Hedonic Regressions ECN741, Urban Economics

Hedonic Regressions ECN741, Urban Economics
Professor John Yinger, The Maxwell School, Syracuse University, 2018

What is a hedonic regression? The Rosen framework
Hedonic Regressions Class Outline What is a hedonic regression? The Rosen framework The endogeneity problem Dealing with omitted variables A new approach: Deriving the bid-function envelope Introduction

Hedonic Regressions Introduction A regression of house value or rent on housing and neighborhood characteristics is called a hedonic regression. Because these regressions can be used to shed light on many important issues and because data on house sales are widely available, hedonic regressions are perhaps the most widely used empirical tool in urban economics and local public finance. Introduction

Hedonic regressions can address three levels of analysis:
Introduction Hedonic regressions can address three levels of analysis: 1. The impact of housing and neighborhood traits on house values (which is a key issue in studying housing markets and property tax assessing). 2. The demand for public services and neighborhood amenities as reflected in house values (which is linked to benefit-cost analysis of these amenities and public services). 3. The sorting of household types with different amenity demands into different neighborhoods (which is determined by their housing bids). Introduction

Hedonic Regressions Hedonic Applications Hedonic regressions for housing have been used, for example, to study household demand for: The quality of public schools Clean air Neighborhood safety Access to worksites Neighborhood ethnic composition Introduction

Hedonic Regressions The Rosen Framework Most studies follow a famous paper by Sherwin Rosen (JPE, Jan./Feb.1974). This paper distinguishes between A household bid function, which is an iso-utility curve for (in our terms) P and S and which is exactly what we have been studying in this class. The observed price function or hedonic, which is the envelope of the underlying bid functions. The Rosen Framework

Hedonic Regressions The Rosen Framework, 2 In Rosen, θ is a bid, z is a trait, u is utility, and p is price (=envelope). His famous picture is: The Rosen Framework

Hedonic Regressions The Rosen Framework, 3 Note that in this picture, the bid functions, the θs, depend on household traits, as indicated by the utility level, ui*. But the hedonic price function, which is the envelope of the bid functions, does not contain any household-level information. Hence, it is impossible to extract much demand information directly from the hedonic. The Rosen Framework

Rosen also models the supply side, with offer curves,Φ.
Hedonic Regressions The Rosen Framework, 4 Rosen also models the supply side, with offer curves,Φ. The Rosen Framework

Hedonic Regressions The Rosen Framework, 5 The market equilibrium p is a “joint envelope” of the bid and offer curves, and hence may be very complicated. Epple (JPE, February1987) presents a joint envelope but it requires an unusual utility function and very strong assumptions about the distribution of bid and offer curves. The Rosen Framework

Indeed, Rosen (p. 40) recognized this link:
Hedonic Regressions The Rosen Framework, 6 This framework is perfectly consistent with the local public finance theory covered in previous classes. Indeed, Rosen (p. 40) recognized this link: “A clear consequence of the model is that there are natural tendencies toward market segmentation, in the sense that consumers with similar value functions purchase products with similar specifications. In fact, the above specification is very similar in spirit to Tiebout’s (1956) analysis of the implicit market for neighborhoods, local public goods being the “characteristics” in this case.” The Rosen Framework

This framework is also consistent with the logic of urban models.
Hedonic Regressions The Rosen Framework 7 This framework is also consistent with the logic of urban models. In studying access to jobs, we derived bid functions and envelopes. Now we are applying these concepts to a broader set of locational traits. Rosen, by the way, cites Alonso, too. The Rosen Framework

The Rosen Framework, 8 Hedonic Regressions
Although consistent with the local public finance theory, the Rosen framework was not specifically designed for housing markets. Hence, the supply side does not fit very well. Housing suppliers are generally not producing new housing; most housing comes from the existing supply. Suppliers (or buyers) can adjust housing traits (in H), but not neighborhood traits (in P). Suppliers want the highest P per unit of H, so sorting should depend only on P (as in urban models). An elaborate model of housing supply is therefore not necessary to apply the Rosen framework—but the distinction between P and H helps. The Rosen Framework

