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 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

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 much 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 (JPE 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 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

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 (or buyers) can adjust housing traits (in H), but not neighborhood traits (in P).

◦ Most scholars assume the implicit prices of housing traits are fixed (= no submarkets).

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 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 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

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 As Epple (JPE 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

Principal Challenge, 2 One way to see this is to look 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

BiddingandSorting

Endogeneity

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, 1A

Epple, Peress, and Sieg (AEJ: Micro 2010)

◦ The latest effort by Epple and colleagues derives a nonparametric model with similar assumptions.

◦ They estimate this model with housing sales data from Pittsburgh.

◦ This technically sophisticated approach to sorting has a single index of public service quality (which combines school, police, and commuting data).

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, and involves some strong assumptions (including the same index assumption as in Epple et al.).

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 (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 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

SPK 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 in a future class.

Endogeneity

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

Border Fixed Effects One strategy made famous by Black (QJE 1999) is

called boundary 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

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 account for sorting; the fact that higher-demand households live on the side of the border (the side with good schools!) suggests that the assumption of equal amenities may not be very accurate.

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!

◦ One cannot solve a bias problem with a method that changes the meaning of the regression!

Omitted Variables

Solution to the BFE Sorting Problem?, 2 It also is not clear that home buyers can

observe their potential neighbors’ incomes.

◦ Do buyers consult census data? Perhaps brokers provide it to them.

Income might be a proxy for neighborhood traits that can be observed (e.g. golf courses).

◦ So why not include observable neighborhood traits, which do not cause trouble, instead of income!

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 (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 The envelope I derive yields most of the parametric

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

( ) 1 if 0 and ln{ } if 0XX X

1 211 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 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

A New Approach

Step 1 for Deriving an Envelope

Bid Function 1 Bid Function 2

SS*

dCdψ

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

{ }

{ }1S S

dC Sd

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 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 estimatedand we can test whether, as predicted, σ2 > 0.

This function was illustrated in a previous figure:

A New Approach

31 2{ } ( )S

BiddingandSorting

Endogeneity

Step 2, Continued

Last summer, I discovered a new way to derive the 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

Step 2, Continued

An example: 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.

A New Approach

1 2{ }S

The Final Envelope

Now 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 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 1

32( ) ( )( )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

Per

Uni

t of

Hou

sing

Ser

vice

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

1 ln{ } logS

= -∞ = 1 21

2 2

12

quadraticS 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.

◦ Despite its apparent generality, a standard Box-Cox rules out sorting because it just has one term.

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

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 XVr r

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.

Let’s go over the key ones:

Hedonic Vices

Hedonic Vices: Specification One-variable hedonic specification rule out

sorting.

The hedonic specification may be inconsistent with the specification of the second-step demand functions.

Hedonic Vices

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-functions slope and there can be no sorting!

Hedonic Vices

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

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