# 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.

# 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:

Year | Number of bonds | Percentage 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 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

mean | standard deviation | mean | standard deviation | mean | standard deviation | mean | standard deviation | mean | standard deviation | ||||

1996 | 164 | 51 | 49.4 | 11.9 | 7312 | 5423 | - | - | - | - | - | 2536 | 2315 |

1997 | 149 | 43 | 48.3 | 11.2 | 7764 | 5670 | - | - | - | - | - | 2497 | 2534 |

1998 | 107 | 41 | 48.5 | 10.7 | 9472 | 7965 | - | - | - | - | - | 2644 | 3806 |

1999 | 117 | 48 | 49.8 | 11.7 | 8348 | 6388 | - | - | - | - | 30 | 2097 | 1783 |

2000 | 117 | 49 | 49.8 | 12.9 | 7694 | 5835 | 24.4 | 5.6 | 2.97 | 2.07 | 36 | 2364 | 2141 |

2001 | 108 | 63 | 53.3 | 12.6 | 7487 | 6093 | 24.7 | 5.2 | 2.89 | 2.09 | 27 | 2469 | 2452 |

2002 | 83 | 59 | 52.3 | 14.8 | 7882 | 6440 | 25.2 | 5.0 | 2.79 | 1.98 | - | 2560 | 2705 |

2003 | 70 | 39 | 46.7 | 14.7 | 9820 | 10590 | 26.0 | 4.4 | 3.01 | 2.29 | - | 3361 | 4096 |

2004 | 71 | 63 | 53.9 | 15.3 | 9243 | 6661 | 24.4 | 5.3 | 2.80 | 1.89 | - | 3034 | 2943 |

2005 | 58 | 40 | 48.9 | 11.8 | 9981 | 10131 | 26.0 | 3.6 | 2.84 | 1.79 | - | 2558 | 2467 |

2006 | 59 | 44 | 48.0 | 11.5 | 7771 | 6827 | - | - | - | - | - | 3740 | 3533 |

2007 | 68 | 47 | 48.2 | 13.4 | 8033 | 8179 | - | - | - | - | - | 2660 | 2395 |

2008 | 44 | 57 | 51.6 | 12.6 | 7598 | 4791 | - | - | - | - | - | 2320 | 3249 |

2009 | 50 | 70 | 54.7 | 12.2 | 6087 | 6375 | - | - | - | - | - | 5386 | 16869 |

Total | 1265 | 50 | 50.0 | 12.7 | 8123 | 6907 | 25.0 | 5.0 | 2.89 | 2.03 | 32 | 2722 | 4327 |

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.

All school district | Never proposed an election | Ever proposed an election | ||||
---|---|---|---|---|---|---|

mean | standard deviation | mean | standard deviation | mean | standard deviation | |

Expenditure per pupil | ||||||

Total | 8849 | 3981 | 8898 | 4083 | 8835 | 3953 |

Current | 7319 | 3038 | 7804 | 3579 | 7183 | 2855 |

Teacher salary | 3194 | 1400 | 3428 | 1544 | 3129 | 1351 |

Capital | 714.7 | 1713 | 399.6 | 1157 | 802.7 | 1828 |

Construction | 625.3 | 1644 | 337.3 | 1103 | 705.6 | 1757 |

Land and structure | 89.48 | 448.6 | 62.24 | 332.8 | 97.08 | 475.7 |

Instructional equipment | 46.31 | 87.50 | 61.53 | 151.2 | 42.06 | 57.73 |

Demography | ||||||

Enrollment | 2937 | 7249 | 1845 | 2818 | 3243 | 8036 |

White students | 87.76 | 18.12 | 87.23 | 19.53 | 87.91 | 17.70 |

Free lunch | 32.71 | 18.89 | 33.05 | 20.95 | 32.62 | 18.26 |

Achievement (proficiency) | ||||||

4th grade reading | 70.15 | 18.54 | 68.72 | 19.98 | 70.49 | 18.18 |

7th grade reading | 61.83 | 18.85 | 61.67 | 19.79 | 61.87 | 18.62 |

Number of districts | 577 | 144 | 433 | |||

Sample size | 7751 | 1693 | 6058 |

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:

Relative year | 4th grade | 7th grade | Capital expenditure | Enrollment |
---|---|---|---|---|

