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Do Capitalization Rates Have a Welfare Interpretation? Hedonic Theory with an Application to School Quality

Nicolai V. Kuminoff ♦ Department of Economics Arizona State University [email protected]

and

Jaren C. Pope Department of Agricultural and Applied Economics Virginia Tech [email protected]

November, 2009

♦

We thank Roger von Haefen, Matt Kahn, H. Allen Klaiber, Ray Palmquist, Chris Parmeter, Jonah Rockoff, V. Kerry Smith, and seminar participants at ASSA for helpful comments on earlier drafts of this paper. Research support from the National Science Foundation and the Virginia Agricultural Experiment Station is gratefully acknowledged.

Do Capitalization Rates Have a Welfare Interpretation? Hedonic Theory with an Application to School Quality

ABSTRACT: This paper explains how the increasingly popular quasi-experimental approach to hedonic estimation relates to Rosen’s (1974) concept of market equilibrium. We demonstrate that the conventional approach to hedonic estimation and the quasi-experimental approach to estimation attempt to measure different phenomena. Comparisons between their estimates are generally invalid. Recent work in microeconometrics has sought to bridge the gap between structural and quasi-experimental methods by identifying situations where treatment effects can serve as “sufficient statistics” for welfare measurement (Chetty 2009). We extend this logic to the hedonic model by formalizing restrictions on the data that make it possible, in special cases, to interpret a quasi-experimental estimate for a capitalization rate as a measure of willingness to pay. Using data on property values and school quality in Detroit, Los Angeles, and Philadelphia, we find that these restrictions are often violated. This helps to explain the differences between our estimates of the capitalization rate for school quality improvements and our estimates of the marginal willingness to pay from a boundary discontinuity model.

KEY WORDS: capitalization, hedonic, quasi-experiment, policy, preferences, welfare

JEL CODES:

C33, Q21, R21, H41

ii

1. Introduction How can we measure the public’s willingness to pay for a public good? This problem has intrigued economists for four decades. 1 Recently, Chay and Greenstone (1998, 2005) proposed a new solution—using quasi-experiments to measure the rates at which public goods are capitalized into property values. Nesting their quasi-experimental design within Rosen’s (1974) hedonic model of market equilibrium allows the capitalization rate to be interpreted as a measure of willingness to pay. The appeal of combining a credible identification strategy with a policy relevant interpretation of the treatment effect has led to a flood of applications. However, this literature has yet to investigate one of the key steps in Chay and Greenstone’s logic. The ability to interpret a capitalization rate as a measure of willingness to pay rests on the maintained assumption that the hedonic price function is stationary. Because the price function is an equilibrium outcome generated by interactions between all of the buyers and sellers in a market, restrictions on its evolution have profound implications for preferences and technology. The goal of this paper is to define testable restrictions on preferences and on the design of quasi-experiments that allow us to interpret capitalization rates as welfare measures. We find that the preference restrictions needed to generate a stationary price function are extraordinary. In particular, the market demand for the public good must be perfectly elastic. If this restriction is not satisfied, a welfare interpretation of the capitalization rate requires a quasi-experiment where the shock to the public good is orthogonal to its baseline. We test these two restrictions using micro data on housing sales and public school quality in Detroit, Los Angeles, and Philadelphia. Violations of each restriction help to explain why our estimates of the willingness

1

Past proposals have included the median voter model (Bergstrom and Goodman 1973, Rubinfeld , Shapiro, and Roberts 1987) the property value capitalization model (Lind 1973, Starrett 1981), the hedonic model of housing market equilibrium (Scotchmer 1985, 1986, Bartik 1987), and equilibrium sorting models of neighborhood formation (Epple and Sieg 1999, Bayer, Ferreira, and McMillan 2007).

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to pay for school quality are quite different from the capitalization rates for recent improvements. To provide context for our analysis, section 2 begins with a brief history of theoretical results and empirical challenges in measuring the willingness to pay for a public good. Then the “conventional” and “quasi-experimental” hedonic models are each defined. Rosen’s (1974) conventional model describes market equilibrium at a single point in time. The quasiexperimental approach to estimating his model measures the rate at which unexpected shocks are capitalized into market prices over time. Thus, it tracks the movement between two different “conventional” equilibria. As a result, the two models generally measure different phenomena. It is the assumption of a stationary price function that creates a mapping between them. Section 3 uses Rosen’s description of equilibrium to examine the causes and consequences of temporal variation in the price function. It is shown that perfectly elastic demand curves will generate stationary price functions. When demand is less than perfectly elastic, the divergence between capitalization rates and marginal values depends on model primitives. A parameterization based on Tinbergen (1959) is used to demonstrate that capitalization rates may understate or overstate willingness to pay. While these findings may seem negative, we find reason for optimism. By changing the shape of the price function, an unexpected event can produce instruments that solve the endogeneity problem in estimating market demand curves (Epple 1987, Bartik 1987). Thus, we argue that quasi-experiments hold a previously unrecognized potential to help attain the original goal of hedonic estimation. In section 4 we move from theory to practice by deriving testable restrictions on the data that are sufficient for capitalization rates to identify the average household’s marginal willingness to pay (MWTP). Consider a large shock to a public good that serves as the basis for a quasi-experiment. One can identify MWTP in the pre-shock equilibrium if the implicit price of

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every characteristic is constant over the duration of the study. If this restriction does not hold, one can still identify MWTP in the post-shock equilibrium if the shock is uncorrelated with all other variables. If this restriction is also violated, we demonstrate the capitalization rate can fall outside the range defined by the true MWTP in the pre-shock and post-shock equilibria. Our approach in this section is consistent with Chetty’s (2009) proposal for bridging the gap between structural and quasi-experimental methods by identifying situations where treatment effects can serve as “sufficient statistics” for welfare measurement. Section 5 applies our framework to micro data on houses that sold in Detroit, Los Angeles, and Philadelphia school districts during the first and fifth years after the federal No Child Left Behind program was enacted (2003 and 2007). During this period, average test scores for math and reading proficiency increased between 15% and 24%. To obtain consistent estimates of the MWTP for school quality in each (year, metro area) we use a boundary discontinuity design based on Black (1999) to control for omitted neighborhood characteristics. Then we compare the results to first-differenced estimates of the rate at which the change in academic performance was capitalized into property values. Capitalization rates are not statistically different from MWTP for the area that comes closest to satisfying the data restrictions we derive (Philadelphia). For the area that is furthest from satisfying these restrictions (Detroit) capitalization rates and MWTP have opposite signs. Our findings add to three distinct literatures. First, our conceptual model helps to clarify the relationship between property value capitalization (Lind 1973, Starrett 1981) and hedonic equilibria (Scotchmer 1985, 1986, Bartik 1987) in the revealed preference literature on using housing market outcomes to evaluate public policy. Second, we provide a welfare theoretic foundation for interpreting the results from quasi-experiments in implicit markets for public

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goods (Davis 2004, Chay and Greenstone 2005, Greenstone and Gallagher 2008, Linden and Rockoff 2008, Pope 2008, Bim, Landry, and Meyer 2009, Horsch and Lewis 2009, Cellini, Ferreira, and Rothstein, forthcoming). Finally, our empirical results add to the literature on valuing school quality by providing the first evidence on the difference between the MWTP and the empirical capitalization rate (Kain and Quigley 1975, Rubinfeld, Shapiro, and Roberts 1987, Black 1999, Figlio and Lucas 2004, Bayer, Ferreira, and McMillan 2007).

2. A Brief History of the Quasi-Experimental Hedonic Model In his seminal 1956 paper, Tiebout hypothesized that freely mobile households will reveal their preferences for public goods through the location choices they make. His reasoning influenced the development of two revealed preference techniques: the capitalization model and the hedonic property value model. Hundreds of applications of these methods over the past 40 years have contributed much of what we currently know about the value of public goods, environmental services, and urban amenities. Capitalization studies use data before and after a market shock to measure its effect on housing prices. 2 The power of this technique is the ability to simultaneously measure a change in asset values and demonstrate that the change was caused by some event. Capitalization models are routinely used by expert witnesses in litigation over private property externalities (Simons 2006). They are also used to measure the market value of risk and uncertainty (Brookshire et al. 1985, Hallstrom and Smith 2005). A limitation of the technique is that it lacks a welfare interpretation. Lind (1973) and Starrett (1981) demonstrated that, under the type of sorting behavior Tiebout envisioned, market capitalization of a shock may understate or

2

The idea for using panel data to measure how changes in quality characteristics influence housing prices dates back at least to Bailey, Muth, and Nourse (1963). Economic applications begin with Palmquist (1982).

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overstate the change in household welfare. 3 In contrast, the hedonic property model based on Rosen (1974) offers a theoretically consistent approach to welfare measurement. The difficulty is econometric identification. Scotchmer (1985, 1986) proved that data from a single market are only sufficient to identify marginal values. To identify a demand curve, one must collect multi-market data on the characteristics of households and their houses, plus instrumental variables for housing characteristics (Bartik 1987, Epple 1987). Unfortunately, barriers to obtaining these data have stymied demand estimation. 4 The vast majority of empirical studies only aspire to recover marginal values. 5 Even the seemingly modest task of estimating marginal values is now believed to be plagued by omitted variable bias. Chay and Greenstone (1998, 2005) were the first to characterize this problem and propose a solution. They replaced the conventional hedonic estimator with an instrumental variables strategy that isolates how property values are affected by unexpected shocks to amenities. From a historical perspective on revealed preference methodology, what is remarkable about Chay and Greenstone’s analysis is that it bridges the capitalization and hedonic literatures. It integrates a quasi-experimental version of the identification strategy from the capitalization literature with the welfare interpretation of Rosen’s hedonic model.

