuals and a population, the expected number of people in each neighborhood matches the number of available dwellings. Housing prices are computed using an adaptation of an algorithm developed by Berry et al. (1995), that is,

p ^ j t + 1 = p ^ j t + In ( s j s ^ j ( p ^ j t ) ) ,

where s _j and s ^ j ( p ^ j t ) are the actual and expected number of people in the yth neighborhood and p _j is a measure of housing prices in the jth neighborhood. To summarize, the new equilibrium population distribution over neighborhoods is computed in the following steps: (1) compute residence probabilities associated with neighborhoods at time t; (2) sum over individuals within neighborhoods to get new values for Z j t ; (3) compute new market clearing prices; (4) repeat 1 through 3 until convergence.

8.3. Agent-based Models

Agent-based models are a third approach to linking individual mobility to neighborhood dynamics (Macy and Willer 2002; Bonabeau 2002). Such models are microsimulations in which hypothetical individuals make choices based on either assumed behavioral rules or a statistical model of behavior. Agent-based models explicitly represent the feedback between individuals’ behavior and aggregate processes (e.g., residential mobility and neighborhood change, mate preferences and marriage market dynamics, decisions to smoke or drink and high school norms around these behavior), and they can allow for detailed geography and individual heterogeneity. Schelling’s (1971, 2006) model of residential tipping is an example of an agent-based model of a social process. Related models have been used to study norms regarding age at first marriage (Todd, Billari, and Simao 2005), income inequality and racial residential segregation (Bruch 2011), and other phenomena.

Agent-based models contain a population of actors who are assigned behaviors appropriate to the substantive application. An agent-based model of residential mobility assumes rules about how agents evaluate the desirability of neighborhoods and decide when and where to move. These rules can be simple heuristics or a more complex model such as that provided by the coefficients of a discrete choice model. If agents’ behavior is grounded in a discrete choice model, they use the values of neighborhood characteristics in their simulated world (as well as their own attributes) in combination with the model parameters to generate transition probabilities for moving among neighborhoods. The agent translates these transition probabilities into a (multinomial) distribution for the probabilities of selecting each neighborhood and “samples” a neighborhood via a draw from this distribution. In practice, the neighborhood-specific probabilities are cumulated, and the agent picks a neighborhood by drawing a number between 0 and 1 and choosing the neighborhood associated with the interval that contains that number. Figure 8 illustrates this process. See Bruch and Mare (2006, 2009) for a more detailed description of how to incorporate discrete choice models of residential mobility into agent-based models of neighborhood dynamics.

OPEN IN VIEWER

Figure 8.

Determining chosen destination for agent using discrete choice framework.

There are a number of software packages available for doing agent-based modeling, including Mason, Swarn, and Ascape. Netlogo (http://ccl.northwestern.edu/netlogo/) and Repast (http://repast.sourceforge.net/) are two of the most widely used options; both are free. Netlogo is a good option for researchers new to object-oriented programming. It is aimed at a less technical audience, has extensive documentation and user support, and contains a number of basic social science models in its model library. Repast requires some knowledge of Java, but can be used to run more complex models with larger populations. The software also allows for parallel processing. Both programs allow the user to import Geographic Information Systems (GIS) data to simulate mobility using realistic geography.

8.4. Comparing Approaches for Micro and Macro Linkage

Each of the three approaches takes a different tack to looking at segregation processes. Both Interactive Markov (IM) models and General Equilibrium (GE) models focus on the aggregate distributions of populations across neighborhoods, whether estimated in practice from aggregate or individual-level data. As they have been applied to residential choice studies, however, GE models are used for comparing equilibria under alternate assumptions, whereas IM models are used to examine the dynamics of residential mobility and neighborhood change (contrast Mare and Bruch 2003 using IM models with papers by Bayer and colleagues). Like interactive Markov models, agent-based models also focus on population dynamics, but they are built up from the actions of simulated individuals. However, there are two key differences between agent-based models and the GE and IM models: (1) Agent-based models have an explicit notion of “vacancies,” where agents can move into an area only if there is an available slot; and (2) individuals in agent-based models make realized—not probabilistic—decisions. These differences may lead to substantively different segregation dynamics (for the same population and behavioral model).