1 | -0.450 | -0.468 | 677.7 *** | 350.8 |

(0.539) | (0.573) | (71.01) | (299.5) | |

2 | 0.390 | -0.375 | 3075 *** | 417.9 |

(0.521) | (0.594) | (175.7) | (329.5) | |

3 | 0.768 | 0.389 | 1937 *** | 558.2 |

(0.559) | (0.567) | (165.5) | (352.4) | |

4 | 0.608 | 0.996 | 355.2 *** | 768.2 ** |

(0.502) | (0.578) | (137.6) | (390.3) | |

5 | 0.493 | -0.283 | -211.6 * | 895.7 ** |

(0.513) | (0.552) | (127.1) | (398.1) | |

6 | 0.244 | 0.515 | -138.3 | 853.8 ** |

(0.563) | (0.639) | (137.5) | (417.8) | |

7 | 0.227 | 0.947 | -129.3 | 682.1 |

(0.547) | (0.594) | (115.1) | (423.9) | |

8 | 0.774 | 1.118 * | -303.4 *** | 624.9 |

(0.527) | (0.593) | (102.1) | (433.3) | |

9 | -0.067 | 0.537 | -95.29 | 685.4 |

(0.573) | (0.653) | (88.59) | (422.7) | |

10 | 0.963 | 0.431 | -95.66 | 540.3 |

(0.625) | (0.741) | (110.2) | (475.1) | |

11 | 1.181 | 0.287 | -46.58 | 442.7 |

(0.758) | (0.808) | (120.8) | (536.1) | |

12 | 1.368 | -0.311 | -157.3 | 484.3 |

(0.884) | (0.802) | (200.3) | (620.9) | |

13 | 1.552 | 1.227 | 68.41 | -675.3 |

(1.305) | (1.110) | (168.2) | (822.5) | |

Sample size | 7244 | 7219 | 7746 | 7751 |

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.

All elections (1) | Passed elections (2) | Failed elections (3) | Difference (2)-(3), t test (4) | |||||
---|---|---|---|---|---|---|---|---|

mean | standard deviation | mean | standard deviation | mean | standard deviation | difference in mean | p-value | |

Bond characteristics | ||||||||

Bond amout per pupil | 9371 | 6858 | 8042 | 5888 | 11350 | 7682 | -3309 | 0.000 |

Repayment time | 25.19 | 4.739 | 24.48 | 5.035 | 26.44 | 3.876 | -1.957 | 0.000 |

Millage rate | 3.375 | 1.886 | 2.879 | 1.707 | 4.255 | 1.874 | -1.376 | 0.000 |

Purpose (new building) | 0.372 | 0.485 | 0.330 | 0.472 | 0.432 | 0.498 | -0.102 | 0.149 |

Referendum characteristics | ||||||||

First in the year | 0.951 | 0.216 | 0.946 | 0.225 | 0.957 | 0.202 | -0.011 | 0.445 |

Date of referendum | 188.4 | 86.50 | 192.7 | 86.09 | 182.1 | 86.83 | 10.62 | 0.066 |

Year of referendum | 2001 | 3.960 | 2001 | 4.009 | 2001 | 3.869 | 0.522 | 0.048 |

Number of voters | 2721 | 4807 | 2725 | 5647 | 2716 | 3174 | 8.528 | 0.979 |

Sample size | 936 | 560 | 376 | 936 |

Notes: Average bond amount per pupil is measured in constant 2000 dollars. Referendum data is measured as day of year.

Relative year | Main Specification | Controlling for Bond and Referendum Characteristics | ||
---|---|---|---|---|

