ECN741, Urban EconomicsProfessor Yinger

Hedonic Regressions:How Housing Prices Reflect the Demand for Public Services and

Neighborhood Amenities

Introduction

A regression of house value or rent on housing and neighborhood characteristics is called a hedonic regression.

Because house values reflect households’ bids for housing in different locations, this type of regression has been used to study:

◦ Household demand for public services and locational amenities

◦ The benefit side in a benefit-cost analysis of these services and amenities

Introduction

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

Lecture Overview

This lecture reviews the literature on hedonic regressions and presents a new approach to hedonics that draws on the theory of local public finance.

Outline

◦ The Rosen framework◦ The endogeneity problem◦ Dealing with omitted variables◦ A new approach: deriving the bid-function

envelope

Introduction

The Rosen Framework

Most studies follow, a famous paper by Sherwin Rosen (JPE 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 I presented in the previous lecture.

◦ The observed price function or hedonic, which is the envelope of the underlying bid functions.

The Rosen Framework

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

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 demand information directly from the hedonic.

The Rosen Framework

The Rosen Framework 4

Rosen also models the supply side, with offer curves,Φ.

The Rosen Framework

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 (1987) presents a joint envelope but it requires an unusual utility function and very strong assumptions about the distribution of bid and offer curves.

◦ This envelope is quadratic with interactions—as is the utility function.

The Rosen Framework

The Rosen Framework 6

This framework is perfectly consistent with the local public finance theory in my first lecture.

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

The Rosen Framework 7

Although it is 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 are not deciding how much of the “characteristic” to supply but are simply providing housing at a given location.

An elaborate model of housing supply is therefore not necessary to apply the Rosen framework.

The Rosen Framework

A Common Misunderstanding

Despite the fame of the Rosen diagram, many scholars estimate a hedonic function (the envelope) and interpret the estimated coefficients as measures of willingness to pay (bids).

As indicated earlier, however, the 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

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 even average this across households.

But this is not very meaningful.

◦ Few policy interventions raise the amenity a tiny amount and larger changes cannot be valued with MWTP at a point.

◦ Few policy interventions have the same impact on everyone.

So this Rosen result must be used carefully!!!

The Rosen Framework

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 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

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).

The Rosen Framework

Principal Challenge: Endogeneity

As Epple 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

Principal Challenge, 2

One way to see this is to look a graph of the slopes. (Ignore the S{ψ} function on the next slide for now.)

Bid-function slopes indicate marginal willingness to pay, so they plot out a demand curve. The offer function slopes plot out a supply curve.

Thus, observed prices do not describe bid or offer functions—i.e. they do not correspond to demand or supply.

Moreover, demand factors that steepen a bid function affect sorting and hence affect the observed price.

Endogeneity

Endogeneity

Figure 3. Supply and Demand for School Quality

0 10 20 30 40 50 60 70 80 90 100

Bid

Per

Un

it o

f H

ou

sin

g S

erv

ices

Percent Passing

0 10 20 30 40 50 60 70 80 90 100

Slo

pe o

f B

id o

r O

ffer

Fu

ncti

on

Percent Passing

Slope of bid function i

Slope of bid function j

Slope of envelope

Slope of offerfunction k Slope of offer function m

S{ψ}

Bid function j

Bid function i

Offer function m

Offer function k

Envelope

Observed Points

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, vol. 3.

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

Epple and Sieg (JPE 1999), Epple, Romer, and Sieg (Econometrica 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, 2

Ekeland, Heckman, and Nesheim (JPE 2004)

◦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, so far as I know.

Endogeneity

Selected Recent Contributions, 3

Bajari and Kahn (J. Bus. and Econ. Stat. 2004)

◦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, then assume unitary price elasticities and estimate the second-step demand functions.

Endogeneity

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

1/( 1)

or

SS

S S

SP

K Y

P S K Y

Selected Recent Contributions, 4

Bayer, Ferreira, McMillan (JPE, 2007)

◦ These authors estimate a fancy multinomial choice model of sorting.

◦ Their econometrics is fancy, but some aspects of their model are simplistic (e.g., linear utility functions).

◦ They also estimate a linear hedonic; more on this later.

Endogeneity

A Second Major Challenge

A 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

Border Fixed Effects

One strategy made famous by Black (QJE 1999) is called boundary fixed effects (BFE).

◦ 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

A Key Problem with Border Fixed Effects

I’ll have more to say about BFEs in a future class, but for now, one issue is key:

BFEs do not control for sorting, which could be a major source of bias, because different groups are willing to pay different amounts for shared neighborhood traits.

Omitted Variables

Solution to the BFE Sorting Problem?

Bayer, Ferreira, and McMillan (JPE 2007) acknowledge that sorting exists and complicates a BFE approach.