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While Lind (1973) does not develop a formal utility theoretical framework, he proves that any welfare interpretation of capitalization requires there be zero consumer surplus. This effectively rules out preference-based sorting by heterogeneous agents, as Starrett (1981) later demonstrated. 4 An alternative strategy to identify demand is to provide additional information about consumer preferences. This information may consist of a parametric representation for the utility function (Epple and Sieg 1999; Bayer et al. 2007; Bishop and Timmins 2008), separability restrictions on preferences (Ekeland, Heckman and Nesheim 2004), or an assumption that consumers in different cities share a common distribution of unobserved tastes (Bartik 1987). 5 That said, the number of studies that aspire to recover marginal values is also vast. To give a rough sense of scale, there are more than 1600 citations of Rosen (1974) in the Social Science Citation Index and approximately 4000 reported by Google Scholar. Property value applications are one of (if not the) most frequent application. See Palmquist and Smith (2002) and Palmquist (2005) for reviews of the property value hedonic literature.

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To illustrate the basic idea, let the price of housing be expressed as p = p(g , h, ξ ) , where g is the public good of interest, h measures all other public goods and housing characteristics observed by the analyst, and ξ represents unobserved variables. It is standard practice to specify a linear-in-parameters price function such as

p1 = g1θ1 + h1η1 + ε (ξ1 ) ,

(1)

where the subscripts indicate the time period. The first order conditions from Rosen (1974) allow us to interpret θ1 as the marginal willingness to pay (MWTP) for the public good in period 1. However, θ1 is not identified if ξ1 is correlated with [g1 , h1 ] . Now suppose p, g, and h are also measured after an unexpected shock that serves as a valid quasi-experiment. First-differencing the data produces a new estimator, ∆p = ∆gφ + ∆hγ + ∆ε ,

(2)

where ∆z = z 2 − z1 for z = [ p, g , h, ε ] . Assuming omitted variables are purged by differencing the data, (2) provides a consistent estimator for φ . 6 Interpreted literally, φ is the rate at which the shock to g was capitalized into property values. Chay and Greenstone (1998, 2005) observe that the capitalization rate will equal MWTP if the shape of the price function is constant over time (i.e. φ = θ1 = θ 2 and γ = η1 = η 2 ). The ability to combine a credible identification strategy with a welfare theoretic interpretation of the capitalization rate has led to a flood of applications. Recently published studies have used quasi-experimental hedonic models to estimate the willingness to pay for changes in cancer risk (Davis 2004), air quality (Chay and Greenstone, 2005), hazardous waste (Greenstone and Gallagher 2008), crime (Linden and Rockoff 2008; Pope 2008), open space 6

This assumption is relaxed in Chay and Greenstone (2005) and Greenstone and Ghallager (2008). They exploit discontinuities in the structure of public policies to develop additional instruments for ∆g .

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(Bim, Landry, and Meyer 2009), invasive species (Horsch and Lewis 2009), and investment in education (Cellini, Ferreira, and Rothstein 2009). In all of these studies, the validity of welfare measures rests on the assumption of a stationary price function. The assumption has been made for areas ranging from a single county to the contiguous United States, for periods of up to 20 years. No theoretical justification or empirical evidence has been presented to support the assumption that the hedonic price function is constant over time.

3.

Hedonic Theory and the Capitalization of Market Shocks

This section uses a simple conceptual framework to examine the causes and consequences of temporal variation in the hedonic price function. After defining the primitives of the model and characterizing market equilibrium, we demonstrate that extraordinary restrictions on preferences are needed to guarantee the price function will be stationary. While a nonstationary price function confounds a welfare theoretic interpretation of the capitalization rate, it can also provide instruments to identify the demand for a public good. These results are explained and illustrated with examples. 3.1.

Demand, Supply, and Market Equilibrium

Price-taking households are assumed to be free to choose a home with any combination of housing characteristics (e.g. bedrooms, bathrooms, sqft) in the neighborhood that provides their desired levels of amenities (e.g. school quality, air quality, racial composition). 7 The utility maximization problem is

7

If households are unable to choose continuous quantities of every attribute, it is not possible to point-identify their MWTP for housing attributes from the hedonic price function. However, it is still possible to identify bounds on MWTP if one is willing to specify the utility function (Bajari and Benkard 2005; Kuminoff 2009).

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max U (g , X , b; α ) subject to y = b + P( g , X ) , g , X ,b

(3)

where X = [h, ξ ] . A household chooses housing characteristics, amenities, and the numeraire composite commodity (b) to maximize its utility, given its preferences ( α ), income ( y ), and the after-tax price of housing P( g , X ) . The first order conditions are ∂P( g , X ) ∂U ∂ g = ≡ D(g ; X , α , y ) , ∂g ∂U ∂b

(4a)

∂P (g , X ) ∂U ∂X = ≡ R( X ; g , α , y ) . ∂X ∂U ∂b

(4b)

The first equality in (4a) implies that each household will choose a neighborhood that provides a quantity of g at which their marginal willingness-to-pay for an additional unit exactly equals its marginal implicit price. Assuming the marginal utility of income is constant for each household, the second equality observes that as g varies the marginal rate of substitution defines its inverse demand curve, conditional on X. Equation (4b) defines analogous first order conditions for X. Producers in this market may include developers, contractors, and individuals selling their homes. Let C (g , M , X ; β ) denote a producer’s cost function, where M is the number of type-(g,X) homes they sell and β is a vector of parameters describing the producer. For a developer or contractor, the cost function will reflect the physical, labor, and regulatory costs of building a home. For a homeowner, the cost function will reflect their psychological attachment to the home as well as the cost of renovation. Variation in β captures differences in costs faced by different producers. Following Rosen (1974), we treat each producer as a price taker who is free to vary the number of units they sell as well as a subset of the characteristics of each unit.

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For notational convenience, g is assumed to be exogenously determined. 8 In this case, the profit maximization problem is max π = M ⋅ P(g , X ) − C (g , M , X ; β ) ,

(5)

X ,M

with the corresponding first order conditions P(g , X ) =

∂C ( g , M , X ; β ) , ∂M

∂P( g , X )  1  ∂C ( g , M , X ; β ) . =  ∂X ∂X M 

(6)

Producers choose M to set the offer price of the marginal home equal to its production costs, and they choose X to set the marginal per unit cost of each attribute equal to its implicit price. Equilibrium occurs when the first order conditions in (4) and (6) are simultaneously satisfied for all households and producers. This system of differential equations implicitly defines the equilibrium hedonic price function that clears the market (Rosen 1974). It will be useful to rewrite the price function to acknowledge its dependence on model primitives,

{

}

P(g , X ) ≡ P g , X [g , F ( y, α ), V (β )] , F ( y, α ), V (β ) .

(7)

Equilibrium levels of X are determined by all of the exogenous variables: g the amenity of interest, F ( y , α ) the joint distribution of household income and preferences, and V (β ) the distribution of producer characteristics. 9

3.2.

Restrictions Needed to Guarantee the Price Function is Stationary

The shape of the equilibrium price function in (7) generally covaries with g, F ( y , α ) , and V (β ) . To obtain stationarity, we have two options: hold g, F ( y , α ) , and V (β ) fixed or restrict the

8

The main results of this section are not altered by allowing firms to choose g or by restricting their ability to choose X. The key restriction needed to relate our model to the new empirical capitalization literature is that g is at least partly determined by forces that are exogenous to our model. 9 M drops out of the expression for X in (6) because it is a function of model primitives.

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shape of P{ ⋅ }. Both options present difficulties. The first option contradicts the premise for the capitalization model. To identify the capitalization rate, changes in g must be observed. Additionally, the distribution of real income in the United States is known to change over time (for example, see Gottschalk 1997). The difficulty with the second option is that it requires strong restrictions on preferences. To ensure the equilibrium price function is invariant to changes in g, F ( y , α ) , and V (β ) , its Hessian must be restricted to be zero. The implication for preferences can be seen from the first order conditions (4). The gradient of the price function provides a mapping to the distribution of marginal values in the consumer population. Thus, a constant gradient implies that demand curves are perfectly elastic. If demand is downward sloping, a change in the level of a characteristic will also change its marginal implicit price. Another way to see that the “constant gradient” restriction is extraordinary is to observe that it is a special case of a linear gradient, which Ekeland, Heckman, and Nesheim (2004) prove is a nongeneric property of hedonic equilibrium.

3.3.

Implications for Welfare Measurement

Figure 1 provides a stylized picture of how the equilibrium price function reveals the distribution of marginal values for g at a single point in time. It relates the marginal price function for g to demand curves for two households and supply curves for two producers. 10 Evaluating

∂P (g , X ) ∂g at a household’s chosen level of g will return their willingness to pay for a marginal increase (MWTP). Combining this information with g identifies exactly one point on

10

Because g is assumed to be exogenous, producers do not have the ability to vary the level of g for an individual home. The upward sloping supply curves can be interpreted as developers selling several homes at different locations.