A Common Misunderstanding
Hedonic Regressions A Common Misunderstanding Despite Rosen’s fame, many scholars estimate an hedonic function (the envelope) and interpret the estimated coefficients as measures of willingness to pay (bids). As indicated earlier, however, Rosen’s diagram clearly shows that the envelope reflects both movement along a bid function and shifts in the bid function due to sorting. Hence, willingness to pay cannot be estimated without separating bidding and sorting. The Rosen Framework

A Common Misunderstanding, 2
Hedonic Regressions A Common Misunderstanding, 2 As Rosen pointed out, the tangency in his picture indicates that a household is setting its MWTP for the amenity equal to the implicit price, which is the slope of the envelope. Thus, we can observe an individual’s MWTP at the level of the amenity they consume—and we can average this across households. But this is not very meaningful. Few policy interventions raise the amenity a tiny amount for every household. Any estimate of average MWTP is dependent on the underlying hedonic equilibrium (and may change if the equilibrium changes, e.g. with immigration). So this Rosen result must be used carefully!!! The Rosen Framework

More Misunderstanding
Hedonic Regressions More Misunderstanding Other scholars think they have solved this problem because they observe changes over time. They regress ΔV on ΔS and claim to have found willingness to pay for the change. This is not true. The change in S involves movement along the demand curve and therefore changes MWTP. Moreover, a change in S could lead to re-sorting so that the people bidding in the second period are different from the people bidding in the first. Hence the change in bids may mix willingness to pay for the change in S with changing to a different set of households with different preferences. The Rosen Framework

Rosen proposes a two-step approach to estimating hedonic models.
Hedonic Regressions The Rosen Two-Step Rosen proposes a two-step approach to estimating hedonic models. Step 1: Estimate a hedonic regression (the envelope) and differentiate the results to find the implicit or hedonic price, ∂V/∂S ≡ VS, for each amenity, S. Step 2: Estimate the demand for amenity S as a function of VS (and other things). Step 2 Alternate: Estimate the inverse demand function, which is VS as a function of S. The Rosen Framework

Principal Challenge: Endogeneity
Hedonic Regressions Principal Challenge: Endogeneity As Epple (JPE, Feb. 1987) and other scholars have pointed out, the main problem facing a 2nd step regression in the Rosen framework is that the implicit price is endogenous. The hedonic function is undoubtedly nonlinear, so households “select” an implicit price when they select a level of S (and if the hedonic is linear, it yields no variation in VS with which to estimate demand!) Households have different preferences, so the level of S, and hence of VS, they select depends on their observed and unobserved traits. Endogeneity

One way to see this is to look at a graph of the slopes.
Hedonic Regressions Principal Challenge, 2 One way to see this is to look at a graph of the slopes. Bid-function slopes indicate marginal willingness to pay, so they plot out a demand curve. But envelope slopes also reflect increases in bid-function steepness as sorting occurs, which is represented in the following picture by the upward shifts in the dotted lines. Endogeneity

Hedonic Regressions Bidding and Sorting Endogeneity

Dealing with Endogeneity in Hedonics
Hedonic Regressions Dealing with Endogeneity in Hedonics Some articles find instruments for VS in the 2nd step, usually from geographic price variation (e.g. using prices in neighboring tracts as instruments). See the review by Sheppard in the Handbook of Urban and Regional Economics. But most scholars are now nervous about this approach because sorting leads to correlations across locations. A variety of alternatives have been proposed…. Endogeneity

Selected Recent Contributions, 1
Hedonic Regressions Selected Recent Contributions, 1 Epple and Sieg (JPE, Aug. 1999), Epple, Romer, and Sieg (Econometrica, Nov. 2001) These scholars solve a general equilibrium model of bidding, sorting, and public service determination with specific functional forms. Their model includes an income distribution and a taste parameter with an assumed distribution. They solve for percentiles of the income distribution (and other things) in a community as a function of the parameters and then estimate the values of the parameters that best approximate the income distribution in the communities in the Boston area. Endogeneity

