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Bid-Function Envelopes or How to Estimate Capitalization

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

Challenges

- 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

- 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

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

Challenges

- Popularized by Black (1999); appears in at least 16 studies.
- Define elementary school attendance zone boundary segments.
- Define a border fixed effect (BFE) for each segment, equal to one for housings within a selected distance from the boundary.
- Drop all observations farther from boundary.

Challenges

- 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

- BFEs have three other weaknesses:
- They shift the focus from across-district differences in S to within-district differences in S, which are smaller and less interesting.
- They require the removal of a large share of the observations (and could introduce selection bias).
- 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

- Two recent articles (Kane et al. and Bayer et al.) find significant differences in demographics across attendance-zone boundaries.
- Bayer et al. then argue that these demographic differences become neighborhood traits and they include them as controls.
- I argue that these differences are implausible as neighborhood traits, but 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

- 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 a panel).
- These approaches account for some unobservable factors, but may also introduce problems.

Challenges

- They all limit the variation in the data for estimating capitalization.
- School district fixed effects, for example, imply that the coefficient of S must be estimated based only on changes in S.
- They may account 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 effectively controls for household income, resulting in the same problems as those caused by BFEs.
- My interpretation is not popular.
- Economists seem to prefer more controls even if they do not make theoretical sense.

Challenges

- 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

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).

Challenges

- 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); 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

- 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 this stage must be corrected for heteroskedasticity.
- The coefficient of each FE is based on a different number of observations.

Challenges

- 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 2012)

Recent Studies

- B/F/M have census data from the San Francisco area.
- They estimate a linear hedonic with BFE’s, pooling sales and rental data.
- 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

Recent Studies

- They estimate a linear hedonic, which rules out sorting (in an article about sorting!) and is inconsistent with their own bid functions.
- They control for neighborhood income (implausible theoretical basis).
- They have only 3 housing traits and 3 other location controls (+ BFE’s).
- One neighborhood control (density) is a function of the dependent variable; I guess they never took urban economics!

Recent Studies

- They use a panel of housing transactions in Connecticut between 1994 and 2004
- They use tract fixed effects to control for neighborhood quality.
- They look at math scores and cost factors (e.g. student poverty)
- They find that tract fixed effects lower the estimate of capitalization even below its level with income and other demographics.

Recent Studies

Recent Studies

- They use a semi-log form with only one term for S, which rules out sorting.
- They control for neighborhood demographics and tract FEs, 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

- 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

- 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:

Recent Studies

Recent Studies

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

- The form derived in my last lecture:

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

- 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

- 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

- 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

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

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

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

- 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

A New Approach

- 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

Conclusions, Empirical Results

- Willingness to pay for some aspects of school quality can be estimated with precision.
- The price elasticity of demand for high school quality is about -1.0 and housing prices are up to 30% higher where high school passing rates are high than where they are low.
- The theory of sorting is strongly supported in some cases.
- Household types with steeper bid functions for high school quality tend to live where school quality is higher.

A New Approach

Conclusions, Empirical, Continued

- Household seem to care about several dimensions of school quality, but precise demand parameters cannot be estimated in many cases.
- The price elasticity and other parameters cannot be precisely estimated for relative elementary scores.
- Results for elementary value added suggest a relationship that is too complex for current specifications; parents appear concerned about schools with low starting scores even when they improve.
- Results for percent minority teachers indicate that many households prefer teacher diversity, which calls for a specification different from any used up to now.

A New Approach

- Once the envelope has been estimated, one can recover its slope with respect to S, which is a function of income and other demand variables (for S and H).
- The theory says that the income coefficient is (-θ/μ - γ).
- Normal sorting requires this coefficient to be positive.
- Recall that the amenity price elasticity,μ, is negative.

A New Approach

- Direct and indirect tests are possible.
- A direct test looks at the income coefficient controlling for all other observable demand determinants.
- An indirect test says that normal sorting for S may arise indirectly through the correlation between S and other amenities (and the impact of income on these other amenities).
- Based on the omitted variable theorem, the indirect test comes from the sign of the income term in a regression omitting all other demand variables.

A New Approach

- Normal sorting is neither supported nor rejected for relative elementary school.
- Normal sorting is strongly supported for high school quality.
- So normal sorting appears to be strong across districts if not within them.

A New Approach

Conclusions, Empirical, Continued

- Preferences for neighborhood racial and ethnic composition are diverse.
- Some households (the majority) prefer all-white neighborhoods.
- But some prefer majority-black neighborhoods.
- Results for Hispanic neighborhoods are similar but less precise.
- I cannot link these results to the race/ethnicity of a household.

Capitalization

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