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their demand curve. The rest of the curve is not identified without further restrictions. It can be seen from the figure that any number of demand curves could pass through the equilibrium points defined by (MWTP1 , g1 ) and (MWTP2 , g 2 ) . Now consider the new equilibrium that follows an unexpected shock to g. The capitalization rate is defined by the difference in the pre and post-shock price functions divided by the change in g,

} { g , X [g , F ( y,α ),V (β )] , F ( y,α ),V (β )} ,

{

P1 g1 , X 1 [g1 , F1 ( y, α ), V1 (β )] , F1 ( y, α ), V1 (β ) − P0

0

0

0

0

0

0

0

g1 − g 0

(8)

where 0 and 1 subscripts denote pre and post-shock equilibria. This difference quotient reduces to the derivative in (4a) in two situations. The first situation occurs when producer and household characteristics are time-constant and the change in g is marginal. From the definition of a derivative, one can see that (8) approaches (4a) as g1 − g 0 approaches zero. The second situation occurs where ∂P ∂g = c , a constant. For example, plugging (1) into (8) returns the MWTP if ∂P ∂g = θ1 = θ 2 = c . More generally, when ∆g is not marginal and the price function is not stationary, (8) does not equal (4a). The direction and magnitude of the bias from misinterpreting the capitalization rate as a measure of MWTP will depend on the shape of the price function and the nature of the shock. This will be illustrated with an example shortly. First, however, it is important to clarify that a nonstationary price function is not a barrier to welfare measurement in general. Changes in the price function can actually help to identify the demand for a public good. Equation (9) presents a nonlinear approximation to the demand function for g with an additive error:

D (g ; X , α , y ) = f (β ; g , X , w, y ) + ν ,

(9)

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where β is a parameter vector, w is a vector of observable demographic characteristics that are correlated with tastes, and ν is an unobserved taste component: w ∪ ν = α . 11 The endogeneity problem characterized by Epple (1987) and Bartik (1987) is that a consumer’s unobserved tastes will be correlated with their choices for g and X. With this in mind, consider an unexpected shock to g, X, F ( y , α ) , or V (β ) that changes the shape of the price function. A change in the price function will affect choices for g and X through the budget constraint (3). Therefore, as long as re-equilibration of the market does not change the distribution of unobserved tastes, a dummy variable for the post shock period provides a valid instrument. 12

3.4.

A Parametric Example

A parametric example may help to clarify ideas. Suppose the housing stock is fixed, utility is quadratic, and heterogeneous preferences and housing characteristics are normally distributed. These assumptions conveniently provide a closed-form expression for the equilibrium price function. 13 Specifically, let the utility from a home defined by attributes k = [g , X ] be parameterized as ′Ω U = −(k − α ) (k − α ) + b , 2

(10)

where Ω is a positive definite diagonal scaling matrix. When k and α are both normally 11

Unobserved tastes area assumed to be separable for notational simplicity. All arguments for identification of the demand parameters can be extended to the case of nonseparable errors. See Epple (1987) for analysis of identifying a hedonic demand system. 12 The importance of this observation is underscored by previous studies. Epple (1987) concludes that “Extending hedonic analysis to dynamic environments in which information and uncertainty play a role is an important problem for future research” (p.79). Bartik (1987) concludes that “The practical problem for empirical hedonic research is finding instruments whose exogeneity can be defended with some plausibility” (p.87). Chay and Greenstone (2005) conclude that “Future research should integrate the credible estimation of the hedonic price schedule with strategies to recover MWTP functions” (p.418). 13 Tinbergen (1959) first used this example to illustrate the properties of equilibria in labor markets with heterogeneous workers. Epple (1987) and Ekeland, Heckman, and Nesheim (2004) use the model to illustrate other features of hedonic equilibria. See Sattinger (1980) for closed form expressions for the hedonic price function under alternative assumptions about market primitives.

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distributed such that k ~ N (µ k , Σ k ) and α ~ N (µα , Σα ) , the price function can be expressed as P(k ) = Ψ ′k + k ′

Γ k, 2

where Ψ = Ω(µα − Σα0.5 Σ 0k.5 µ k ) and Γ = −Ω(I − Σα0.5 Σ 0k.5 ) .

(11)

Notice the reduced-form parameters of the price function (Ψ, Γ ) are functions of the structural parameters that describe the distributions of household preferences (µα , Σα ) and housing characteristics (µ k , Σ k ) . Consider a shock to g that alters µ k and Σ k . Before the shock, MWTP is Ψ0 + Γ0 k . After the shock, MWTP is Ψ1 + Γ1k . It follows from (11) that, in general, Ψ0 ≠ Ψ1 and Γ0 ≠ Γ1 . The rate at which the shock is capitalized into property values is P1 { ⋅ } − P0 { ⋅ } = g1 − g 0

Ψ1′k1 + k1′

Γ Γ1 k1 − Ψ0′k 0 + k 0′ 0 k 0 2 2 . g1 − g 0

(12)

As (g1 − g 0 ) → 0 , one can see from (11) that Ψ1 → Ψ0 , Γ1 → Γ0 , and (12 ) → ∂P ∂g . Thus, for an infinitesimal change in g, MWTP in the pre-shock equilibrium will equal MWTP in the postshock equilibrium which will equal the capitalization rate. For a non-marginal change in g, the capitalization rate will generally differ from both MWTP0 and MWTP1 . The size and magnitude of the difference will depend on the correlation between the change in g and its initial level. We illustrate this with a numerical example: k = [g , x1 , x 2 ] , 2 0 0 Σα = 0 1 0  , 0 0 3

µα = [20 50 25] , 1 0 0 Ω = 0 2 0  , 0 0 3

µ k = [5 10 0] ,

(13)

 2 0 0 and Σ k = 0 1 0 . 0 0 1

With these parameter values, all three characteristics are normal goods, the demand for each is downward sloping, and the equilibrium price function is linear. The multivariate normal

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distributions (13) are used to take 1,000,000 random draws. After evaluating the initial price function, we shock g using ∆g ~ N (3,0.25) and cov(∆g , x1 ) = cov(∆g , x2 ) = 0 . Then we evaluate the new price function and measure the bias conveyed by interpreting the capitalization rate from (2) as a measure of average MWTP. Figure 2 reports results for two different values of cov(∆g , g1 ) . Each panel shows the implicit price functions for g before and after the shock, as well as demand curves for two households. Increasing g increases the price of housing but decreases MWTP because demand is less than perfectly elastic. In panel A, g1 and ∆g are negatively correlated so that areas with the lowest baseline levels of g receive the largest improvements. This results in a sufficiently large upward bias on the capitalization-based estimate for MWTP ($18.12) that it exceeds the true average MWTP in the pre-shock equilibrium ($15). Intuitively, this is gentrification. The households who value g the most drive up prices in improved areas by more than the average resident is willing to pay.

Panel B demonstrates the opposite case where areas with the highest baseline levels of g receive the largest improvements. In this case, the capitalization rate ($6.03) substantially underestimates average MWTP in the new equilibrium ($11.85). This example of preferential attachment is consistent with Starrett’s (1981) observation that there is little upward pressure on prices when highest quality neighborhood experiences a further improvement. Households who previously chose to live in lower quality areas did so because they have a lower marginal valuation of g. They are simply not in the market for homes in higher quality areas. Finally, notice that the changes in the price functions in figure 2 identify two points on the demand curve for each household. This illustrates the intuition for instrumental variables estimation. The point of implication is that the policy discontinuities and unexpected events that 14

served as the basis for quasi-experiments in Chay and Greenstone (2005), Davis (2004), and other recent studies also offer the potential to help identify the demand for an amenity in a single metro area.

4.

Sufficient Statistics for Quasi-Experimental Measurement of Marginal Values

The theoretical model provided several reasons to expect the price function to change over time. Yet, no predictions were made for the speed of its evolution or the magnitude of the changes. If adjustment is slow or changes are small, capitalization rates may provide good approximations to welfare measures. Investigating this possibility requires a pragmatic approach. We use a common specification for the empirical price function to derive testable restrictions on the data that are sufficient for capitalization rates to measure welfare. Our approach is consistent with Chetty’s (2009) proposal for identifying “sufficient statistics” for quasi-experimental welfare measurement.

4.1.

A Simple Econometric Model

Empirical studies typically specify the hedonic price function to be linear in parameters.

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We

follow this approach and abstract from potential econometric complications such as measurement error and approximation error in the choice of functional form. The reason for abstraction is to focus attention on the relationship between capitalization and welfare. That said, the analysis in this section could be repeated under any specific choices for the quasiexperimental estimator and the true equilibrium price function.

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The true functional form is unknown. The prevalence of the linearity assumption in empirical work is partly due to Cropper, Deck, and McConnell (1988). Working with simulated data, they found that linear specifications for the price function tended to provide more accurate predictions for MWTP than a more flexible Box-Cox quadratic model in the presence of unobserved variables and errors in variables.

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We begin by repartitioning X into observed (h) and unobserved ( ξ ) components. Using this partition, the linear price functions that describe market equilibria before and after an unexpected shock to g are p1 = g1θ1 + h1η1 + ε 1 (ξ1 ) and p2 = g 2θ 2 + h2η 2 + ε 2 (ξ 2 ) . 15 Parameter subscripts recognize that the shape of the function may have been altered by the shock to g and by concomitant changes in h, ξ , F ( y , α ) , and V (β ) . Subtracting the old price function from the new one yields a general time-differenced estimator,

∆P = (g 2θ 2 − g1θ1 ) + (H 2η 2 − H 1η1 ) + ∆ε .

(14)

In the special case where θ1 = θ 2 and η1 = η 2 , equation (14) reduces to the quasi-experimental capitalization model introduced in section 2, ∆p = ∆gφ + ∆hγ + ∆ε . Our interest lies in the general relationship between the capitalization rate ( φ ) and MWTP ( θ1 ,θ 2 ), which can be expressed as:

[]

′ r ′g1 (θ 2 − θ1 ) + r h1 (η2 − η1 ) , E φˆ = θ 2 + r ′r r ′r

(15)

where r = ∆g − ∆h (∆h ′∆h ) ∆h ′∆g . −1

The expected value of the first-differenced estimator for φ is a function of the parameters of the true price functions that precede and follow the shock. Put differently, (15) reports what we can expect to learn about welfare from estimating (2) when (14) is the true model and the quasiexperiment is valid in the sense that the data satisfy the zero conditional mean assumption,

E [∆ε | ∆g , ∆h ] = 0 . 16 The h1 matrix of control variables may also include a vector of ones so that η includes an intercept. There is reason to question this assumption. If unobserved features of the urban landscape are endogenously determined through household location choices, the zero conditional mean assumption may be systematically violated (Kahn [2006]). For example, Banzhaf and Walsh (2008) find that an increase in toxic emissions leads to 15 16

16

The estimate for the capitalization rate is a function of ex-ante MWTP, ex-post MWTP, and correlations between housing characteristics. The second term to the right of the equality in (15) is a “price effect” that arises from a change in the implicit price of g between the initial equilibrium and the new equilibrium. The third term is a “substitution effect” that arises from changes in the implicit prices of other housing characteristics that affect utility and, in some sense, serve as substitutes for g.

