Kai Hong

# Abstract

In the United States, the reform of the financial system of capital expenditure is under consideration, as people believe the current system through local referenda contributes to inequality in student achievement across school districts. Several studies using a regression discontinuity design (RDD) find zero to modest positive effects of capital expenditure on student achievement; however, these studies identify only the effect of capital expenditure financed by a marginally passed bond with a vote share at the cutoff. In this paper I estimate the average effect of capital expenditure on student achievement by incorporating a latent factor model into the existing RDD framework, and comparing school districts that are similar in their underlying confounding variables, namely preferences for educational investment. The results show that, on average, capital expenditure financed by a passed bond does not have significant effect on student achievement.

JEL Classification: I22; H54

# Acknowledgements:

An early version of this paper was presented to the departmental seminar at Vanderbilt University. I thank participants of the seminar for helpful suggestions and discussions. I am grateful to Kathryn Anderson, Andrew Dustan, Evan Elmore, Federico Gutierrez, Hendrik Jürges, Pedro Sant’anna, Peter Savelyev, Adam Shriver, Ron Zimmer and two anonymous referees for detailed and productive comments.

# Appendix

## A Appendix

### Figure 5:

Proficiency trends relative to bond passage.

Notes: Each dot represents the average proficiency rates in the corresponding year relative to an election.

Table 6:

School bond summary statistics.

YearNumber of bondsPercentage passed (%)Average vote share (%)Average bond amount per pupil ($)Average repayment time (year)Average millage rate (amount per$1000)Percentage of new building (%)Number of voters
meanstandard deviationmeanstandard deviationmeanstandard deviationmeanstandard deviationmeanstandard deviation
19961645149.411.973125423-----25362315
19971494348.311.277645670-----24972534
19981074148.510.794727965-----26443806
19991174849.811.783486388----3020971783
20001174949.812.97694583524.45.62.972.073623642141
20011086353.312.67487609324.75.22.892.092724692452
2002835952.314.87882644025.25.02.791.98-25602705
2003703946.714.798201059026.04.43.012.29-33614096
2004716353.915.39243666124.45.32.801.89-30342943
2005584048.911.899811013126.03.62.841.79-25582467
2006594448.011.577716827-----37403533
2007684748.213.480338179-----26602395
2008445751.612.675984791-----23203249
2009507054.712.260876375-----538616869
Total12655050.012.78123690725.05.02.892.033227224327
1. Notes: The sample includes bonds with non-missing values in both passage and vote share. Average bond amount per pupil is measured in constant 2000 dollars. Repayment time, millage rate and building type are not reported consistently.

Table 7:

District descriptive statistics of expenditure, demography and achievement.

All school districtNever proposed an electionEver proposed an election
meanstandard deviationmeanstandard deviationmeanstandard deviation
Expenditure per pupil
Total884939818898408388353953
Current731930387804357971832855
Teacher salary319414003428154431291351
Capital714.71713399.61157802.71828
Construction625.31644337.31103705.61757
Land and structure89.48448.662.24332.897.08475.7
Instructional equipment46.3187.5061.53151.242.0657.73
Demography
Enrollment293772491845281832438036
White students87.7618.1287.2319.5387.9117.70
Free lunch32.7118.8933.0520.9532.6218.26
Achievement (proficiency)
Number of districts577144433
Sample size775116936058
1. Notes: Estimation sample includes districts with no missing information in proficiency, election time and vote share. All background variables are defined in the same way as Table 1.

### Figure 6:

Distribution of preferences for educational investment by bond passage.

Notes: The width of each bin is 0.1. Only elections with non-missing vote share are included. Sample sizes are 936, 376 and 560 for the whole sample, the failed group, and the passed group, respectively.

Table 8:

Average effect of bond passage on achievement, subsequent capital expenditure and enrollment.

