Hedonics

1. Public Finance SeminarSpring 2013, Professor John YingerDepartment of Public AdministrationThe Maxwell School, Syracuse University Hedonics These notes are provided for teaching purposes only and are not to be quoted without the permission of the author. Comments are welcome.

2. Hedonics Class Outline • What Are Hedonic Regressions? • The Rosen Framework • Methods for Separating Bidding and Sorting • Hedonic Vices

3. Hedonics Hedonic Regressions • A regression of house value or rent on housing and neighborhood traits is called a hedonic regression. • Hedonic regressions appear in marketsfor other products with multiple attributes, such as automobiles or computers. • But these notes focuses on the application of hedonic analysis to housing markets.

4. Hedonics Hedonic Regressions, 2 • Many of the outcomes of interest in public finance, such as public services (e.g. education) or neighborhood amenities (e.g. air quality) are not traded in private markets. • As a result, we cannot directly observe demand for these outcomes or determine the value households place on changes in these outcomes (as in benefit-cost analysis, for example). • However, households have to compete for access to locations with desirable attributes, and their bids reveal something about the “missing” demand. • Thus, hedonic regressions provide the main way for scholars to study household demand for public services and neighborhood amenities.

5. Hedonics Hedonic Regressions, 3 • Hedonic regressions have been used, for example, to study household demand for: • School quality • Clean air • Protection from crime • Neighborhood ethnicity • Distance from toxic waste sites • Distance from public housing

6. Hedonics Hedonic Regressions, 4 • As we know from the theory of local public finance, different household types also compete against each other for entry into the most desirable neighborhoods (see Ross and Yinger’s Handbook chapter, 1999) • This competition takes place through bids on housing. • So hedonic regressions also can, in principle, tell us something about the way different household types sort across locations—and therefore can give us insight into the nature of our highly decentralized federal system.

7. Hedonics The Rosen Framework • A famous paper by Sherwin Rosen (JPE 1974) gives a framework everyone uses based on a bid functions and their mathematical envelope. • A bid function is the amount a household type would pay for housing at different levels of an amenity, holding their utility constant. • The hedonic envelope is the set of winning bids; each point on the envelope is tangent to one of the underlying bid functions.

8. Hedonics The Rosen Picture • Here is the the Rosen picture, where z1 is a product attribute, θindicates a bid function, and p is the hedonic price function, that is, the envelope.

9. Hedonics Household Heterogeneity • Note that Rosen’s framework is designed to consider heterogeneous households—that is, households with different demands for the amenity as indicated by utility, u*, in the bids. • If all households are alike, the hedonic price function is the same as the bid function. • But with heterogeneous households, we only observe one point—the tangency point—on each household type’s bid function.

10. Hedonics Rosen and Housing • Rosen’s paper is about hedonics in general, not just about housing. • But it is consistent with the bidding and sorting framework developed at about the same time in local public finance (as Rosen mentions). • However, many housing applications make two changes in the Rosen framework: • They ignore the supply side. • They look at bids per unit of housing services.

11. Hedonics Rosen’s Supply Side • In Rosen’s framework, each firm has an offer function, which is the amount it would pay to supply a given level of an attribute, holding profits constant. • With heterogeneous firms and households, the hedonic price function is the joint envelope of firm offer and household bid functions. • Because amenities are not supplied by firms (and because most house sales are of existing housing), the supply side is not central in most housing applications and is simply ignored. • That is, the distribution of amenities is taken as given.

12. Hedonics Housing Services • Rosen treats all attributes the same way. • But many housing scholars distinguish between housing services, H, which are a function of structural housing traits, X, and the price per unit of housing services, P, which is a function of public services and neighborhood amenities, A. • The value of a house is V = P{A}H{X}/r • The index of housing services, H{X}, may be assumed not to vary across households. • You need to look for this distinction when reading the literature.

