Bid-Function Envelopes or How to Estimate Capitalization

1. Bid-Function Envelopes or How to Estimate Capitalization ECN741, Urban Economics Professor Yinger

2. Lecture Outline • Methodological Challenges • Examples • Recent Publications • My Cleveland Application Outline

3. Methodological Challenges • 1. Functional form • 2. Defining School Quality (S) • 3. Controlling for neighborhood traits • 4. Controlling for housing characteristics Challenges

4. Functional Form • As discussed in previous classes, simply regressing V on S (with or without logs) is not satisfactory. • Regressing ln{V} on S and S2is pretty reasonable—but cannot yield structural coefficients. • To obtain structural coefficients, one must use either nonlinear regression or the Rosen 2-step method (with a general form for the envelope and a good instrument for the 2nd step). • Both approaches are difficult! Challenges

5. Defining School Quality • Most studies use a test score measure. • A few use a value-added test score. • A few use a graduation rate. • Some useinputs (spending or student/teacher ratio). • A few use both, but the use of multiple output measures is rare (but sensible!). Challenges

6. Neighborhood Controls • Data quality varies widely; some studies have many neighborhood controls. • Many fixed-effects approaches are used (sometimes inappropriately) to account for unobservables, e.g.: • Border fixed effects (cross section) • Neighborhood fixed effects (panel) • Another approach is to use an instrumental variable. Challenges

7. Border Fixed Effects • BFEs were popularized by Black (1999); they appear in at least 16 studies. • BFEs define elementary school attendance zone boundary segments. • Define a border fixed effect (BFE) for each segment, equal to one for houses within a selected distance from the boundary. • Drop all observations farther from boundary. Challenges

8. Border Fixed Effects, 2 School House Sale Boundary Segment Challenges

9. Border Fixed Effects, 3 • The idea is that the border areas are like neighborhoods, so the BFEs pick up unobservables shared by houses on each side of the border. • But bias comes from un-observables that are correlated with S; by design, BFEs are weakly correlated (i.e. take on the same value for different values of S). Challenges

10. Border Fixed Effects, 4 • BFEs have three other weaknesses: • They shift the focus from across-district differences in S to within-district differences in S, which are likely to be smaller and less interesting (since many elementary schools feed a single high school). • They require the removal of a large share of the observations (and could introduce selection bias if demand for S is different for people who select border neighborhoods). • They ignore sorting; that is, they assume that neighborhood traits are not affected by the fact that sorting leads to people with different preferences on either side of the border bid. Challenges

11. BFE and Sorting • Two recent articles (Kane et al. and B/F/M) find significant differences in demographics across attendance-zone boundaries. • B/F/M then argue that these demographic differences become neighborhood traits and they include them as controls. • As discussed in a previous class, these differences are measures of demand—which do not belong in an hedonic. • As Rosen argued long ago, the envelope is not a function of demand variables. • Including demand variables re-introduces the endogeneity problem and changes the meaning of the results. Challenges

12. Other Fixed Effects • Other types of fixed effects are possible, e.g., • Tract fixed effects (with a large sample or a panel) • School district fixed effects (with a panel). • House fixed effects (with double-sales data and a panel). • These approaches account for some unobservable factors, but may also introduce problems. Challenges

13. Problems with Fixed Effects • They all limit the variation in the data for estimating capitalization. • With panel data, these house or school district fixed effects, imply that the coefficient of S must be estimated based only on changes in S. • In a cross section (but not a panel), they partially control for demand factors, such as income, that should not be included in a hedonic. • Because household and tract income are highly correlated, including tract dummies at least partially controls for household income, resulting in the same problems as those caused by BFEs. • My interpretation is not popular. • Economists seem to embrace fixed effects even if they do not make theoretical sense while at the same time accepting studies with few control variables. Challenges

14. The IV Approach • With omitted variables, the included explanatory variables are likely to be correlated with the error term. • A natural correction is to use an instrumental variable—and 2SLS. • However, credible IVs are difficult to find. • For example, the well-known 2005 Chay/Greenstone article in the JPE estimates a hedonic for clean air using a policy announcement as an instrument. • But many studies (some mentioned below) show that announcements affect house values so the C/G instrument fails the exogeneity test. Challenges