1-0.450-0.468677.7 ***350.8
(0.539)(0.573)(71.01)(299.5)
20.390-0.3753075 ***417.9
(0.521)(0.594)(175.7)(329.5)
30.7680.3891937 ***558.2
(0.559)(0.567)(165.5)(352.4)
40.6080.996355.2 ***768.2 **
(0.502)(0.578)(137.6)(390.3)
50.493-0.283-211.6 *895.7 **
(0.513)(0.552)(127.1)(398.1)
60.2440.515-138.3853.8 **
(0.563)(0.639)(137.5)(417.8)
70.2270.947-129.3682.1
(0.547)(0.594)(115.1)(423.9)
80.7741.118 *-303.4 ***624.9
(0.527)(0.593)(102.1)(433.3)
9-0.0670.537-95.29685.4
(0.573)(0.653)(88.59)(422.7)
100.9630.431-95.66540.3
(0.625)(0.741)(110.2)(475.1)
111.1810.287-46.58442.7
(0.758)(0.808)(120.8)(536.1)
121.368-0.311-157.3484.3
(0.884)(0.802)(200.3)(620.9)
131.5521.22768.41-675.3
(1.305)(1.110)(168.2)(822.5)
Sample size7244721977467751
1. Notes: The specification is the regression described in eq. (6). Clustered standard errors by school district are shown in parentheses. *, ** and *** indicate the statistical significance at 10%, 5% and 1% levels, respectively.

Table 9:

Bond and referendum descriptive statistics.

All elections (1)Passed elections (2)Failed elections (3)Difference (2)-(3), t test (4)
meanstandard deviationmeanstandard deviationmeanstandard deviationdifference in meanp-value
Bond characteristics
Bond amout per pupil9371685880425888113507682-33090.000
Repayment time25.194.73924.485.03526.443.876-1.9570.000
Millage rate3.3751.8862.8791.7074.2551.874-1.3760.000
Purpose (new building)0.3720.4850.3300.4720.4320.498-0.1020.149
Referendum characteristics
First in the year0.9510.2160.9460.2250.9570.202-0.0110.445
Date of referendum188.486.50192.786.09182.186.8310.620.066
Year of referendum20013.96020014.00920013.8690.5220.048
Number of voters2721480727255647271631748.5280.979
Sample size936560376936
1. Notes: Average bond amount per pupil is measured in constant 2000 dollars. Referendum data is measured as day of year.

Table 10:

Average effects of bond passage on reading proficiency controlling for bond and referendum characteristics.

Relative yearMain SpecificationControlling for Bond and Referendum Characteristics
1-0.450-0.468-0.453-0.872
(0.539)(0.573)(0.563)(0.606)
20.390-0.3750.507-0.699
(0.521)(0.594)(0.560)(0.628)
30.7680.3890.7840.134
(0.559)(0.567)(0.621)(0.602)
40.6080.9960.7871.262 **
(0.502)(0.578)(0.532)(0.595)
50.493-0.2830.781-0.334
(0.513)(0.552)(0.586)(0.584)
60.2440.5150.3890.245
(0.563)(0.639)(0.644)(0.695)
70.2270.9470.6001.095 *
(0.547)(0.594)(0.613)(0.636)
80.7741.118 *0.9281.139 *
(0.527)(0.593)(0.629)(0.626)
9-0.0670.5370.2960.310
(0.573)(0.653)(0.659)(0.712)
100.9630.4311.0900.704
(0.625)(0.741)(0.691)(0.807)
111.1810.2871.3190.384
(0.758)(0.808)(0.834)(0.863)
121.368-0.3111.306-0.299
(0.884)(0.802)(0.962)(0.864)
131.5521.2271.5551.192
(1.305)(1.110)(1.333)(1.126)
Sample size7244721972447219
1. Notes: The specification is based on the regression described in eq. (6). Columns (1) and (2) show the results from the main specification. Columns (3) and (4) show the results from the specification that also controls for bond and referendum characteristics, which are summarized in Table 6. * and ** indicate the statistical significance at 10% and 5% levels, respectively.

### Figure 7:

Total expenditure per pupil relative to reading proficiency.

Notes: The figure represents the average total expenditure per pupil in the past six years. Total expenditure per pupil is measured in constant year 2000 dollars.

Table 11:

Average effects of bond passage on 4th grade reading proficiency controlling for factors of Pre-existing achievement.