[]

Without any restrictions on the data, E φˆ may fall outside the range of values for MWTP defined by θ1 and θ 2 . For example, consider a quality improvement that decreases MWTP but has no effect on the control variables or their marginal implicit prices:

[]

∆h = η 2 − η1 = 0 . In this case, (15) implies that E φˆ < θ 2 < θ1 if cov(∆g , g1 ) > 0 . Alternatively,

[]

θ 2 < θ1 < E φˆ if var(∆g ) < − cov(∆g , g1 ) . Thus, it is clear that additional restrictions on the data are needed to give the estimated capitalization rate a welfare theoretic interpretation. Two sets of testable restrictions can serve as sufficient statistics for the capitalization model to provide an unbiased estimate of MWTP.

Sufficient Statistic #1: A Randomized Shock If the shock to g is not correlated with its initial level, or with the initial levels of the control variables, or with changes in those variables, the capitalization rate measures MWTP in the postshock equilibrium, even if the shape of the hedonic price function changes. More formally, it

[]

can be seen from (15) that E φˆ = θ 2 if ∆h′∆g = h1∆g = g1′∆g = 0 . If these three restrictions hold, φˆ provides a consistent measure of ex-post MWTP. If they do not hold, we must restrict emigration of wealthier households. Changes in wealth may lead to changes in potentially unobserved amenities that matter to households (crime, public school quality, racial composition, etc).

17

the price function to be stationary.

Sufficient Statistic #2: Stationary Implicit Prices If the demand for every housing characteristic is perfectly elastic over the range of the changes in g and h and constant over time, then MWTP for each housing characteristic must be constant

[]

such that θ1 = θ 2 and η1 = η 2 . In this case, (15) reduces to E φˆ = θ1 = θ 2 . Thus, the capitalization model in (2) provides an unbiased estimator of ex ante MWTP which equals ex post MWTP. This assumption of stationary implicit prices can be tested if instruments are available to support consistent estimation of the single-period price functions.

4.2.

Discussion

In a true experiment, the researcher could measure ex post MWTP by holding the control variables constant while randomly varying the amenity of interest (sufficient statistic #1). Unfortunately, real world data rarely satisfies this requirement. Catastrophic events that have been exploited as natural experiments in the past, including earthquakes, hurricanes, and floods, tend to impact continuous geographic regions, which may induce correlation between the event and spatially correlated amenities. This would imply h1′∆g ≠ 0 . Meanwhile, the public policies that have served as the basis for quasi-experiments, such as the Clean Air Act, the Superfund program, and No Child Left Behind have been explicitly targeted on the basis of existing quality, so that g1′∆g ≠ 0 by design. Without a truly randomized shock, the researcher must invoke the “stationary implicit price” assumption in order to interpret the capitalization rate as a welfare measure (sufficient statistic #2). For example, in order to estimate the benefits from large air quality improvements

18

in the United States during the 1970s, Chay and Greenstone (2005) assume the implicit price of air quality was constant throughout the decade. Davis (2004) assumes the implicit price of pediatric leukemia risk was unaffected by a six-fold increase in that risk. Greenstone and Gallagher (2008) assume the implicit price of Superfund-sponsored cleanup of hazardous waste sites was unchanged between 1980 and 2000. The stationary implicit price assumption may provide a reasonable approximation for small changes that occur over short time periods. However, it is natural for quasi-experimental analysis to focus on large shocks and/or long time periods. It takes a large shock to explain reequilibration of the housing market, especially if we acknowledge moving costs, imperfect information, and other sources of friction that are likely to limit mobility in response to small changes. Furthermore, it can take many years for markets to re-equilibrate following a shock. Over a longer interval the MWTP for an amenity may change in response to changes in wealth, information, substitute goods, or preferences. Costa and Kahn (2003) provide preliminary evidence that MWTP does change over time. They use national housing and wage data from 1970 to 1999 to assess the degree of temporal variation in the marginal implicit price of living in a metropolitan area with a temperate climate. Their results imply that the implicit price doubled between 1970 and 1980, and then doubled again between 1980 and 1990. Thus, over a ten-year period when climate itself is relatively stable, Costa and Kahn observe a large change in its implicit price. It seems reasonable to expect similar trends for other public goods and environmental amenities which are conveyed through the location of a home.

5. Sufficient Statistic Tests: Housing Characteristics and School Quality

19

We apply our logic of sufficient statistics to a unique set of micro data on property values and school quality. The data were originally collected from assessors in 161 counties in 23 states. 17 They describe the sale prices and structural characteristics of approximately 7 million singlefamily residential properties sold between 1998 and 2009. Each transaction includes information on the square feet of the dwelling, the number of bathrooms, the number of bedrooms, the year built, and the size of the lot. Addresses have been geocoded to the street level, enabling us to relate them to Census geography (i.e. block group, county). As a proxy for neighborhood demographic composition, we attach to each home selected variables from the 2000 Census that describe the block group in which the home is located (percent non-white, percent under 18, percent owner occupied housing, percent vacant housing, and population per square mile). Table 1 provides summary statistics. Our analysis proceeds in two stages. As a first pass, we use the full dataset to test for stationarity of the implicit prices of structural housing characteristics. Then we narrow our analysis to three metropolitan areas (Detroit, Los Angeles, and Philadelphia) where recent federal legislation has reportedly improved public school quality. For each metro area, we evaluate sufficient statistics #1 and #2 and compare econometric estimates for MWTP with estimates for the rates at which observed school quality improvements were capitalized.

5.1. Are the Implicit Prices of Housing Characteristics Constant over Time? As a preliminary test on the stationarity of implicit prices we estimate the following model ln( P) = hη1 + d 2 hη 2 + d 2 + tract + ε ,

(16)

ten times for each county. Each regression uses data from two years: 1998 (the base year) and

17

This data was purchased from the housing data vendor Dataquick.

20

one subsequent year (between 1999 and 2008). 18 The objective is to test for changes in the implicit prices between the two years. In the equation, h includes basic structural characteristics (square feet, bath, bedrooms, year built, and lot acres), d 2 is a dummy variable for the test year (1999,…,2008), and tract is a set of dummy variables for Census tracts. A Wald test of η 2 = 0 provides some intuition for whether sufficient statistic #2 is likely to be satisfied. Table 2 reports the share of counties failing to reject the null hypothesis that η 2 = 0 at the 90th, 95th, and 99th significance levels. If the price function were stationary, we would expect to see 10%, 5%, and 1% of the counties rejecting the null (falsely). Actual rejection rates are much higher. The first row of the table reports rejection rates of 51%, 42%, and 28% for a comparison between price functions estimated in 1998 and 1999. Increasing the interval between the base year and the comparative year usually increases the rate of rejection. For example, the last row reports rejection rates of 81%, 75%, and 71% for a comparison between implicit prices in 1998 and 2008. 19 There are two potential explanations for the pattern of results in table 2. One is that omitted public goods and neighborhood amenities are varying within counties over time. The other is that the hedonic price function is nonstationary. We suspect there is some truth to both explanations. By narrowing the geographic resolution of the fixed effects in our analysis of school quality we can begin to distinguish between the two explanations.

5.2. NCLB and the Difference between Capitalization and MWTP for School Quality

18

Some of these counties were missing a year of data between 1998 and 2008 or had so few observations that they dropped out of the analysis. Therefore in Table 2 the sample size of counties ranges between 150 and 161. 19 To check if the statistical significance for failing to reject the null is driven by counties with large numbers of housing transactions, we split the sample into counties with different numbers of housing transactions (less than 1k, between 1k and 2k, between 2k and 5k, between 5k and 10k, and greater than 10k). Small counties are somewhat less likely to reject the hypothesis of time-constant prices than large counties, especially the further the test year is from the base year. However, we found for individual implicit prices the mean difference between the 1998 base year estimate and the estimate in other years was actually larger for the smaller counties.

21

In several areas of the United States, students in the public school system are required to attend schools located within the boundaries of the school districts in which their parents live. This assignment creates a link between the choice of a home and public school quality. Several previous studies have exploited this link as a way to measure parents’ willingness to pay for school quality improvements. Early work used conventional hedonic analysis without an explicit strategy to control for omitted neighborhood characteristics (e.g. Kain and Quigley 1975). More recent studies have exploited discontinuities in time and space to mitigate potential confounding. Notably, Black (1999) was the first to suggest exploiting the spatial discontinuities that arise from the geography of school attendance zone boundaries. Her cross-section regression used data on homes that sold within small “neighborhoods” around each attendance zone boundary. Intuitively, including dummy variables for each boundary in the regression allowed her to identify the implicit price of school quality from variation in the prices of similar homes located on opposite sides of a particular attendance zone boundary. In subsequent work, Figlio and Lucas (2004) exploited temporal changes in quality associated with the release of a state’s grading of schools. Their panel data analysis attempts to identify the value of school quality by looking at property values before and after the release of grades. We combine the identification strategies developed by Black (1999) and Figlio and Lucas (2004) to estimate the implicit price of school quality, analyze its time constancy, and compare it to capitalization rates. We first explore the time-constancy of implicit prices using Black’s boundary discontinuity approach to identify the implicit prices of school quality in cross-sections of our data for the 2003 and 2007 school years. The spatial discontinuity approach increases our confidence that we are controlling for omitted variables. This is what allows us to identify the implicit price of school quality in 2003 and 2007 and then test the null hypothesis of θ1 = θ 2

