2. 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

3. 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

4. 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

5. 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.

6. 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:

7. The Rosen Framework 3  Note that in this picture, the bid functions, the θs, depend on household traits, as indicated by the utility level, u i *.  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.

8. The Rosen Framework 4  Rosen also models the supply side, with offer curves, Φ.

9. 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.

10. 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.”

11. 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.

12. 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.

13. 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!!!

14. 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.

15. 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 ≡ V S, for each amenity, S. ◦ Step 2: Estimate the demand for amenity S as a function of V S (and other things). ◦ Step 2 Alternate: Estimate the inverse demand function, which is V S as a function of S.

16. Principal Challenge: Endogeneity  As Epple (JPE 1987) and other scholars have pointed out, the main problem facing a 2 nd 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 V S with which to estimate demand!) ◦ Households have different preferences, so the level of S, and hence of V S, they select depends on their observed and unobserved traits.

17. 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.

18. Bidding and Sorting

19. Dealing with Endogeneity in Hedonics  Some articles find instruments for V S in the 2 nd 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….

20. 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.

21. 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).

22. 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.).

23. 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.

24. The Bajari/Kahn Assumption  The Rosen two-step method estimates P S, which a household sets equal to MB S. With constant elasticity demand and μ = -1,  This equation does not have an endogenous variable on the right side.

25. 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.

26. A Second Major Challenge  Another major challenge in estimating Rosen’s 1 st 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.

27. 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.

28. 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.

29. 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!

30. 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!

31. 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.

32. 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.

33. 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

34. 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.

35. 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*.

36. Step 1 for Deriving an Envelope

37. 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

38. 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.

39. 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, σ 2 > 0.  This function was illustrated in a previous figure:

40. Bidding and Sorting

41. 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!

42. 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 σ 3 opens the door to many other distributions and transformations.

43. 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, σ 1, σ 2, σ 3, ) to be estimated.

44. 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.

45. The Envelope Equation  The envelope that results has Box-Cox forms: where

47. 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 σ 3.

48. Special Cases, Continued Note: μ = -∞ implies a horizontal demand curve σ 3 = ∞ implies no sorting

49. 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 2 nd step estimate of the price elasticity will contradict the 1 st step assumption.

50. Extension to Multiple Amenities  So long as S i is not directly a function of S j, 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 S i holding other amenities constant.  Highly correlated amenities may need to be combined into an index.

51. 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 ).

52. 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:

53. 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.

54. 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!

55. 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).

56. 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.