Selected Recent Contributions, 1A
Hedonic Regressions Selected Recent Contributions, 1A Epple, Peress, and Sieg (AEJ: Micro, Nov ) The latest effort by Epple and colleagues estimates a semi-parametric equation derived from the same type of general equilibrium model. They estimate this model with housing sales data from Pittsburgh—a huge empirical improvement. This technically sophisticated approach involves sorting based on income and a single taste parameter (each with an assumed distribution). Endogeneity

Hedonic Regressions E/P/S, 2 One of the great advantages of this approach is that it can account for “inexact” sorting in which one household type lives in many locations or many types share a single location. Their model clearly specifies the conditions for locational equilibrium, which they call “boundary indifference,” “stratification,” and “ascending bundles.” These are just formal versions of the sorting conditions at the heart of local public finance. Recent Studies

Hedonic Regressions E/P/S, 3 However, E/P/S assume households value a single amenity index (based on school quality, air quality, and crime). Suppose A = α1A1 + α2A2. Then 100/α1 units of A1 and no units of A2 yield the same utility as 100/α2 units of A2 and no units of A1—or as an appropriately weighted mix of the two. Moreover, their estimation method does not yield parameters that describe the sorting equilibrium—at least not ones that are easy to interpret. Recent Studies

Hedonic Regressions Selected Recent Contributions, 2 Ekeland, Heckman, and Nesheim (JPE , Feb ); and Heckman, Matzkin, and Nesheim (Econometrica, Sept. 2010) They use fancy nonparametric techniques to estimate the hedonic equation. They then identify the bid function based on the fact that the bid function and the envelope have different curvature. This complex approach has not been applied to housing and involves some strong assumptions (including the same index assumption as in Epple et al.). Endogeneity

Hedonic Regressions Selected Recent Contributions, 3 Bajari and Kahn (J. Bus. and Econ. Stat. 2005) They show that the endogeneity can be eliminated when the price elasticity of demand for the amenity equals -1. They estimate a general form for the first- step hedonic (without showing what it looks like!), then assume unitary price elasticities and estimate the second-step demand functions. These two steps may not be consistent. Endogeneity

The Bajari/Kahn Assumption
Hedonic Regressions The Bajari/Kahn Assumption The Rosen two-step method estimates PS, which a household sets equal to MBS. With constant elasticity demand and μ = -1, This equation does not have an endogenous variable on the right side. Endogeneity

An Envelope for the Bajari/Kahn Assumption
Hedonic Regressions An Envelope for the Bajari/Kahn Assumption A method discussed later can derive an envelope using the B/K assumption: μ = -1. In the case of a linear sorting equilibrium (to be explained later), the form is This equation does not look a lot like local linear regression! Endogeneity

Hedonic Regressions Selected Recent Contributions, 4 Bayer, Ferreira, McMillan (JPE, August 2007) These authors estimate a fancy multinomial choice model of sorting. Their econometrics is fancy and they can do some cool simulations with their model, but some aspects of their model are simplistic (e.g., linear utility functions!). They also estimate a linear hedonic; more on this in a future class. Endogeneity

A Second Major Challenge
Hedonic Regressions A Second Major Challenge Another major challenge in estimating Rosen’s 1st step is omitted variable bias. Many variables influence house values and leaving out key variable can obviously bias estimated implicit prices and coefficients of interest. One approach is to devise various fixed- effects strategies. Another is to collect extensive information on housing and neighborhood traits. Omitted Variables

Hedonic Regressions Border Fixed Effects One strategy made famous by Black (QJE , May ) is called border fixed effects (BFEs). Identify houses near school attendance zone boundaries and define a fixed effect for each boundary segment. Regress house value on school quality controlling for these BFEs. These BFEs account for neighborhood traits that spill over each boundary. See if the results depend on distance from the boundary. Omitted Variables

Hedonic Regressions Border Fixed Effects, 2
BFEs are is a clever strategy to remove fixed effects with cross-section data. It has since been used by dozens of housing hedonic studies. We will return to BFEs in the next class when we look at some specific studies, but it should be noted from the onset that BFEs have some serious limitations. First, they may do little to eliminate the bias from omitted neighborhood traits, Second, they greatly cut the sample size, Third, they change the question addressed by a hedonic regression and are therefore easy to misinterpret. Omitted Variables