4th grade (1) | 7th grade (2) | 4th grade (3) | 7th grade (4) | |

1 | -0.450 | -0.468 | -0.453 | -0.872 |

(0.539) | (0.573) | (0.563) | (0.606) | |

2 | 0.390 | -0.375 | 0.507 | -0.699 |

(0.521) | (0.594) | (0.560) | (0.628) | |

3 | 0.768 | 0.389 | 0.784 | 0.134 |

(0.559) | (0.567) | (0.621) | (0.602) | |

4 | 0.608 | 0.996 | 0.787 | 1.262 ** |

(0.502) | (0.578) | (0.532) | (0.595) | |

5 | 0.493 | -0.283 | 0.781 | -0.334 |

(0.513) | (0.552) | (0.586) | (0.584) | |

6 | 0.244 | 0.515 | 0.389 | 0.245 |

(0.563) | (0.639) | (0.644) | (0.695) | |

7 | 0.227 | 0.947 | 0.600 | 1.095 * |

(0.547) | (0.594) | (0.613) | (0.636) | |

8 | 0.774 | 1.118 * | 0.928 | 1.139 * |

(0.527) | (0.593) | (0.629) | (0.626) | |

9 | -0.067 | 0.537 | 0.296 | 0.310 |

(0.573) | (0.653) | (0.659) | (0.712) | |

10 | 0.963 | 0.431 | 1.090 | 0.704 |

(0.625) | (0.741) | (0.691) | (0.807) | |

11 | 1.181 | 0.287 | 1.319 | 0.384 |

(0.758) | (0.808) | (0.834) | (0.863) | |

12 | 1.368 | -0.311 | 1.306 | -0.299 |

(0.884) | (0.802) | (0.962) | (0.864) | |

13 | 1.552 | 1.227 | 1.555 | 1.192 |

(1.305) | (1.110) | (1.333) | (1.126) | |

Sample size | 7244 | 7219 | 7244 | 7219 |

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:

Relative year | Main Specification (1) | Adding Pre-Existing Proficiency | Alternative 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) | |

2 | 0.390 | 0.027 | -0.265 | 0.059 | 0.102 | 0.030 |

(0.521) | (0.471) | (0.482) | (0.468) | (0.503) | (0.502) | |

3 | 0.768 | 0.382 | 0.185 | 0.360 | 0.636 | 0.445 |

(0.559) | (0.490) | (0.519) | (0.490) | (0.537) | (0.549) | |

4 | 0.608 | 0.422 | 0.179 | 0.546 | 0.478 | 0.358 |

(0.502) | (0.426) | (0.454) | (0.425) | (0.478) | (0.487) | |

5 | 0.493 | 0.308 | 0.065 | 0.533 | 0.322 | 0.067 |

(0.513) | (0.458) | (0.470) | (0.468) | (0.514) | (0.515) | |

6 | 0.244 | 0.087 | -0.170 | 0.294 | 0.083 | -0.182 |

(0.563) | (0.508) | (0.523) | (0.508) | (0.557) | (0.551) | |

7 | 0.227 | -0.008 | -0.217 | 0.221 | -0.061 | -0.206 |

(0.547) | (0.483) | (0.487) | (0.481) | (0.540) | (0.524) | |

8 | 0.774 | 0.476 | 0.305 | 0.536 | 0.396 | 0.408 |

(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) | |

10 | 0.963 | 0.675 | 0.569 | 0.693 | 0.567 | 0.786 |

(0.625) | (0.534) | (0.553) | (0.523) | (0.589) | (0.509) | |

11 | 1.181 | 0.686 | 0.467 | 0.765 | 0.742 | 0.901 |

(0.758) | (0.693) | (0.719) | (0.687) | (0.723) | (0.722) | |

12 | 1.368 | 0.921 | 0.703 | 0.571 | 1.137 | 1.172 |

(0.884) | (0.810) | (0.819) | (0.836) | (0.836) | (0.837) | |

13 | 1.552 | 1.189 | 0.815 | 1.481 | 1.416 | 1.494 |

(1.305) | (1.122) | (1.143) | (1.089) | (1.188) | (1.200) | |

F Test (p-value) | 0.395 | 0.688 | 0.806 | 0.473 | 0.531 | 0.587 |

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:

Relative year | Latent factor (main) | Pooled OLS 1 (1) | Pooled OLS 2 (2) | Fixed effect (3) |
---|---|---|---|---|