Relative yearMain Specification (1)Adding Pre-Existing ProficiencyAlternative Polynomial
Level (2)Change (3)Level and Change (4)Quadratic (5)Cubic (6)
1-0.450-0.382-0.762-0.305-0.566-0.614
(0.539)(0.467)(0.475)(0.466)(0.505)(0.512)
20.3900.027-0.2650.0590.1020.030
(0.521)(0.471)(0.482)(0.468)(0.503)(0.502)
30.7680.3820.1850.3600.6360.445
(0.559)(0.490)(0.519)(0.490)(0.537)(0.549)
40.6080.4220.1790.5460.4780.358
(0.502)(0.426)(0.454)(0.425)(0.478)(0.487)
50.4930.3080.0650.5330.3220.067
(0.513)(0.458)(0.470)(0.468)(0.514)(0.515)
60.2440.087-0.1700.2940.083-0.182
(0.563)(0.508)(0.523)(0.508)(0.557)(0.551)
70.227-0.008-0.2170.221-0.061-0.206
(0.547)(0.483)(0.487)(0.481)(0.540)(0.524)
80.7740.4760.3050.5360.3960.408
(0.527)(0.478)(0.489)(0.484)(0.515)(0.505)
9-0.067-0.390-0.514-0.430-0.586-0.338
(0.573)(0.511)(0.517)(0.511)(0.550)(0.549)
100.9630.6750.5690.6930.5670.786
(0.625)(0.534)(0.553)(0.523)(0.589)(0.509)
111.1810.6860.4670.7650.7420.901
(0.758)(0.693)(0.719)(0.687)(0.723)(0.722)
121.3680.9210.7030.5711.1371.172
(0.884)(0.810)(0.819)(0.836)(0.836)(0.837)
131.5521.1890.8151.4811.4161.494
(1.305)(1.122)(1.143)(1.089)(1.188)(1.200)
F Test (p-value)0.3950.6880.8060.4730.5310.587
1. Notes: The specification is based on the regression described in eq. (6). The first column shows the results from the main specification. Column (2) to (4) show the results from the specification that also controls for a latent factor of pre-existing proficiency, a latent factor of the change in pre-existing proficiency, and both factors, respectively. Column (5) and (6) show the results when quadratic and cubic preferences of educational investment is controlled, respectively.

### Figure 8:

Proficiency relative to preferences of educational investment.

Notes: Each dot represents the average proficiency rates at the corresponding preferences of educational investment.

Table 12:

Average effects of bond passage on 4th grade reading proficiency using alternative specifications.

Relative yearLatent factor (main)Pooled OLS 1 (1)Pooled OLS 2 (2)Fixed effect (3)
1-0.450-0.6131.819 ***1.838 ***
(0.539)(0.997)(0.679)(0.686)
20.3900.4182.477 ***2.407 ***
(0.521)(1.017)(0.732)(0.708)
30.7682.078 **2.939 ***2.776 ***
(0.559)(0.986)(0.711)(0.695)
40.6083.287 ***3.026 ***2.816 ***
(0.502)(1.028)(0.700)(0.679)
50.4934.291 ***3.014 ***2.897 ***
(0.513)(1.044)(0.704)(0.676)
60.2445.094 ***2.480 ***2.326 ***
(0.563)(1.118)(0.742)(0.733)
70.2276.531 ***1.974 ***1.860 ***
(0.547)(0.977)(0.693)(0.692)
80.77410.26 ***2.561 ***2.421 ***
(0.527)(0.962)(0.655)(0.651)
9-0.06711.27 ***1.497 **1.392 **
(0.573)(1.078)(0.684)(0.700)
100.96313.06 ***1.701 **1.640 **
(0.625)(1.196)(0.720)(0.754)
111.18113.68 ***2.152 **1.964 **
(0.758)(1.371)(0.872)(0.909)
121.36812.07 ***2.149 **2.097 **
(0.884)(1.575)(1.040)(1.051)
131.55212.68 ***1.8652.219
(1.305)(2.150)(1.466)(1.457)
Year fixed effectYesNoYesYes
District fixed effectYesNoNoYes
Latent preferenceYesNoNoNo
Sample size7244724472447244
1. Notes: The specification is based on the regression described in eq. (6). The first column shows the results from the main specification. In all other columns I do not control for the latent preferences for educational investment. In addition, Columns (1) does not include year fixed effect or district fixed effect. Columns (2) does not include district fixed effect. Clustered standard errors by school district are shown in parentheses. ** and *** indicate the statistical significance at 5% and 1% levels, respectively.

## B Estimation of the Factor Model

By eq. (4), the covariance between the two observed measures of preferences on financing education is

(11)Cov(Pljt,Pljt)=ψlψlVar(ΘjtP)+ψlCov(ΘjtP,ηljt)+Cov(ηljt,ηljt).