Border Fixed Effects, 3 Hedonic Regressions
Limited controls for omitted variables BFEs limit bias in the coefficient of elementary school quality only of that variable is correlated with the unobserved neighborhood traits that are shared across a border. But differences in elementary school quality lead to sorting, with higher income people on the side with the higher school quality. This sorting may also lead to a differences in other amenities across attendance zone boundaries. BFEs pick up shared amenities, if any, but do not control for amenities that differ across the borders. Omitted Variables

Border Fixed Effects, 4 Hedonic Regressions Restricted sample size
BFEs only use house sales close to an attendance zone boundary, which may be a small share of total sales. Most studies vary the distance from the boundary as a robustness check. Sample selection problems could arise if people housing bids are affected by distance from a boundary, because people expect boundaries to move. Omitted Variables

Border Fixed Effects, 5 Hedonic Regressions Changing the question
A regression with BFEs estimates the impact of differences in elementary school quality on house values within a school district. BFEs capture all sources of variation across school districts, so these differences do not influence the school quality coefficient. Asking whether variation in elementary school quality within a school affects house values is a perfectly legitimate question, but it is not the same as asking about variation in school quality across districts; parents may be more or less concerned, e.g., about the quality of the end point (high school) than about the starting point (elementary school). Omitted Variables

An Alternative Approach: More Data!
Hedonic Regressions An Alternative Approach: More Data! Most hedonic studies have only a few (2 or 3!) control variables for public services and neighborhood amenities. Many studies appear to use BFEs as an excuse not to collect more data. But BFEs provide limited controls, and a better approach is to collect data on many public services and neighborhood amenities. Of course, many variable describing the structural traits of housing are also needed, but it is the amenity variables that usually fall short. Omitted Variables

Hedonic Regressions More Data, A Caveat
As Rosen and others have pointed out, an envelope is a market function that excludes individual demand traits. This is analogous to long-run average cost functions in production theory: the short run functions include plant size, but the long-run function does not. It is a common mistake to include small-neighborhood (e.g. block group) income or education in a hedonic regression, but these are highly correlated with individual demand traits (due to sorting!). It is fine to include amenities correlated with income, such as parks and golf courses. Omitted Variables

Hedonic Regressions A Caveat, 2
If individual or small-area demand traits are included, one is estimating a bid function regression, not an envelope. Following the production analogy, this is like estimating short-run cost functions. This is a legitimate approach in principle, but note that it requires two steps that are never taken: The classic endogeneity problem now appears in the first-stage bid-function regression; demand traits and bids are simultaneously determined; good luck finding instruments! The regression must include interactions between the amenity and the demand traits; otherwise, every household, regardless of demand traits, has the same bid-function slope and there can be no sorting! Omitted Variables

An Alternative Approach: Double Sales
Hedonic Regressions An Alternative Approach: Double Sales Some studies have panel data and can identify houses that sold twice. This makes it possible to switch to a change form of the regression. All time-invariant housing and neighborhood traits are differenced out. But the results are then just based on variation over time, which might differ from cross-section effects. and changes in housing and neighborhood traits could be a source of bias. Omitted Variables

Hedonic Regressions The Yinger Approach The Yinger approach draws on standard models of local public finance to solve several of these problems. The key insight is that once a bid function is specified, it may be possible to derive and estimate the envelope of the bid functions for heterogeneous households (given certain assumptions!). This envelope provides information about the underlying bids of individual households. But it also contains information about the way different types of households sort into different neighborhoods. A New Approach

The Payoff Hedonic Regressions
The resulting envelope yields most of the parametric forms in the literature as special cases. Moreover, this approach Avoids the endogeneity problem in the Rosen two-step approach; Does not require extreme assumptions; Eliminates inconsistency between the functional forms of the envelope and of the underlying bid functions; Characterizes household heterogeneity in a general way and makes it possible to test hypotheses about the sorting process. A New Approach

where C is a constant and
Hedonic Regressions Bidding Review Recall that with constant-elasticity demand functions for a public service (S) and housing services (H), the before-tax bid for H is where C is a constant and A New Approach