22

from equation (15). Results from 2003 and 2007 are then compared to a first-difference panel data model that estimates how changes in average test scores between 2003 and 2007 were capitalized into property values. The 2003 to 2007 interval was chosen because it immediately follows the federal No Child Left Behind act (NCLB), which provided new incentives for poorperforming schools to improve. 20

5.2.1

Background and Data

Five criteria were used to select metropolitan areas for our analysis: (i) a sufficient number of unified school districts to conduct boundary discontinuity estimation; (ii) a sufficient number of housing transactions for estimation; (iii) school districts that reported their test scores under NCLB in every school year since the program was implemented; (iv) no inter-district open enrollment policy that allows students to enroll in schools other than those located in the district where the student lives; and (v) geographic diversity. 21,22,23 Three metro areas satisfied these criteria—Philadelphia, Los Angeles, and Detroit. For each metro area, table 3 reports summary statistics for houses that sold during the 2003 and 2007 “school years”. We defined the 2003 school year as October 1, 2003 to

20

Since the enactment of No Child Left Behind, states have been required to implement statewide accountability systems that measure student’s performance in reading and math. This standardized state testing is done in grades 3 through 8 and at least once during high school. These test scores are used to determine if a public school is making “Adequate Yearly Progress” (AYP) toward the goal of having 100% of students attaining the state standards in reading and mathematics by 2014. Schools that do not meet AYP face a series of repercussions that provide incentives for the schools to improve. See http://www.ed.gov/nclb/overview/intro/execsumm.html for more details on the NCLB act. 21 Boundary discontinuity analysis is extremely data-intensive because it discards housing transactions that occur beyond small distances from the school district boundaries. 22 We use unified school districts rather than individual schools as in the case of Black (1999) for several reasons. First the digital maps of the school districts are made available by the 2000 census whereas the individual school attendance zones are not. Second, most states now have mandatory intra-district open enrollment policies that allow students to enroll in any school within the school district. Finally, district-level school data has less idiosyncratic variability from year to year which is useful for our analysis. 23 States were not required to start reporting test scores until 2006 and so many states did not have test score data available early enough for the analysis.

23

September 30, 2004, and similarly for 2007. These date ranges were chosen because the NCLB test scores and school grades for the preceding school year are typically announced at the end of August or the beginning of September. Thus we want to allow time for our proxy for school quality—test scores—to influence home buyers’ decisions. The proxy variable, “lmathread” is the natural log of a combined measure of proficiency rates for math and reading in the school district for the year preceding the housing transactions. 24 Table 3 summarizes “lmathread” and all of the other housing characteristics for three subsets of the data in each metro area: (i) the full sample; (ii) sales that occurred within 0.35 miles of a school district boundary; and (iii) sales that occurred within 0.20 miles of a boundary. Notice that, within a metro area, there is little difference between the typical homes in each subset. The only systematic difference appears to be that homes located closer to a school district boundary tend to be slightly smaller and older. Figure 3 illustrates the spatial relationship between housing transactions, school districts, and the 0.35 mile buffers around school district boundaries for a portion of our Los Angeles metro area. Table 4 provides summary statistics for the panel data used in our capitalization analysis. To create the dataset, we average all characteristics (including school quality) over the housing transactions that occurred in each block group within a metro area. 25 Table 4 reveals that there are 1529, 6977, and 1499 block groups for which we have data in 2003 and 2007 in Philadelphia, Los Angeles and Detroit respectively. Summary statistics are provided for both the 2003 levels and the 2007-2003 differences. Focusing on the differences in housing prices (d_price), it can be

24

The school quality information was obtained from www.schooldatadirect.org . The combined measure of reading and math is an overall measure (calculated by Standard & Poor’s) that provides an average of the proficiency rates achieved across all reading and math tests, weighted by the number of tests taken, such that proficiency rates on tests with greater numbers of test takers have more influence on the measure than proficiency rates on tests with fewer test takers. 25 Aggregation to the block group level provides hupigher geographic resolution that some previous capitalization studies (e.g. Chay and Greenstone 2005, Greenstone and Ghallagher 2008) and lower resolution than others (e.g. Davis 2004, Pope 2008). Unfortunately, we lack sufficient data on repeated sales of individual homes to conduct a micro-level analysis.

24

seen that average housing prices were increasing in both Philadelphia and Los Angeles, but decreasing in Detroit, reflecting its recent economic upheaval. In contrast, average scores in math and reading proficiency (d_mathread) increased in all three metro areas (15.7% in Philadelphia, 23.8% in Los Angeles, and 15.9% in Detroit). 26

5.2.2

Is NCLB a Randomized Shock to School Quality? (Sufficient Statistic #1)

President George W. Bush announced his “No Child Left Behind” framework for education reform three days after taking office, and within a year the NCLB Act had been passed. Passage of this sweeping set of reforms was rapid and appears to have accelerated improvement in standardized test scores. But does the shock meet the criteria for randomization defined by sufficient statistic #1? Table 5 reports correlations between the 2003 levels and 2007-2003 differences in school quality and each housing characteristic. These correlations allow us to evaluate the three testable restrictions imposed by sufficient statistic #1 ( ∆h′∆g = h1∆g = g1′∆g = 0 ). First, consider the correlation between the change in housing characteristics and the change in school quality ( ∆h′∆g ). Looking at the three rows for “d_lmathread” it can be seen that the correlations are all close to zero. The largest individual value (-.08) is for the correlation with “d_age” in Detroit. Now focus on the “d_lmathread” column. The absolute values of correlations between changes in school quality and the 2003 levels of housing characteristics ( h1∆g ) range from 0.12 to 0.46. Interestingly, the correlations are systematically largest in Detroit. Finally, the correlations between the changes in school quality and their 2003 levels ( g1′∆g ) range from − .84 to − .91 . As one might expect, the schools that initially performed the worst were the ones that 26

The measure of math and reading proficiency is not directly comparable across states because each state uses its own testing standards.

25

experienced the largest improvements. There are two implications of the pattern of results in table 5. The first is that, without further restrictions, we cannot expect capitalization rates to reveal ex-post MWTP for school quality. Second, based on the expression in (15), we would expect the difference between capitalization rates and MWTP to be especially sensitive to changes in the implicit price of school quality. This follows from the relatively small correlation between ∆h and ∆g and the relatively large correlation between g1 and ∆g .

5.2.3

Is the MWTP for School Quality Constant over Time? (Sufficient Statistic #2)

Our boundary discontinuity approach to estimating the MWTP for school quality is based on the following equation

ln( P) = hη1 + d 2 hη 2 + d 2 + lmathreadθ1 + d 2 lmathreadθ 2 + FE + ε ,

(17)

for each metro area, using the 2003 and 2007 cross-section data. In the equation, h includes both the basic structural housing characteristics and the block group demographics from the 2000 census, d 2 is a dummy variable for transactions that occurred in the 2007 school year, lmathread denotes the log of math and reading proficiencies for the district the year prior to the

housing transaction, and FE is either a set of county fixed effects or a set of school district boundary fixed effects, depending on the particular regression. Table 6 summarizes our regression results. Throughout the table, the results are broadly consistent with previous hedonic analyses in terms of the size and magnitude of the coefficients on housing characteristics and the R-squared that ranges from 0.673 to 0.834. We will say no more about these features of the results. Our interest lies in the relative magnitudes of the coefficients on lmathread in the various specifications.

26

Consider the results for Philadelphia. The coefficient on lmathread in column (1) indicates that, in 2003, an increase in average math and reading proficiencies of 1% would lead to an increase of approximately 0.597% in property values in the corresponding school district. The coefficient on lmathread2 suggests that in 2007 the increase would be approximately 1.038% (0.597 + 0.441). The difference between the implicit price for school quality in 2003 and 2007 is not only economically large; it is statistically significant at the 1% level. This can be seen from the Wald test on the coefficient for lmathread2 reported at the bottom of the table. However, it is natural to be concerned that these striking results may simply be driven by correlation between test scores and omitted neighborhood characteristics. At present, the spatial fixed effects in the model are quite aggregate (counties). Column (2) continues to use county fixed effects, but restricts the sample to houses located within 0.35 miles of a school district boundary. The coefficients on both lmathread and lmathread2 are reduced, but not dramatically so. This suggests that moving to the restricted sample is not likely to drive any differences that we find in the results once we add the school boundary fixed effects. Column (3) introduces fixed effects for 121 school district boundaries. They absorb the price effect of the unobserved neighborhood amenities around each boundary. Intuitively, including them in the model forces the identification of the implicit price of school quality to come from variation in the sale prices of structurally similar homes, with demographically similar neighborhoods, that are located “near” one another but are separated by a school district boundary. 27 The coefficient on lmathread is now substantially reduced to 0.209 and the coefficient on lnmathread2 is statistically insignificant and close to zero. These results are unchanged when we restrict the sample to houses that are within 0.20 miles of a school boundary 27

For further discussion of this identification strategy, see Black (1999) or Bayer, Ferreira, and McMillan (2007).

27

(column 4). Overall, the results in columns (3) and (4) provide our most plausible estimates. They suggest that the MWTP for school quality in Philadelphia was essentially unchanged between 2003 and 2007. We now turn our attention to the results for Los Angeles and Detroit. Comparing columns (5)-(6) with (7)-(8) and columns (9)-(10) with (11)-(12) reveals that our estimates for the implicit prices for school quality in Los Angeles and Detroit fall once the school district boundary fixed effects are added, just like in Philadelphia. However, unlike the Philadelphia results, statistically significant differences in the implicit prices for 2003 and 2007 persist. For example, in Los Angeles the coefficient on lmathread in column (7) suggests that a 1% increase in school quality in 2003 leads to a 0.172% increase in housing prices whereas the increase for 2007 is 0.271% (0.172 + 0.099). In Detroit this difference is even more dramatic. The elasticity in 2003 was 0.213 (column 11) compared to an elasticity of 0.648 (0.213 + 0.435) four years later. Thus, our results suggest that the MWTP for school quality in both Los Angeles and Detroit changed between 2003 and 2007. 28

5.2.4

Comparing Capitalization Rates with MWTP

Finally, table 7 reports our estimates for capitalization rates based on first-difference regression of panel data for each metro area. Looking at the results for “Philly,” our point estimate of the capitalization rate is 0.137 suggesting that a 1% change in school quality results in a 0.137% change in property values. This is somewhat lower than the estimate of 0.209% obtained from the cross-sectional models that employed the school district boundary fixed effects. However, 28

Our elasticity estimates (generated using the boundary discontinuity approach for the 0.35 mile sample) of school quality (as measured by the math-reading proficiency measure) on property values ranges from 0.172 in 2003 in Los Angeles, to 0.648 in Detroit in 2007. Black (1999) finds an elasticity of approximately 0.46 for her estimates that use the 0.35 mile sample. Despite the similarity of these results, it should be noted that test scores are not directly comparable across states and our measure of school quality is somewhat different than Black’s (1999).