They claim to solve the problem by including neighborhood demographics, including income, as controls; this approach, they say, picks up higher bids in neighborhoods that, due to sorting, have higher-income residents.

But neighborhood income is a demand factor, which belongs in a bid function, not an envelope.

◦ Because of sorting, income is endogenous and a regression that includes income is not a bid-function envelope!

Moreover, home buyers cannot observe their potential neighbors’ incomes (although they may get clues).

Omitted Variables

The Yinger Approach

The approach I am working on 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 is possible to derive and estimate the envelope of the bid functions for heterogeneous households.

◦ 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

The envelope I derive yields most of the forms in the literature as special cases.

Moreover, my 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

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

1 2( ) ( )ˆ{ }P S C S

( ) 1if 0 and ln{ } if 0

XX X

1 2

11 and

11/ ( / )S HK N K M Y

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

We can now derive the bid-function envelope in two steps.

The first step recognizes that when two bid functions cross there is an explicit mathematical link between the difference in their slopes and the difference in their intercepts.

Consider, as in the following diagram, two bid functions that cross at S = S*.

A New Approach

A New Approach

Step 1 for Deriving an Envelope

*ˆ{ }P S

dC

d

S*

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

1

{ }

{ }1

S S

dCS

d

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

Step 2 Continued

We do not know the form of this relationship, so my strategy is 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 estimatedand we can test whether, as predicted, σ2 > 0.

This function was illustrated in a previous figure:

A New Approach

31 2{ } ( )S

A New Approach

Figure 3. Supply and Demand for School Quality

0 10 20 30 40 50 60 70 80 90 100

Bid

Per

Un

it o

f H

ou

sin

g S

erv

ices

Percent Passing

0 10 20 30 40 50 60 70 80 90 100

Slo

pe o

f B

id o

r O

ffer

Fu

ncti

on

Percent Passing

Slope of bid function i

Slope of bid function j

Slope of envelope

Slope of offerfunction k Slope of offer function m

S{ψ}

Bid function j

Bid function i

Offer function m

Offer function k

Envelope

The Final Envelope

Now with the help of this approximation for S{ψ} we can solve the above differential equation for C.

First, we solve the approximation for ψ = S{ψ}, and substitute the result (plus the solution for C) into expression for a bid function.

The result is the envelope, which is the relationship between P and S with the demand factors (ψ) removed and five parameters (C, μ, σ1, σ2, σ3, ) to be estimated.

Approach

A Note on the Supply Side The S{ψ} function approximates the market equilibrium,

so it captures both supply and demand.

Regardless of what happens on the supply side, the market price function is an envelope of the underlying bid functions; remember that 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

A New Approach

The Envelope Equation

The envelope that results has Box-Cox forms:

where

132

( )( )( )1

02 2

1ˆ EP C S S

1

2

3 2 3

( )

( )

1 ;

(1 ) / ;

1/ ;

( 1) / if 0; and

ln{ } if 0.

X X

X X

A New Approach

10 20 30 40 50 60 70 80 90 100

Example of A Bid Function Envelope for School Quality

Percent Passing

Bid

Pe

r U

nit

of

Ho

usin

g S

erv

ice

s

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

A New Approach

Special Cases, Continued

Note: μ = -∞ implies a horizontal demand curve σ3 = ∞ implies no sorting

Value of μ Value of σ3 Formula

= -0.5 = ∞ 1

2

1 1 inverseS

= -1 = ∞ 1

2

1ln{ } logS

= -∞ = 1 21

2 2

1

2quadraticS S

= -∞ = ∞ 1

2

1 linearS

< 0; ≠ -1

= ∞

(1 )/1

2

1 1

(1 ) /Box-CoxS

Sorting and Specification

Note that any specification that is consistent with sorting requires two terms.

◦ The quadratic special case is an example.

◦ The general result with σ3 < ∞ requires 2 Box-Cox terms.

The Bayer et al. article misses this point.

◦ The article develops a discrete-choice approach to sorting.

◦ But also estimates a linear hedonic, which rules out sorting.

◦ And interprets this mis-specified hedonic as an indicator of median preferences—an example of the misunderstanding discussed earlier.

A New Approach

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.

A New Approach

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

ˆ{ , } { } { } { }E EP S H X P S H XV

r r

Remaining Challenges

Avoiding extreme assumptions, such as a linear utility function (Bayer et al.) or unitary price elasticities (Bajari and Kahn).

Avoiding inconsistency between the forms of the envelope and of the underlying bid functions (present in many studies).

Testing hypotheses about sorting, which after all is central to the theories of local public finance and hedonics.

A New Approach

Preview

In later classes, 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.

And I will present my own results for school quality capitalization using my new method and my data set from the Cleveland area.

Preview

Hedonics