13. Hedonics The Rosen Challenge • The Rosen framework is valuable because it brings together bidding and sorting. • Just as in the local public finance literature, households with steeper bid functions end up at places with higher amenities. • But this framework also reveals an enormously difficult challenge: • The relationship we can observe—the hedonic price function—reflects both (a) the underlying bid functions and (b) the factors that determine sorting across household types. • Because we only observe one point on each bid function, there is no simple way to separate these two phenomena, that is, no simple way to study either the determinants of household demand for amenities (bidding) or the determinants of household sorting across locations.

14. Hedonics Interpreting the Envelope • Many studies simply estimate the hedonic, that is, the bid-function envelope. • For example, there is a large literature on school quality “capitalization” in which most studies simply regress house values on a measure of school quality (and controls). • Two types of information can be gained from these studies.

15. Hedonics Interpreting the Envelope, 2 • First, a positive, significant coefficient for an amenity indicates that people value that amenity. • If people do not care about an amenity, their bid functions will be flat—and so will the envelope, regardless of sorting. • Results must be interpreted with care. • One can say that a 1 unit increase in school quality leads to an x% increase in house values, all else equal. • One cannot say that people are willing to pay x% more for housing when school quality increases one unit, because the result does not refer to any particular household type and it mixes bidding and sorting.

16. Hedonics Interpreting the Envelope, 3 • Second, Rosen shows that a utility-maximizing household sets its marginal willingness to pay (MWTP) for an attribute equal to the slope of the hedonic price function. • This slope is called the implicit price of the attribute. • In our terms, it is (∂V/∂S) = (∂PE/∂S)H/r. • Hence, we can observe every household’s MWTP at the level of the attribute they consume—and we can calculate the average MWTP. • But this average MWTP has a limited interpretation. • It indicates only what people would pay on average for a small, equal increase in the amenity at all locations, starting from the current equilibrium. • This average cannot be compared across places or across time because the equilibrium is not the same.

17. Hedonics Separating Bidding and Sorting • To go beyond this limited information from the hedonic itself, scholars must separate bidding and sorting. • As just noted, this is inherently a very difficult issue because we only observe one point on each bid function. • Thus, there is no general solution to this problem. • It is impossible to separate bidding and sorting without making some strong assumptions!

18. Hedonics Separating Bidding and Sorting • Scholars have come with 5 different approaches to separating bidding and sorting, based on different assumptions with different strengths and weaknesses. • 1. The Rosen two-step method. • 2. The Epple et al. general-equilibrium method. • 3. Fancy econometric methods (often linked to Heckman and co-authors). • 4. Discrete-choice methods. • 5. The Yinger “derive the envelope” method.

19. Hedonics Method 1: The Rosen Two-Step • Rosen proposes a two-step approach to estimating hedonic models. • Step 1: Estimate a hedonic regression using a general functional form (the envelope) and differentiate the results to find the implicit or hedonic price, VS, for each amenity, S. • Step 2: Estimate the demand for amenity S as a function of VS (and of income and other things).

20. Hedonics Rosen’s First Step • The idea of the first step is to use as general an estimating method as possible to approximate the hedonic envelope. • Early studies used a Box-Cox form, which has linear, double-log, and semi-log forms as special cases. (We will return to this form later.) • Some more recent studies use a nonparametric technique, such as “local linear regression.” • Some studies still just use linear, log, or semi-log, which are undoubtedly not non-linear enough.

21. Hedonics Endogeneity in the Rosen Second Step • As later scholars pointed out (e.g. Epple, JPE 1987), the main problem facing the 2nd step regression in the Rosen framework is that the implicit price is endogenous. • The hedonic function is almost certainly nonlinear, so households “select” an implicit price when they select a level of S. • Households have different preferences, so the level of S, and hence of VS, they select depends on their observed and unobserved traits.

