Department of Defense HPSP Medical Student Guide by Capt. Kenneth Stewart, DO
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ECN741, Urban Economics ECN741, Urban Economics Professor John Yinger, The Maxwell School, Syracuse University, 2018
Class Outline 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
Class Outline 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 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. hedonic regression.
Introduction Introduction Hedonic regressions can address three levels of analysis: 1. The impact on house values (which is a key issue in studying housing markets and property tax assessing). 2. The demand neighborhood amenities as reflected in house values (which is linked to benefit-cost analysis of these amenities and public services). 3. The sorting amenity demands into different neighborhoods (which is determined by their housing bids). impact of housing and neighborhood traits demand for public services and sorting of household types with different
Hedonic Applications 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
Class Outline 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
The Rosen Framework The Rosen Framework Most studies follow a famous paper by Sherwin Rosen (JPE, Jan./Feb.1974). This paper distinguishes between A household bid curve for (in our terms) P and S and which is exactly what we have been studying in this class. bid function function, which is an iso-utility The observed price function or hedonic, which is the envelope envelope of the underlying bid functions.
The Rosen Framework, 2 The Rosen Framework, 2 In Rosen, is a bid, z is a trait, u is utility, and p is price (=envelope). His famous picture is:
The Rosen Framework, 3 Note that in this picture, the bid functions, the s, 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 Hence, it is impossible to extract much demand information directly from the hedonic. The Rosen Framework, 3 does not contain any household- -level information level information.
The Rosen Framework, 4 The Rosen Framework, 4 Rosen also models the supply side, with offer curves, .
The Rosen Framework, 5 The Rosen Framework, 5 The market equilibrium p is a joint of the bid and offer curves, and hence may be very complicated. joint envelope envelope 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, 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, 6
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 7
The Rosen Framework, 8 The Rosen Framework, 8 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.
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 and shifts Hence without separating A Common Misunderstanding movement along a bid function shifts in the bid function due to sorting. Hence, willingness to pay cannot be estimated without separating bidding and sorting , willingness to pay cannot be estimated bidding and sorting. .
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!!! A Common Misunderstanding, 2
More More Misunderstanding Misunderstanding Other scholars think they have solved this problem because they observe changes over time. They regress V on S 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 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. re- -sorting sorting so that
Class Outline 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
The Rosen Two The Rosen Two- -Step 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 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. implicit or hedonic price, V/ S VS, for each
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 VSwith 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. Principal Challenge: Endogeneity
Principal Challenge, 2 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.
Bidding and Sorting
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 1999 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 . Dealing with Endogeneity in Hedonics
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. Selected Recent Contributions, 1
Selected Recent Contributions, 1A Epple, Peress, and Sieg (AEJ: Micro, Nov. 2010) 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). Selected Recent Contributions, 1A
E/P/S, 2 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.
E/P/S, 3 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 = 1A1 + 2A2. Then 100/ 1 units of A1 and no units of A2 yield the same utility as 100/ 2 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.
Selected Recent Contributions, 2 Ekeland, Heckman, and Nesheim (JPE , Feb. 2004); 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.). Selected Recent Contributions, 2
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. Selected Recent Contributions, 3
The The Bajari Bajari/Kahn Assumption /Kahn Assumption The Rosen two-step method estimates PS, which a household sets equal to MBS. With constant elasticity demand and = -1, = 1/( 1) S P S K Y S or P S = K Y S S This equation does not have an endogenous variable on the right side.
An Envelope for the An Envelope for the Bajari Bajari/Kahn Assumption /Kahn Assumption A method discussed later can derive an envelope using the B/K assumption: = -1. In the case of a linear sorting equilibrium (to be explained later), the form is ln{ } P b = + + ln{ } b S b S 0 1 2 This equation does not look a lot like local linear regression!
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. Selected Recent Contributions, 4
Class Outline 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
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. A Second Major Challenge omitted variable bias.
Border Fixed Effects Border Fixed Effects One strategy made famous by Black (QJE , May 1999) 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.
Border Fixed Effects, 2 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.
Border Fixed Effects, 3 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. Border Fixed Effects, 3
Border Fixed Effects, 4 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. Border Fixed Effects, 4
Border Fixed Effects, 5 Border Fixed Effects, 5 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).
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. An Alternative Approach: More Data!
More Data, A Caveat 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.
A Caveat, 2 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!
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. An Alternative Approach: Double Sales
Class Outline 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
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 heterogeneous households (given certain assumptions!). This envelope provides information about the underlying bids But it also contains information about the way different types of households sort The Yinger Approach derive and estimate the envelope of the bid functions for bids of individual households. sort into different neighborhoods.
The Payoff The Payoff 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.
Bidding Review 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 ( ) { } P S = + ( ) C S 1 2 where C is a constant and ( ) X X 1if = = = 0 and ln{ } if X 0 + 1 ( = + = 1 and 1 2 ) ( 1 ) 1/ ( / ) + = K N K M Y S H
Bidding Review 2 Bidding Review 2 In these formulas, the price elasticity of demand for public services, , is the main parameter of interest, And 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.
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 . We must derive C{ } 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*. Deriving the Envelope: Step 1