And ψ is an index of the relative slope of a household’s bid function.
Hedonic Regressions Bidding Review 2 In these formulas, the price elasticity of demand for public services, μ, is the main parameter of interest, And ψ is an index of the relative slope of a household’s bid function. It contains all the information from a household’s demand functions for S and H that influences the slope of the bid function and is not shared by other households at a given S. A New Approach

Deriving the Envelope: Step 1
Hedonic Regressions Deriving the Envelope: Step 1 We can now derive the bid-function envelope in two steps. The first step recognizes that the bid function derived above does not have an envelope as written because all households have the same intercept. To ensure than an envelope exists, we need to make the intercept a function of ψ . We must derive C{ψ} such that, at a point where two bid-functions cross, the difference in C between the two bid functions is consistent with the difference in their slopes. Consider, as in the following diagram, two bid functions that cross at S = S*. A New Approach

Step 1 for Deriving an Envelope
Hedonic Regressions Step 1 for Deriving an Envelope A New Approach

Step 1: Solving for the Constant
Hedonic Regressions Step 1: Solving for the Constant To find dC/dψ we must differentiate the bid function with respect to C and ψ holding S constant and set the result equal to zero. With the above form for the bid function, the result is A New Approach

Step 2: Bringing in some Economics
Hedonic Regressions Step 2: Bringing in some Economics This result is a differential equation in ψ. Because it includes S{ψ}, we cannot solve this differential equation unless we know how S and ψ are related. This is where the theory of local public finance comes in. The most basic theorem from the consensus model is that people sort according to the slopes of their bid functions, which implies that S is a monotonic, upward-sloping function of ψ. A New Approach

This function was illustrated in a previous figure:
Hedonic Regressions Step 2 Continued We do not know the form of this relationship, so my original strategy was to write down the most general approximation for a monotonic relationship that results in a tractable differential equation. This form is: where the σ’s are parameters to be estimated and we can test whether, as predicted, σ2 > 0. This function was illustrated in a previous figure: A New Approach

Hedonic Regressions Bidding and Sorting Endogeneity

Hedonic Regressions Step 2, Continued
It turns out that there is also a theoretical basis for this equilibrium relationship. Define one-to-one matching as an equilibrium in which each household type, defined by ψ, has its own value of S. This is, of course, an extreme case, but it may be approximately right in many settings. With one-to-one matching, the equilibrium relationship between ψ and S does not depend on the distributions of ψ and S, but instead depends on the nature of the transformation from one of these distributions to the other. Epple et al. do not assume one-to-one matching (that is, they allow one household type to live with different values of S or many household types to have the same value of S), but their method is complex! A New Approach

Suppose both distributions are normal.
Hedonic Regressions Step 2, Continued Suppose both distributions are normal. Then a linear transformation will convert the distribution of ψ into the distribution of S (or vice versa), and the equilibrium relationship between ψ and S will be linear: Adding σ3 opens the door to many other distributions and transformations. See Yinger (JHE, September 2015). A New Approach

Hedonic Regressions The Final Envelope With the help of this approximation for S{ψ} we can solve the above differential equation for C. Then we substitute the solution for C and the market equilibrium expression for ψ into the expression for the bid function. The result is the envelope, which is P as a function of S with the demand factors (ψ) removed and four parameters (μ, σ1, σ2, σ3, ) to be estimated for each amenity. Approach

A Note on the Supply Side
Hedonic Regressions A Note on the Supply Side The S{ψ} function approximates the market equilibrium, so it captures both supply and demand. Regardless of the supply side, the market price function is an envelope of the underlying bid functions: Rosen’s p is a joint envelope. Moreover, the sorting theorem (that sorting depends on bid function slopes) does not require any assumptions about the supply side. The supply side affects the number of people in a jurisdiction, but this connection does not alter the sorting theorem. The supply side surely affects the parameters of the equilibrium approximation, the σ’s, but it does not alter the interpretation of the estimated μ’s. A New Approach