22. Hedonics Addressing Endogeneity in the 2nd Step • Scholars have identified two ways to deal with this endogeneity in the second-step of the Rosen procedure. • The first is to find an instrument, which is, of course, easier said than done. • Some early studies pooled data for metropolitan areas and used construction cost as an instrument; however, hedonics operate at the level of a metropolitan area and this pooling is not appropriate. • Other studies (see the Handbook chapter by Sheppard in 1999) use nearby prices as an instrument; however, nearby prices may, because of sorting, reflect the same unobservables that cause the problem in the first place. • No general solution through instruments has been identified.

23. Hedonics Addressing Endogeneity, 2 • The other approach is to assume that the price elasticity of demand for the amenity equals -1.0. See, for example, Bajari and Kahn (J. Bus. and Econ. Stat.2004). • In this case the demand for S can be written as follows: where Y is income and K is other demand factors. • Because the implicit price now appears on the left-side, nothing endogenous remains on the right side and endogeneity bias disappears. • The problem with this approach, of course is that the price elasticity is the main thing we are trying to estimate!

24. Hedonics Method 2: General Equilibrium • Epple and Sieg (JPE 1999); Epple, Romer, and Sieg (Econometrica 2000). • These scholars derive a general equilibrium model of bidding, sorting, and public service determination with a specific functional form for the utility function. • 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. • The second paper includes the demand for public services through voting.

25. Hedonics Method 2: General Equilibrium, 2 • This approach obviously requires considerable technical skill on modeling and econometrics. • It also makes some strong assumptions. • For our purposes, the most important weaknesses are (1) that the model contains a single index of public services and amenities and (2) that it requires a certain distribution of preferences and income. • Their index includes school quality and the crime rate, with an estimated weight—a very restrictive approach. • So far, the only applications of this method are to community-level data.

26. Hedonics Method 3: Fancy Econometrics • One approach is to use models of random coefficients (i.e. different bid-function slopes) with endogenous variation (due to sorting)—but no theory; the required instruments are hard to find! • The models were developed by Heckman and Vytlacil (JHR 1998), among others. • A hedonic application is Chay and Greenstone (JPE 2005) on air pollution. • Another approach is to use non-parametric techniques that recognize the difference in curvature between bid functions and their envelope. • Ekeland, Heckman, and Nesheim (JPE 2004) • This approach has not yet been applied to housing, so far as I know, and is, like the Epple et al. approach, limited to a single amenity index.

27. Hedonics Method 4: Discrete Choice • Several scholars have pointed out that a hedonic price function can be estimated with a discrete-choice model of the allocation of household types to housing types as a function of housing traits. • A recent article by Bayer, Ferreira, and McMillan (JPE 2007) takes this logic a step farther. • They estimate a multinomial logit model of the allocation of heterogeneous households to individual houses. • This model directly addresses sorting—and makes it possible to simulate new sorting equilibria with other assumptions, such as equal income distributions for black and white households.

28. Hedonics Method 4: Discrete Choice, 2 • This method is clever, but requires strong assumptions. • They assume that utility functions are linear! • They assume that housing prices 3 miles away are a good instrument for actual housing prices. • They assume that the hedonic price function is linear.

29. Hedonics Method 5: Derive the Envelope • Yinger (2012) derives a new form for the hedonic equation using the standard bidding model with constant-elasticity demand functions for the amenity and housing. • This paper draws on the standard theory of sorting to derive a bid-function envelope across households with different preferences. • This approach can be generalized to any number of amenities and can be applied to the housing-commuting trade-off in a standard urban model.

30. Hedonics Method 5: Derive the Envelope, 2 • This approach estimates the price elasticity of demand for S, μ, in the hedonic equation itself. • Thus the main parameter of interest, μ, can be estimated without encountering the standard endogeneity problem. • This approach allows for a general treatment of household heterogeneity, but then integrates out the determinants of this heterogeneity in deriving the bid-function envelope. • So this approach accounts for household heterogeneity without requiring data on household characteristics to estimate μ.