28

their 95% confidence intervals clearly overlap. The relatively large confidence interval on the capitalization rate partly reflects the order of magnitude decrease in sample size from moving to a panel setting. However, it is also consistent with our finding of a stationary MWTP for school quality in Philadelphia between 2003 and 2007. For the “L.A.” sample, the first-difference estimate for the capitalization rate (0.217%) falls right in the middle of the cross-section estimates of 0.172% in 2003 and 0.271% in 2007. The endpoints of its confidence interval just barely reach the pre-NCLB and post-NCLB results. In stark contrast to the other two areas, the capitalization rate for Detroit ( − 0.222 ) which is approximately the same magnitude as the cross-section estimates, has the wrong sign! The fact that Detroit has the largest “capitalization bias” in measuring MWTP should come as no surprise. It experienced the largest change in the implicit prices of school quality (table 6) as well as the largest correlations between changes in school quality and other characteristics (table 5). Omitted variables may be another contributing factor. Even over the short interval we consider, it is possible that Detroit’s recent economic turmoil led to changes in the supply of unobserved public goods and neighborhood amenities within census block groups. In summary, our empirical analysis confirms the potential for divergence between capitalization rates and welfare measures. We have demonstrated that the implicit prices for both housing characteristics and school quality can vary over a short period—such that sufficient statistic #2 is not realized. Similarly, our analysis of the NCLB act illustrated that sufficient statistic #1 will not be realized if the treatment outcome has a systematic component that covaries with its baseline level. The further we are from realizing these requirements, the greater the divergence between capitalization rates and welfare measures.

29

6. Conclusion We have argued that, in theory, the conventional approach to hedonic estimation and the newer quasi-experimental approach measure phenomena that are fundamentally different. It is well known that the conventional approach offers the potential to measure consumer welfare from a marginal change in a public good. The quasi-experimental approach offers the potential to measure the rate at which changes in that good are capitalized into property values. Only under special circumstances will these two measures be equal. Our analysis of the implicit markets for school quality in Detroit, Los Angeles, and Philadelphia demonstrated that the empirical differences between capitalization and welfare can be quite large. These examples also illustrated that the data restrictions we developed can serve as preconditions to quasiexperimental welfare measurement. As part of our analysis, we observed that an unexpected shock that changes the shape of the price function can generate instrumental variables to identify the market demand for a public good. Recovering demand curves from a reduced form description of market equilibrium was the original goal of hedonic estimation. The prospect of attaining this goal through quasiexperiment analysis is worthy of further research. Perhaps the greatest difficulty will be obtaining consistent “first stage” estimates of the price function from cross-section data. In our application, we were fortunate to be able to exploit Black’s (1999) boundary discontinuity solution to omitted variable contamination. A more widely applicable (but less precise) alternative is the prudent use of spatial fixed effects. Finally, a structural model of the transition dynamics between pre-shock and post-shock equilibria could help to guide future efforts to develop a “single-market, multi-period” approach to “second stage” demand estimation.

30

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Chetty, Raj. 2009. “Sufficient Statistics for Welfare Analysis: A Bridge Between Structural and Reduced-Form Methods,” Annual Review of Economics 1: 451-488. Costa, Dora L and Matthew E. Kahn. 2003. "The Rising Price of Nonmarket Goods." American Economic Review, 93(2): 227-32. Cropper, Maureen L., Leland B. Deck, and Kenenth E. McConnell. 1988. "On the Choice of Functional Form for Hedonic Price Functions." Review of Economics and Statistics, 70(4): 668-75. Davis, Lucas. 2004. "The Effect of Health Risk on Housing Values: Evidence from a Cancer Cluster." American Economic Review, 94(5): 1693-704. Ekeland, Ivar, James J. Heckman, and Lars Nesheim. 2004. "Identification and Estimation of Hedonic Models." Journal of Political Economy, 112(1): S60-S109. Epple, Dennis. 1987. "Hedonic Prices and Implicit Markets: Estimating Demand and Supply Functions for Differentiated Products." Journal of Political Economy, 95(1): 59-80. Figlio, David N., and Maurice E. Lucas. "What's in a Grade? School Report Cards and the Housing Market." American Economic Review 94, no. 3 (2004): 591-604. Gottschalk, Peter. "Inequality, Income Growth, and Mobility: The Basic Facts." Journal of Economic Perspectives 11, no. 2 (1997): 21-40. Greenstone, Michael and Justin Gallagher. 2008. "Does Hazardous Waste Matter? Evidence from the Housing Market and the Superfund Program." Quarterly Journal of Economics, 123(3): 951-1003. Horsch, Eric J., and David J. Lewis. "The Effects of Aquatic Invasive Species on Property Values: Evidence from a Quasi-Experiment." Land Economics 85, no. 3 (2009): 391-409. Kahn, Matthew E. 2006. "Environmental Valuation Using Cross-City Hedonic Methods," in Environmental Valuation: Interregional and Intraregional perspectives. John I. Carruthers and Bill Mundy eds. Hampshire: Ashgate. Kain, John F., and John M. Quigley. "Housing Markets and Racial Discrimination." New York: National Bureau of Economic Research (1975). Keane, Michael P. Forthcoming. "Structural vs. Atheoretic Approaches to Econometrics." Journal of Econometrics. Kuminoff, Nicolai V. "Decomposing the Structural Identification of Nonmarket Values." Journal of Environmental Economics and Management 57, no. 2 (2009): 123-39.

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Linden, Leigh and Jonah E. Rockoff. 2008. "Estimates of the Impact of Crime Risk on Property Values from Megan's Laws." American Economic Review, 98(3): 1103-27. Lind, Robert C. 1973. "Spatial Equilibrium, the Theory of Rents, and the Measurement of Benefits from Public Programs." Quarterly Journal of Economics, 87(2): 188-207. Palmquist, Raymond B. 1982. "Measuring Environmental Effects on Property Values without Hedonic Regressions." Journal of Urban Economics, 33(3): 333-47. Palmquist, Raymond B. and V. Kerry Smith. 2002. "The Use of Hedonic Property Value Techniques for Policy and Litigation," in The International Yearbook of Environmental and Resource Economics 2002/2003. Tom Tietenberg and Henk Folmer eds. Cheltenham, UK: Edward Elgar, pp. 115-64. Palmquist, Raymond B. 2005. "Property Value Models," in Handbook of Environmental Economics, Volume 2. Karl-Göran Mäler and Jeffery Vincent eds. Amsterdam: North Holland Press. Pope, Jaren C. 2008. "Fear of Crime and Housing Prices: Household Reactions to Sex Offender Registries." Journal of Urban Economics, 64(3): 601-14. Rosen, Sherwin. 1974. "Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition." Journal of Political Economy, 82(1): 34-55. Rubinfeld, Daniel L., Perry Shapiro, and Judith Roberts. "Tiebout Bias and the Demand for Local Public Schooling." Review of Economics and Statistics 69, no. 3 (1987): 426-37. Sattinger, Michael. 1980. Capital and the Distribution of Labor Earnings. Amsterdam: North Holland. Scotchmer, Suzanne. "Hedonic Prices and Cost/Benefit Analysis." Journal of Economic Theory 37, no. 1 (1985): 55-75. Scotchmer, Suzanne. 1986. "The Short-Run and Long-Run Benefits of Environmental Improvement." Journal of Public Economics, 30(1): 61-81. Simons, Robert. 2006. When Bad Things Happen to Good Property. Washington D.C.: Environmental Law Institute. Starrett, David A. 1981. "Land Value Capitalization in Local Public Finance." Journal of Political Economy, 89(2): 306-27. Tinbergen, Jan. 1959. "On the Theory of Income Distribution," in Selected Papers of Jan Tinbergen. L.H. Klaassen, L.M. Koych and H.J. Witteveen eds. Amsterdam: North Holland.

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$

S 2 (g | X , β 2 ) S1 (g | X , β1 ) ∂P (g , X ) ∂g

MWTP2 MWTP1

D2 ( g | X , α 2 , y2 )

D1 ( g | X , α1 , y1 )

g1

g2

g

FIGURE 1. Demand curves for two households, supply curves for two producers, and the equilibrium marginal price function for g.

34

Pg (g1 )

Pg (g1 )

Pg (g1 + ∆g )

Pg (g1 + ∆g )

D2 (g )

D2 (g )

D1 (g )

D1 (g )

cov(∆g , g1 ) = −0.5

cov(∆g , g1 ) = 0.5

pre-shock MWTP: $15.00 post-shock MWTP: $11.97 capitalization rate: $18.12

pre-shock MWTP: $15.00 post-shock MWTP: $11.85 capitalization rate $6.03

PANEL A: Gentrification

PANEL B: Preferential attachment

Note: Each panel depicts equilibria before and after an unexpected shock to g in the linear-quadratic-normal hedonic model. All other variables are held constant. D1 and D2 are demand curves for two households, Pg ( g1 ) is the

marginal implicit price function for g in the pre-shock equilibrium, and Pg ( g1 + ∆g ) is the implicit price function in the post-shock equilibrium. The post-shock MWTP and capitalization rate are based on 1,000,000 random draws on preferences and ∆g .