The envelope that results has Box-Cox forms:
Hedonic Regressions The Envelope Equation The envelope that results has Box-Cox forms: where A New Approach

Hedonic Regressions A New Approach

Hedonic Regressions Special Cases This general Box-Cox specification includes most of the parametric estimating equations in the literature as special cases. On the left side, the assumption that the price elasticity of demand for housing, ν, equals -1 leads to a log form, which is used by most studies. Studies that use this form do not recognize that they are making this assumption about ν . On the right side, a wide range of functional forms are possible depending on the values of μ and σ3. A New Approach

Special Cases, Continued
Hedonic Regressions Special Cases, Continued A New Approach

Sorting and Specification
Hedonic Regressions Sorting and Specification Note that any specification that is consistent with sorting (except the last) requires two terms. The quadratic special case is an example. Despite its apparent generality, a standard Box-Cox rules out sorting unless S is proportional to ψ. The price elasticity cannot be estimated without a non-linear specification. Simple forms are based on an assumption about the price elasticity. A simple form therefore does not make sense for the Rosen two-step approach because any 2nd step estimate of the price elasticity will contradict the 1st step assumption. A New Approach

Extension to Multiple Amenities
Hedonic Regressions Extension to Multiple Amenities So long as Si is not directly a function of Sj, this approach can be extended to multiple amenities, and the LaFrance results about underlying utility functions still hold. This approach assumes that amenity space is dense enough so that we can pick up bidding for Si holding other amenities constant. Highly correlated amenities may need to be combined into an index, by assumption or using something like principal components. A New Approach

Combining bids and housing services yields
Hedonic Regressions The Hedonic Equation Combining bids and housing services yields To estimate this equation: Extend the envelope to multiple amenities. Assume a multiplicative form for H{X} Introduce the property tax rate (τ) and the degree of property tax capitalization (β). A New Approach

Hedonic Regressions Hedonic Vices The huge hedonic literature seems to have lost touch with this theory. As a result, most studies, including those in leading journals, have a series of “hedonic vices,” that is, weaknesses in their theoretical underpinnings. For more, see Nguyen-Hoang and Yinger (JBCA, July 2016) Hedonic Vices

Hedonic Vices: Specification A
Hedonic Regressions Hedonic Vices: Specification A One-variable hedonic specifications rule out sorting. Suppose households have linear bid functions— i.e., bid functions with constant slopes. Sorting arises when higher-demand households win the competition at higher amenity levels because they have steeper bid functions, which implies that the slope of the hedonic rises with the amenity level. If the hedonic is estimated with a constant slope, this sorting process is ruled out. Hedonic Vices

Hedonic Vices: Specification B
Hedonic Regressions Hedonic Vices: Specification B The hedonic specification may be inconsistent with the specification of the second-step demand functions. With Yinger’s assumptions, a quadratic envelope implies an infinite price elasticity of demand. Estimating the price elasticity using implicit prices from a quadratic envelope is therefore inconsistent. Hedonic Vices

Hedonic Vices: Control Variables
Hedonic Regressions Hedonic Vices: Control Variables Hedonic envelopes should not include demand variables. Including demand variables turns the regression into a bid-function regression. A bid function regression must deal with the fundamental endogeneity between prices and amenities. A bid function regression must interact demand variables with amenities—or else everyone has the same bid-function slope and there is no sorting! Hedonic Vices

Hedonic Vices: Interpretation
Hedonic Regressions Hedonic Vices: Interpretation A properly specified hedonic yields average MWTP. But this only applies to an equal marginal change at all levels of the amenity starting from the current equilibrium. A change form of the regression does not yield average MWTP, despite the claims in famous papers. Adding border fixed effects changes the meaning of the regression; this step does not simply lower omitted variable bias (and it does not address the main bias issues). Hedonic Vices

In the next class, we will turn to specific empirical studies.
Hedonic Regressions Preview In the next class, we will turn to specific empirical studies. With the insights obtained from the literature on local public finance and hedonics, we will review the methods and findings of several key studies. We will also review school capitalization results based on the Yinger envelope method applied to the Cleveland area data set. Preview

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