Since the latent factor is standardized (VarΘjtP=1) and it is assumed that the latent factor is uncorrelated with the error terms (Cov(ΘP,ηljt)=0) and the error terms are mutually uncorrelated (Cov(ηljt,ηljt)=0), I have Cov(Pljt,Pljt)=ψlψl. Thus, I have the following equation system with three unknown factor loadings:

(12){Cov(P1jt,P2jt)=ψ1ψ2Cov(P1jt,P3jt)=ψ1ψ3Cov(P2jt,P3jt)=ψ2ψ3

Solving the system, I get

(13)ψ^1=(r12r13r23)12,ψ^2=(r12r23r13)12,ψ^3=(r13r23r12)12.

By eq. (4), the variance of the observed measures of preferences on financing education, which is standardized to one, can be written as

(14)Var(Pljt)=ψl2Var(ΘjtP)+Var(ηljt)=1.

Thus, given Var(ΘjtP)=1, Var(ηljt)^=1ψ^l2, for l=1,2,3.

I then use the regression scoring method (Thomson 1951) to estimate the preferences for educational investment for each observation of district-year, and get Θ^jtP=ψ^(Σ)1Pjt, where ψ^=(ψ^1ψ^2ψ^3) and Pjt=(P1jtP2jtP3jt). The regression scoring method minimizes mean squared error, thus produce the most accurate prediction. However, the estimated factor itself may be biased (Hershberger 2005). Alternatively, I can also use the Bartlett scoring method (Bartlett 1937), which produces estimated factors that are less accurate but unbiased. Specifically, I estimate the preferences for educational investment by minimizing the sum of squared residuals that are weighted by the specific variances:

(15)Θ^jtP=argminl=13ηljt2El,

where El is the variance of ηl, the error term specific to the l-th observed measure Pl. Thus, the estimated preferences for educational investment is Θ^jtP=(ψ^E^1ψ^)1ψ^E^1Pjt, where E is the variance-covariance matrix of η, the measurement errors in the latent factor model (4). Table 13 shows that the results by the regression scoring method and the Bartlett method are quite similar.

Table 13:

TOT effect of bond passage on achievement, regression scoring and Bartlett method.

Relative yearRegression scoringBartlett method
1-0.450-0.468-0.796-0.882
(0.539)(0.573)(0.537)(0.574)
20.390-0.3750.237-0.472
(0.521)(0.594)(0.521)(0.597)
30.7680.3890.6940.149
(0.559)(0.567)(0.564)(0.571)
40.6080.9960.5550.896
(0.502)(0.578)(0.505)(0.578)
50.493-0.2830.388-0.393
(0.513)(0.552)(0.515)(0.553)
60.2440.5150.1280.371
(0.563)(0.639)(0.565)(0.640)
70.2270.9470.1460.799
(0.547)(0.594)(0.550)(0.594)
80.7741.118 *0.7301.078 *
(0.527)(0.593)(0.529)(0.591)
9-0.0670.537-0.1280.490
(0.573)(0.653)(0.574)(0.651)
100.9630.4310.9440.359
(0.625)(0.741)(0.627)(0.743)
111.1810.2871.1610.172
(0.758)(0.808)(0.758)(0.808)
121.368-0.3111.338-0.294
(0.884)(0.802)(0.887)(0.801)
131.5521.2271.4481.196
(1.305)(1.110)(1.302)(1.113)
Sample size7244721972447219
1. Notes: The specification is the TOT regression described in eq. (6). Clustered standard errors by school district are shown in parentheses. * indicates the statistical significance at 10% level.

## C Intent-to-Treat Analysis

The most important difference between the ITT estimation in this section and the estimation in the main text is that the ITT estimation estimates the effect of a passed bond without isolating the impacts of bond authorizations in other years. In other words, if an initial bond passage affects the probability of passing another bond in subsequent years, the ITT estimation incorporates the effects of the affected subsequent bonds as parts of the effect of the initial bond passage. Compared with the effect obtained in the main text, the estimated ITT effect has weaker policy implications as it provides insights for the initial bond passage only. But as in the previous literature using RDD, the ITT analysis is helpful for balance and falsification tests because it can estimate the effect of a bond authorization on pre-existing outcomes without controlling for bond authorizations in other years unnecessarily.