31. Hedonics Method 5: Derive the Envelope, 3 • Yinger’s first key assumption is that households have constant-elasticity demand functions for the amenity and housing: where the “^” indicates a before-tax housing price.

32. Hedonics Method 5: Derive the Envelope, 4 • The bid functions that result take a Box-Cox form. • The Box-Cox form is

33. Hedonics Method 5: Derive the Envelope, 5 • To be more specific, the bid functions are: where C is a constant of integration and

34. Hedonics Method 5: Derive the Envelope, 6 • The next step is to bring in sorting. • A fundamental theorem is that sorting depends on the slope of the bid function: a household class with a steeper slope sorts into a jurisdiction with a higher value of S. • The slope is ; ψcontains all non-shared terms in and is thus an index of this slope. • Soa steeper slope (= higherψ) is associated with a higherSas illustrated in the following graphs.

35. Hedonics

36. Hedonics

37. Hedonics Method 5: Derive the Envelope, 7 • Yinger’s approach is to assume that the equilibrium relationship between ψand Scan be approximated with the following equation: • The σs are parameters to be estimated. • The sorting theorem predicts that so long as σ3> 0, as expected,σ2 > 0, which can be tested.

38. Hedonics Method 5: Derive the Envelope, 8 • This assumption leads to the following form for the pre-tax hedonic envelope: where

39. Hedonics Extension to Multiple Amenities • So long as Si is not directly a function of Sj, this approach can be extended to multiple amenities, with two terms in the bid function for each amenity. • 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.

40. Hedonics The Final Hedonic Equation • The final estimating equation is • To estimate this (nonlinear!) equation: • Use the form just derived for . • Assume a multiplicative form for H{X}. • Introduce the degree of property tax capitalization.

41. Hedonics 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 μand σ3.

42. Hedonics Special Cases, 2 Note: μ= -∞ implies a horizontal demand curve; σ3 = ∞ implies no sorting

43. Hedonics Hedonic Vices • Despite the fame of the Rosen article, many scholars have forgotten some of its key messages. • What follows is my guide to “hedonic vices,” that is, to approaches that are not consistent with the Rosen framework and related literature.

44. Hedonics Functional Form Vices • 1. The use of a linear (or semi-log or log-linear) form for the hedonic equation. • It is difficult to come up with assumptions that yield these forms. • With constant-elasticity demands, a linear form arises, for example, with no sorting based on bid-function slopes and the assumption that μ= -2; see the following figure.

45. Hedonics Functional Form Vices, 2

46. Hedonics Functional Form Vices, 3 • 2. Contradictions between Rosen’s 2 steps. • The envelope is mathematically connected to the bid functions. • It makes no sense to estimate a hedonic based on one assumption about the price elasticity and then to estimate the price elasticity in the next step. • For example, a quadratic form assumes that μ= -∞, so it makes no sense to get implicit prices from this form and then to estimate μ.

47. Hedonics Control Variable Vices • 1. Using demand variables as controls. • The Rosen framework implies that bid functions depend on demand traits, such as income, but the hedonic envelope does not. • Adding demand variables therefore changes the estimation into a bid-function estimation, not an envelope estimation.

48. Hedonics Control Variable Vices, 2 • There is an exact analogy here to cost functions, as pictured on the next slide. • The family of short-run average cost functions has plant size as a variable. • The long-run average cost function, which is the envelope of the short-run functions, does not have plant size as an argument. • Short-run average cost functions are analogous to bid functions; plant size is analogous to demand factors; and long-run average cost functions are analogous to the hedonic envelope.

49. Hedonics Control Variable Vices, 3

50. Hedonics Control Variable Vices, 4 • This change in interpretation has two critical implications. • First, included demand variables are endogenous; they bring the 2nd stage endogeneity into the 1st stage. • Second, included demand variables must be interacted with the amenity variables; otherwise the coefficients of the amenities do not vary with demand traits and there can be no sorting!