FIGURE 2. Numerical examples of the difference between capitalization rates and marginal willingness to pay.

35

FIGURE 3. Housing Transactions, School Districts and the 0.35 Mile Buffer Distance in Part of the Los Angeles Metropolitan Area. 36

Table 1: Summary Statistics of Housing Data Variable sale price square feet bath bedrooms year built lot acres % non-white % under 18 % owner occupied % vacant population density (norm.)

Mean

Stan. Dev.

Num. Obs.

301604 1825 2.19 3.20 1973 0.34 0.21 0.26 0.72 0.06 0.03

284947 806 0.87 0.82 26 0.52 0.21 0.07 0.19 0.07 0.03

6,977,560 6,977,560 6,977,560 6,977,560 6,977,560 6,977,560 6,977,558 6,977,558 6,977,560 6,977,560 6,977,560

Note: Data represents micro-level housing data for 158 counties in 23 states throughout the U.S., for the years 1998-2008. All data is geocoded to the latitude/longitude and has been cleaned as described in the text. Demographic variables starting with "%" were obtained from the 2000 U.S. Census at the block group level.

Table 2: Test of Time Constancy of Implicit Prices Share of counties failing to reject at: years

90th

95th

99th

N

1998, 1999 1998, 2000 1998, 2001 1998, 2002 1998, 2003 1998, 2004 1998, 2005 1998, 2006 1998, 2007 1998, 2008

0.49 0.38 0.28 0.25 0.25 0.19 0.14 0.13 0.13 0.19

0.58 0.45 0.35 0.28 0.29 0.25 0.17 0.17 0.17 0.25

0.72 0.59 0.49 0.39 0.37 0.30 0.23 0.25 0.26 0.29

158 157 155 155 150 159 159 161 161 161

Note: Table shows the share of counties that fail to reject the null hypothesis that the implicit prices of housing characteristics are constant between pairs of years. 1998,1999 denotes that 1998 is the base year and 1999 is the comparison year in the regressions from equation (16) in the text

37

Table 3: Summary Statistics for School Quality Metro Areas Full Sample Variable

Obs

Mean

0.35 mile

Std. Dev.

Obs

Mean

0.20 mile Std. Dev.

Obs

Mean

Std. Dev.

4,191 4,191 4,191 4,191 4,191 4,191 4,191 4,191 4,191 4,191 4,191 4,191 4,191 4,191 4,191

299,597 12.48 19.98 2.26 47.32 0.43 3.33 0.11 0.25 0.79 0.03 0.03 0.43 73.42 4.28

178,065 0.52 902.31 1.00 27.00 0.56 0.78 0.13 0.05 0.18 0.02 0.02 0.50 12.69 0.21

12,682 12,682 12,682 12,682 12,682 12,682 12,682 12,682 12,682 12,682 12,682 12,682 12,682 12,682 12,682

532,557 12.97 17.07 2.15 44.95 0.20 3.21 0.43 0.28 0.70 0.04 0.06 0.46 46.15 3.77

471,268 0.59 792.48 0.90 21.44 0.21 0.86 0.23 0.07 0.20 0.07 0.04 0.50 16.66 0.36

6,443 6,443 6,443 6,443 6,443 6,443 6,443 6,443 6,443 6,443 6,443 6,443 6,443 6,443 6,443

184,594 11.93 15.90 1.98 47.22 0.30 3.10 0.12 0.25 0.81 0.03 0.03 0.48 72.72 4.27

133,477 0.64 786.90 1.00 21.08 0.42 0.71 0.16 0.06 0.16 0.02 0.02 0.50 12.16 0.18

Philadelphia, PA Sample price lnprice sqft (in 100's) bath age lotacre bedrms perc_nwhite perc_und18 perc_ownocc perc_vacant norm_popdens year_2007 mathread lmathread

29,333 29,333 29,333 29,333 29,333 29,333 29,333 29,333 29,333 29,333 29,333 29,333 29,333 29,333 29,333

312,040 12.49 20.87 2.37 42.03 0.49 3.38 0.11 0.26 0.78 0.03 0.02 0.42 72.30 4.26

204,259 0.56 948.25 1.00 27.85 0.65 0.77 0.14 0.05 0.18 0.03 0.03 0.49 13.92 0.24

6,796 6,796 6,796 6,796 6,796 6,796 6,796 6,796 6,796 6,796 6,796 6,796 6,796 6,796 6,796

301,235 12.48 20.10 2.29 46.45 0.43 3.35 0.11 0.25 0.78 0.03 0.03 0.43 72.88 4.27

183,972 0.52 905.84 0.99 27.73 0.56 0.78 0.12 0.05 0.18 0.02 0.02 0.49 13.02 0.21

Los Angeles, CA Sample price lnprice sqft (in 100's) bath age lotacre bedrms perc_nwhite perc_und18 perc_ownocc perc_vacant norm_popdens year_2007 mathread lmathread

147,112 147,112 147,112 147,112 147,112 147,112 147,112 147,112 147,112 147,110 147,110 147,112 147,112 147,109 147,109

472,373 12.85 17.07 2.13 43.03 0.25 3.18 0.37 0.28 0.67 0.06 0.05 0.46 43.92 3.73

407,611 0.63 767.41 0.86 22.98 0.38 0.87 0.22 0.07 0.21 0.10 0.04 0.50 13.97 0.31

20,886 20,886 20,886 20,886 20,886 20,886 20,886 20,886 20,886 20,886 20,886 20,886 20,886 20,886 20,886

528,224 12.97 16.98 2.14 44.92 0.20 3.20 0.42 0.28 0.70 0.04 0.06 0.46 46.12 3.77

458,725 0.59 781.55 0.89 21.61 0.22 0.86 0.22 0.07 0.20 0.06 0.04 0.50 16.42 0.35

Detroit, MI Sample price lnprice sqft (in 100's) bath age lotacre bedrms perc_nwhite perc_und18 perc_ownocc perc_vacant norm_popdens year_2007 mathread lmathread

32,523 32,523 32,523 32,523 32,523 32,523 32,523 32,523 32,523 32,523 32,523 32,523 32,523 32,523 32,523

194,602 11.99 16.57 2.06 46.05 0.36 3.15 0.13 0.25 0.80 0.04 0.03 0.48 73.83 4.29

134,342 0.64 779.06 1.00 23.25 0.52 0.73 0.18 0.06 0.18 0.03 0.02 0.50 12.34 0.18

9,948 9,948 9,948 9,948 9,948 9,948 9,948 9,948 9,948 9,948 9,948 9,948 9,948 9,948 9,948

184,270 11.93 15.88 1.98 47.52 0.30 3.10 0.12 0.25 0.81 0.03 0.03 0.48 72.56 4.27

131,028 0.64 775.47 1.00 21.25 0.42 0.72 0.16 0.06 0.16 0.02 0.02 0.50 12.17 0.18

38

Table 4: Summary Statistics of 2003 Levels and 2008-2003 Differences in Block Groups Philadelphia, PA Sample Variable

Obs

Mean

Std. Dev.

Los Angeles, CA Sample Obs

Mean

Detroit, MI Sample

Std. Dev.

Obs

Mean

Std. Dev.

350,326 0.56 607.33 0.66 19.69 0.29 0.60 14.14 0.33

1,477 1,477 1,477 1,477 1,477 1,477 1,477 1,477 1,477

214,626 12.16 1626.63 2.01 46.65 0.39 3.14 68.01 4.20

123,303 0.48 609.00 0.75 18.57 0.49 0.48 12.18 0.20

1,477 1,477 1,477 1,477 1,477 1,477 1,477 1,477 1,477

-44,797 -0.27 40.22 0.04 0.45 0.00 0.04 10.80 0.16

70,139 0.27 355.06 0.46 9.50 0.28 0.48 3.70 0.07

2003 Levels price lnprice sqft bath age lotacre bedrms mathread lnmathread

1,529 1,529 1,529 1,529 1,529 1,529 1,529 1,529 1,529

258,477 12.33 1956.24 2.14 49.10 0.41 3.39 64.78 4.13

144,337 0.53 619.25 0.75 22.10 0.45 0.47 15.01 0.30

6,977 6,977 6,977 6,977 6,977 6,977 6,977 6,977 6,977

486,474 12.92 1625.04 1.99 53.08 0.23 3.07 39.11 3.61

2007 - 2003 Differences d_price d_lnprice d_sqft d_bath d_age d_lotacre d_bedrms d_mathread d_lnmathread

1,529 1,529 1,529 1,529 1,529 1,529 1,529 1,529 1,529

42,496 0.16 -39.79 -0.03 0.63 0.00 -0.04 10.16 0.17

65,481 0.22 344.17 0.39 11.67 0.24 0.40 5.70 0.14

6,977 6,977 6,977 6,977 6,977 6,977 6,977 6,976 6,976

65,123 0.09 -18.95 -0.03 0.05 -0.01 0.00 9.28 0.24

208,848 0.24 367.78 0.46 9.69 0.14 0.54 2.37 0.09

39

Table 5: Correlations Between 2007- 2003 Differences and 2003 Levels Philly Correlations d_lnprice d_sqft d_bath d_age d_lotacre d_bedrms d_lnmathread LNPRICE_03 SQFT_03 BATH_03 AGE_03 LOTACRE_03 BEDRMS_03 LMATHREAD_03

d_lnprice d_sqft d_bath d_age d_lota~e d_bedrms d_lmathreaLNPRICE_SQFT_03 BATH_03 AGE_03 LOTACREBEDRMS_LNMATHR 1.00 0.45 1.00 0.40 0.60 1.00 -0.23 -0.22 -0.42 1.00 0.26 0.25 0.12 0.04 1.00 0.20 0.49 0.40 0.02 0.12 1.00 0.08 -0.01 -0.01 0.01 -0.01 -0.02 1.00 -0.33 -0.11 -0.09 0.03 -0.03 -0.01 -0.31 1.00 -0.24 -0.29 -0.18 0.06 -0.05 -0.10 -0.17 0.78 1.00 -0.25 -0.18 -0.27 0.10 -0.01 -0.09 -0.39 0.79 0.83 1.00 0.17 0.07 0.13 -0.24 0.01 -0.04 0.12 -0.48 -0.39 -0.58 1.00 -0.16 -0.10 -0.06 -0.01 -0.27 -0.02 -0.16 0.48 0.50 0.44 -0.40 1.00 -0.13 -0.18 -0.15 -0.05 -0.03 -0.37 0.14 0.38 0.62 0.40 0.01 0.15 1.00 -0.11 -0.01 -0.01 0.00 0.02 0.01 -0.91 0.47 0.28 0.50 -0.18 0.26 -0.11 1.00