For the ITT estimation I use the stacked sample as defined in Cellini, Ferreria, and Rothstein (2010). Specifically, to generate the stacked sample, I first “stack” all district-year observations for the district that has an election in year t in a window from t2 through t+13, for each election. Secondly, the stacked data sets for each separate election are combined into one large panel data set.[28] Then I estimate the average ITT effect of bond passage through the following equation:

(16)Yjt=βτgτBj,tτ+ωτgτΘj,tτP+Gτ+Tt+Ej,tτ+ejt,1996t2009,2τ13,

where Θj,tτP is identified in the latent factor model (4).[29] All other parameters are the same as in eq. (6) and footnote 14. βτ is the ITT effect of passing an initial bond τ years ago on the current student achievement in school district j. The conditional independence assumption for the ITT analysis is modified accordingly to the following:

(17)(Yjt0,Yjt1)Bj,tτ|(Θj,tτP,Gτ,Tt,Ej,tτ),l=1,2,3,1996t2009,

where Yjt0 is the outcome τ years later if Bj,tτ is not passed and Yjt1 is the outcome τ years later if Bj,tτ is passed.

Figure 9 presents the estimated average ITT effects from eq. (16) (see Columns (1) and (2) of Table 14 for details.). The ITT effects show little evidence of effects of bond passage on subsequent proficiency (Panel (A) and (B)). As I show below, these effects may understate the true effects of bond passage because an initial bond passage decreases the possibility of passing subsequent bonds, which indicates that, in other words, we may not observe any effect because an initial bond does not increase cumulative subsequent expenditures on capital.

### Figure 9:

ITT effect of bond passage on achievement, subsequent bond passage and capital expenditure.

Notes: The specification is the ITT regression described in eq. (16). The sample sizes are 9,665, 9,663, 2,665 and 9,829, respectively.

Passing a bond can potentially affect various subsequent outcomes. Figure 9 also shows how passing an initial bond affects the probability of passing subsequent bonds and total capital expenditures on construction, land and structure (see Columns (3) and (4) of Table 14 for details.). Results regarding subsequent bonds in Panel (C) are consistent with the findings of Cellini, Ferreria, and Rothstein (2010) in California and Martorell, McFarlin Jr, and Kevin (2016) in Texas. Passing an initial bond decreases the probability of passing another bond in the short term –- two to five years –- by about 20–30%, although the effects are not always significant. There is no clear longer-term effect on the probability of passing subsequent bonds, except for a positive effect of 0.3 emerging eight years later. The cumulative short-run effect of an initial bond passage in five years is about 0.55, and the cumulative long-run effect of an initial bond passage in 13 years is about -0.58. These results indicate that on average failing an initial bond election reduces the expected total number of passed bond by 0.45 in the short run and 0.42 in the long run.

A bond passage significantly increases total capital expenditure in the short run (Panel (D)). Total capital expenditure starts increasing in the year of bond passage, and peaks after 2 years, when the maximum expenditure on construction per pupil is about $3,000 higher than the expenditure in the districts that fail a bond. The effects start declining in the third year and become negative after the fourth year because of the short-run negative effects on subsequent bonds. The effects diminish after nine years. The effects on capital investment confirm the findings in Panel (C) about subsequent bond. In the middle term an passed initial bond decreases capital expenditure through its negative impacts on the subsequent bond passage in the short run. Table 14: ITT effect of bond passage on achievement, subsequent bond passage and capital expenditure. Relative year4th grade7th gradeSubsequent bond passageCapital expenditure 1-0.459-0.5050.083728.0 *** (0.620)(0.575)(0.145)101.90 2-0.089-0.867-0.259 **2921 *** (0.667)(0.652)(0.128)(213.4) 3-0.1360.168-0.1691001 *** (0.705)(0.663)(0.125)(203.4) 40.0430.629-0.120-873.4 *** (0.707)(0.700)(0.150)(184.3) 5-0.128-0.679-0.287 **-1151 *** (0.689)(0.685)(0.143)(161.3) 6-0.550-0.290-0.191-738.5 *** (0.743)(0.794)(0.141)(193.7) 7-0.665-0.003-0.178-527.4 *** (0.720)(0.813)(0.159)(157.3) 8-0.494-0.2170.306 *-505.6 *** (0.787)(0.780)(0.167)(156.2) 9-1.291 *-0.5950.070-171.6 (0.832)(0.829)(0.164)(153.9) 10-0.828-0.9260.01242.24 (0.860)(0.919)(0.201)(172.2) 11-0.810-0.189(15.07) (0.997)(0.985)(0.245)(194.3) 12-0.643-1.972 *-0.421-264.6 (1.163)(0.984)(0.315)(259.5) 13-0.5650.170 *-0.335 *-210.6 (1.446)(1.279)(0.202)(302.0) Sample size9665966326659829 1. Notes: The specification is the ITT regression described in eq. (16). 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Published Online: 2017-9-29

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