15. Controlling for Housing Traits • A housing hedonic requires control variables for the structural characteristics of housing. • Because housing, neighborhood, and school traits are correlated, good controls for housing traits are important (but surprisingly limited in many studies). • As discussed later, the widely cited B/F/M article in the JPE (2007) has only 2 housing traits: number of rooms and year built, plus one (endogenous?) trait correlated with housing type: whether owner-occupied. Challenges

16. Housing Traits, 2 • If good data on housing traits are available, one strategy for a cross-section is to estimate the hedonic in two stages. • Stage 1: Define fixed effects for the smallest observable neighborhood type (e.g. block group or tract); in a sample of house sales, regress V on housing traits and these FE’s—with no neighborhood traits. • Stage 2: Use the coefficients of the FE’s as the dependent variable in a second stage with neighborhood traits on the right side; the number of observations equals the number of neighborhoods. Challenges

17. Housing Traits, 3 • This approach has two advantages: • The coefficients of the housing traits cannot be biased due to missing neighborhood variables. • The second stage need not follow the same form as the first, so this approach adds functional-form flexibility. • Note that the standard errors in the 2nd stage must be corrected for heteroskedasticity. • The coefficient of each FE is based on a different number of observations. Challenges

18. Selected Recent Examples • Bayer, Ferriera, and MacMillan (JPE 2007) • Clapp, Nanda, and Ross (JUE 2008) • Bogin (Syracuse dissertation 2011), building on Figlio and Lucas (AER 2004) • Yinger (working paper 2013) Recent Studies

19. B/F/M • B/F/M have census data from the San Francisco area. • They estimate a linear hedonic with BFE’s, pooling sales and rental data. (The also estimate a fancy multinomial choice model, which is not considered here.) • They find that adding the BFE’s cuts the impact of school quality on housing prices. • They find that adding neighborhood income cuts the impact of school quality even more. Recent Studies

20. B/F/M Hedonic Recent Studies

21. B/F/M Problems • They estimate a linear hedonic, which rules out sorting (in an article about sorting!) and is inconsistent with their own (linear) bid functions. • They control for neighborhood income, which is not consistent with the Rosen framework. • In addition to BFEs, they have only 2 housing traits, 1 sort-of housing trait, and 4 location controls, one of which is a set of land uses. • One neighborhood control (density) is a function of the dependent variable; I guess they never took urban economics! Recent Studies

22. C/N/R • They use a panel of housing transactions in Connecticut between 1994 and 2004 • They use tract fixed effects to control for neighborhood quality in their panel data. • They look at math scores and cost factors (e.g. student poverty) • They find that tract fixed effects have little impact on the estimate of capitalization when income and other demographics are included. Recent Studies

23. C/N/R Hedonic Recent Studies

24. C/N/R Problems • They use a semi-log form with only one term for S, which rules out sorting. • They control for neighborhood demographics, which raises the same issue as B/F/M: Should demand variables be included? • They have only 4 housing traits and 2 non-demand neighborhood traits. • They include fraction owner-occupied, which appears to be endogenous. Recent Studies

25. Bogin • The Florida school accountability program hands out “failing” grades to some schools. The Figlio/Lucas paper (AER 2004) looks at the impact of this designation on property values. • The national No Child Left Behind Act also hands out “failing” grades. The 2011 Bogin essay looks at the impact of this designation on property values around Charlotte, North Carolina. • In both cases, the failing grades are essentially uncorrelated with other measures of school quality. Recent Studies

26. Bogin 2 • Bogin finds that a failing designation lowers property values by about 6%. • This effect peaks about 7 months after the announcement and fades out after one year. • Bogin also provides a clear interpretation of results with this “change” set-up. • Because of possible re-sorting, the change in house values cannot be interpreted as a willingness to pay. • A failing designation might change the type of people who move into a neighborhood. • Consider the following figure from his paper: Recent Studies

27. Bogin 3 Recent Studies

28. Specification Summary • Many studies estimate the following and interpret β as the average MWTP: • Problems: • A linear specification is not appropriate; Nguyen-Hoang and I find a 46 percent error in one simulation! • Average MWTP is a limited concept, even with the correct specification; it cannot be compared across place or time because it is affected by the sorting equilibrium. Recent Studies

29. Specification, 2 Recent Studies

30. Specification, 3 • Panel: • Problems: • A linear specification is still not appropriate. • The β coefficient does not measure average MWTP unless S does not change (in which case β cannot be estimated!), there is no change in the distribution of demand for S, and no re-sorting! • This does not stop many studies, including Chay/Greenstone and Clapp et al. Recent Studies