LA Correlations d_lnprice d_sqft d_bath d_age d_lotacre d_bedrms d_lnmathread LNPRICE_03 SQFT_03 BATH_03 AGE_03 LOTACRE_03 BEDRMS_03 LMATHREAD_03

d_lnprice d_sqft d_bath d_age d_lota~e d_bedrms d_lmathreaLNPRICE SQFT BATH AGE LOTACREBEDRMS LNMATH~ 1.00 0.39 1.00 0.29 0.73 1.00 -0.14 -0.31 -0.36 1.00 0.18 0.15 0.07 0.03 1.00 0.25 0.61 0.58 -0.19 0.05 1.00 0.10 0.04 0.04 -0.02 0.00 0.02 1.00 -0.08 -0.18 -0.13 0.06 0.00 -0.11 -0.30 1.00 -0.09 -0.38 -0.26 0.09 -0.04 -0.23 -0.29 0.69 1.00 -0.13 -0.33 -0.39 0.14 0.00 -0.24 -0.32 0.62 0.88 1.00 0.28 0.05 0.06 -0.23 0.01 0.02 0.32 -0.06 -0.35 -0.51 1.00 -0.02 -0.01 0.00 -0.05 -0.23 0.01 -0.14 -0.03 0.24 0.18 -0.25 1.00 -0.22 -0.32 -0.27 0.09 -0.03 -0.45 -0.25 0.41 0.69 0.72 -0.44 0.09 1.00 -0.11 -0.05 -0.05 0.04 0.02 -0.05 -0.86 0.45 0.33 0.36 -0.32 0.08 0.29 1.00

Detroit Correlations d_lnprice d_sqft d_bath d_age d_lotacre d_bedrms d_lnmathread LNPRICE_03 SQFT_03 BATH_03 AGE_03 LOTACRE_03 BEDRMS_03 LMATHREAD_03

d_lnprice d_sqft d_bath d_age d_lota~e d_bedrms d_lmathreaLNPRICE SQFT BATH AGE LOTACREBEDRMS LNMATH~ 1.00 0.40 1.00 0.32 0.62 1.00 -0.22 -0.19 -0.29 1.00 0.15 0.12 0.07 0.02 1.00 0.17 0.45 0.43 -0.10 0.00 1.00 -0.04 0.01 0.06 -0.08 0.01 0.01 1.00 -0.02 -0.08 -0.06 0.07 -0.07 -0.07 -0.56 1.00 0.03 -0.24 -0.12 0.07 -0.07 -0.11 -0.38 0.78 1.00 0.06 -0.14 -0.24 0.10 -0.05 -0.12 -0.46 0.79 0.90 1.00 -0.01 -0.01 0.05 -0.25 0.03 0.04 0.38 -0.46 -0.47 -0.56 1.00 0.05 0.05 0.01 0.05 -0.35 0.00 -0.18 0.29 0.30 0.31 -0.39 1.00 0.01 -0.12 -0.11 0.06 -0.02 -0.39 -0.26 0.51 0.69 0.68 -0.33 0.13 1.00 0.12 0.01 -0.01 0.06 -0.01 -0.02 -0.84 0.71 0.45 0.51 -0.42 0.23 0.30 1.00

40

Table 6: Regression Results for Implicit Prices of School Quality (1)

COEFFICIENT sqft (in 100's) sqft2 (in 100's) bath bath2 age age2 lotacre lotacre2 bedrms bedrms2 year_2007 lmathread lmathread2 Constant County FE's Boundary FE's # of Boundaries Wald Test lmathread2 (Prob > F) Observations R-squared

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Philadelphia, PA Sample

Los Angeles, CA Sample

Detroit, MI Sample

All houses 0.35 miles 0.35 miles 0.20 miles

All houses 0.35 miles 0.35 miles 0.20 miles

All houses 0.35 miles 0.35 miles 0.20 miles

0.024*** (0.001) 0.001 (0.001) 0.078*** (0.005) 0.002 (0.007) -0.002*** (0.000) 0.001*** (0.000) 0.064*** (0.005) 0.028*** (0.008) 0.060*** (0.005) -0.019*** (0.007) -1.882*** (0.123) 0.597*** (0.014) 0.441*** (0.027) 9.439*** (0.064)

0.024*** (0.001) 0.004** (0.002) 0.091*** (0.009) -0.023* (0.013) -0.002*** (0.000) 0.001** (0.000) 0.064*** (0.014) 0.047** (0.020) 0.061*** (0.011) -0.028** (0.014) -1.606*** (0.235) 0.551*** (0.032) 0.383*** (0.051) 9.462*** (0.143)

X

X

NA

0.021*** 0.020*** (0.001) (0.002) 0.003** 0.004** (0.002) (0.002) 0.063*** 0.063*** (0.008) (0.011) -0.010 -0.004 (0.012) (0.015) -0.004*** -0.004*** (0.000) (0.000) 0.001** 0.001** (0.000) (0.001) 0.114*** 0.099*** (0.013) (0.019) 0.030* 0.053** (0.018) (0.025) 0.057*** 0.071*** (0.010) (0.014) -0.026** -0.047*** (0.012) (0.017) -0.003 0.215 (0.228) (0.319) 0.209*** 0.209*** (0.037) (0.056) 0.011 -0.021 (0.050) (0.070) 11.094*** 10.994*** (0.165) (0.250)

0.033*** (0.000) 0.006*** (0.001) 0.078*** (0.003) 0.013*** (0.005) 0.000*** (0.000) 0.003*** (0.000) -0.078*** (0.006) 0.053*** (0.009) -0.007*** (0.002) -0.024*** (0.003) -0.372*** (0.037) 0.243*** (0.004) 0.064*** (0.008) 11.614*** (0.020)

0.030*** 0.026*** 0.025*** (0.000) (0.000) (0.000) 0.006*** 0.004*** 0.005*** (0.002) (0.001) (0.002) 0.069*** 0.030*** 0.030*** (0.006) (0.005) (0.007) 0.024** 0.026*** 0.033*** (0.011) (0.009) (0.010) 0.001*** -0.001*** -0.001*** (0.000) (0.000) (0.000) 0.003*** 0.004*** 0.004*** (0.000) (0.000) (0.000) 0.050** 0.137*** 0.147*** (0.020) (0.020) (0.025) 0.072** 0.104*** 0.128*** (0.033) (0.032) (0.044) -0.017*** 0.003 0.008 (0.005) (0.004) (0.005) -0.032*** -0.025*** -0.035*** (0.007) (0.006) (0.008) -0.517*** -0.549*** -0.491*** (0.089) (0.069) (0.086) 0.258*** 0.172*** 0.159*** (0.009) (0.010) (0.012) 0.097*** 0.099*** 0.074*** (0.018) (0.015) (0.018) 11.866*** 12.095*** 12.098*** (0.047) (0.045) (0.056)

X

X

NA

NA

X 153

X 149

0.028*** (0.001) 0.003** (0.001) 0.094*** (0.005) 0.032*** (0.008) -0.002*** (0.000) -0.001*** (0.000) 0.033*** (0.007) 0.012 (0.010) 0.019*** (0.005) -0.012 (0.009) -3.181*** (0.193) 0.762*** (0.019) 0.629*** (0.043) 8.508*** (0.084)

0.029*** (0.001) 0.003 (0.002) 0.096*** (0.009) 0.022 (0.014) -0.003*** (0.000) -0.002*** (0.001) 0.063*** (0.018) -0.016 (0.022) 0.023** (0.009) 0.011 (0.016) -2.942*** (0.315) 0.588*** (0.031) 0.559*** (0.071) 9.217*** (0.135)

0.026*** 0.026*** (0.001) (0.002) 0.003 0.001 (0.002) (0.003) 0.078*** 0.081*** (0.008) (0.010) 0.024* 0.022 (0.013) (0.017) -0.004*** -0.004*** (0.000) (0.000) -0.002*** -0.002*** (0.000) (0.001) 0.114*** 0.144*** (0.020) (0.026) -0.020 -0.018 (0.021) (0.027) 0.019** 0.017 (0.009) (0.011) 0.018 0.014 (0.015) (0.019) -2.329*** -2.532*** (0.303) (0.366) 0.213*** 0.150*** (0.040) (0.048) 0.435*** 0.475*** (0.068) (0.082) 10.797*** 11.114*** (0.181) (0.215)

X

X

NA

NA

X 180

X 167

NA

X 121

X 120

0.000

0.000

0.835

0.766

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

29,333 0.721

6,796 0.693

6,796 0.761

4,191 0.741

147,107 0.705

20,886 0.746

20,886 0.827

12,682 0.834

32,523 0.673

9,948 0.671

9,948 0.716

6,443 0.725

Notes: Census characteristics at the block group level from the 2000 census were also included in the regressions but are omitted from the table. Robust standard errors in parentheses; *** p