31. Specification, 4 • Cross-Section with Demand Variables: • Problems: • This is a bid-function regression, not an envelope. • The income term, Y, is endogenous. • Without an interaction between S and Y, the slope of the bid function (dV/dS) is the same for everyone and there is no sorting!! Recent Studies

32. Estimates with a Derived Envelope • Finally, I would like to present some results for both the hedonic and the underlying bid functions from the application of the method I have developed using data from a large metropolitan area. • This method has several advantages: • It avoids the endogeneity problem in the Rosen 2-step approach. • It avoids inconsistency between the bid functions and their envelope (the hedonic equation). • It includes most parametric forms for a hedonic as special cases. • It allows for household heterogeneity. • It leads to tests of key sorting theorems. A New Approach

33. My Envelope • The form derived in an earlier class: and X(λ)is the Box-Cox form. • A starting point is a quadratic form, which corresponds to μ = -∞ and σ3 = 1 A New Approach

34. The Brasington Data • All home sales in Ohio in 2000, with detailed housing characteristics and house location; compiled by Prof. David Brasington. • Matched to: • School district and characteristics • Census block group and characteristics • Police district and characteristics • Air and water pollution data • I focus on the 5-county Cleveland area and add many neighborhood traits. A New Approach

35. My Two-Step Approach • Step 1: Estimate the envelope using my functional form assumptions to identify the price elasticity of demand, μ. • Step 1A: Estimate hedonic with neighborhood fixed effects • Step 1B: Estimate PE{S, t} for the sample of neighborhoods with their coefficients from Step 1A as the dependent variable. • Step 2: Estimate the impact of income and other factors (except price) on demand. A New Approach

36. Neighborhood Fixed Effects • Start with Census block groups containing more than one observation. • Split block-groups in more than one school district. • Total number of “neighborhoods” in Cleveland area sub-sample: 1,665. A New Approach

37. Step 1A: Run Hedonic Regression with Neighborhood Fixed Effects • Dependent variable: Log of sales price in 2000. • Explanatory variables: • Structural housing characteristics. • Corrections for within-neighborhood variation in seven locational traits. • Neighborhood fixed effects. • 22,880 observations in Cleveland subsample. A New Approach

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40. Step 1B: Run Envelope Regression • Dependent variable: coefficient of neighborhood fixed effect. • Explanatory variables: • Public services and neighborhood amenities • Commuting variables • Income and property tax variables • Neighborhood control variables A New Approach

41. School Variables Variable Definition ---------------------------------------------------- Elementary Average percent passing in 4th grade in nearest elementary school on 5 state tests (math, reading, writing, science, and citizenship) minus the district average (for 1998-99 and 1999-2000). High School The share of students entering the 12th grade who pass all 5 tests (= the passing rate on the tests, which reflects students who do not drop out, multiplied by the graduation rate, which indicates the share of students who stay in school) averaged over 1998-99 and 1999-2000. Value Added A school district's sixth grade passing rate (on the 5 tests) in 2000-2001 minus its fourth grade passing rate in 1998-99. Minority Teachers The share of a district’s teachers who belong to a minority group A New Approach

42. Cleveland and East Cleveland • The Cleveland School District is unique in 2000 because: • It was the only district to have private school vouchers • It was the only district to have charter schools (except for 1 in Parma). • The private and charter schools tend to be located near low-performing public schools. • The East Cleveland School District is unique in 2000 because • It received a state grant for school construction in 1998-2000 that was triple the size of its operating budget. • No other district in the region received such a grant. A New Approach

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49. Estimated Impacts • In the case of the High School variable, housing prices are about 30% higher in a district with the highest value (77% passing) compared to a district with a 20% passing rate. • Prices are also about 3% higher at a 13 percent passing rate than at a 20% passing rate, but this result is not statistically significant (and involves only a few observations). • The results for the Elementary variable in Cleveland are consistent with the view that parents care about elementary school quality, but also care about education opportunities, which are clustered in the neighborhoods with the worst regular public schools. A New Approach

50. Conclusions, Theory • The envelope derived in my paper: • Is based on a general characterization of household heterogeneity. • Makes it possible to estimate demand elasticities (and program benefits) from the first-step equation—avoiding endogeneity. • Ensures consistency between the envelope and the underlying bid functions. • Sheds light on sorting. A New Approach