1 | -0.450 | -0.613 | 1.819 *** | 1.838 *** |

(0.539) | (0.997) | (0.679) | (0.686) | |

2 | 0.390 | 0.418 | 2.477 *** | 2.407 *** |

(0.521) | (1.017) | (0.732) | (0.708) | |

3 | 0.768 | 2.078 ** | 2.939 *** | 2.776 *** |

(0.559) | (0.986) | (0.711) | (0.695) | |

4 | 0.608 | 3.287 *** | 3.026 *** | 2.816 *** |

(0.502) | (1.028) | (0.700) | (0.679) | |

5 | 0.493 | 4.291 *** | 3.014 *** | 2.897 *** |

(0.513) | (1.044) | (0.704) | (0.676) | |

6 | 0.244 | 5.094 *** | 2.480 *** | 2.326 *** |

(0.563) | (1.118) | (0.742) | (0.733) | |

7 | 0.227 | 6.531 *** | 1.974 *** | 1.860 *** |

(0.547) | (0.977) | (0.693) | (0.692) | |

8 | 0.774 | 10.26 *** | 2.561 *** | 2.421 *** |

(0.527) | (0.962) | (0.655) | (0.651) | |

9 | -0.067 | 11.27 *** | 1.497 ** | 1.392 ** |

(0.573) | (1.078) | (0.684) | (0.700) | |

10 | 0.963 | 13.06 *** | 1.701 ** | 1.640 ** |

(0.625) | (1.196) | (0.720) | (0.754) | |

11 | 1.181 | 13.68 *** | 2.152 ** | 1.964 ** |

(0.758) | (1.371) | (0.872) | (0.909) | |

12 | 1.368 | 12.07 *** | 2.149 ** | 2.097 ** |

(0.884) | (1.575) | (1.040) | (1.051) | |

13 | 1.552 | 12.68 *** | 1.865 | 2.219 |

(1.305) | (2.150) | (1.466) | (1.457) | |

Year fixed effect | Yes | No | Yes | Yes |

District fixed effect | Yes | No | No | Yes |

Latent preference | Yes | No | No | No |

Sample size | 7244 | 7244 | 7244 | 7244 |

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

Since the latent factor is standardized (

Solving the system, I get

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

Thus, given

I then use the regression scoring method (Thomson 1951) to estimate the preferences for educational investment for each observation of district-year, and get

where

Relative year | Regression scoring | Bartlett method | ||
---|---|---|---|---|

4th grade | 7th grade | 4th grade | 7th grade | |

1 | -0.450 | -0.468 | -0.796 | -0.882 |

(0.539) | (0.573) | (0.537) | (0.574) | |

2 | 0.390 | -0.375 | 0.237 | -0.472 |

(0.521) | (0.594) | (0.521) | (0.597) | |

3 | 0.768 | 0.389 | 0.694 | 0.149 |

(0.559) | (0.567) | (0.564) | (0.571) | |

4 | 0.608 | 0.996 | 0.555 | 0.896 |

(0.502) | (0.578) | (0.505) | (0.578) | |

5 | 0.493 | -0.283 | 0.388 | -0.393 |

(0.513) | (0.552) | (0.515) | (0.553) | |

6 | 0.244 | 0.515 | 0.128 | 0.371 |

(0.563) | (0.639) | (0.565) | (0.640) | |

7 | 0.227 | 0.947 | 0.146 | 0.799 |

(0.547) | (0.594) | (0.550) | (0.594) | |

8 | 0.774 | 1.118 * | 0.730 | 1.078 * |

(0.527) | (0.593) | (0.529) | (0.591) | |

9 | -0.067 | 0.537 | -0.128 | 0.490 |

(0.573) | (0.653) | (0.574) | (0.651) | |

10 | 0.963 | 0.431 | 0.944 | 0.359 |

(0.625) | (0.741) | (0.627) | (0.743) | |

11 | 1.181 | 0.287 | 1.161 | 0.172 |

(0.758) | (0.808) | (0.758) | (0.808) | |

12 | 1.368 | -0.311 | 1.338 | -0.294 |

(0.884) | (0.802) | (0.887) | (0.801) | |

13 | 1.552 | 1.227 | 1.448 | 1.196 |

(1.305) | (1.110) | (1.302) | (1.113) | |

Sample size | 7244 | 7219 | 7244 | 7219 |

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 ^{[28]} Then I estimate the average ITT effect of bond passage through the following equation:

where ^{[29]} All other parameters are the same as in eq. (6) and footnote 14.

where

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:

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

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.

Relative year | 4th grade | 7th grade | Subsequent bond passage | Capital expenditure |
---|---|---|---|---|

1 | -0.459 | -0.505 | 0.083 | 728.0 *** |

(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 | -0.136 | 0.168 | -0.169 | 1001 *** |

(0.705) | (0.663) | (0.125) | (203.4) | |

4 | 0.043 | 0.629 | -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 | -0.217 | 0.306 * | -505.6 *** |

(0.787) | (0.780) | (0.167) | (156.2) | |

9 | -1.291 * | -0.595 | 0.070 | -171.6 |

(0.832) | (0.829) | (0.164) | (153.9) | |

10 | -0.828 | -0.926 | 0.012 | 42.24 |

(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 | -0.565 | 0.170 * | -0.335 * | -210.6 |

(1.446) | (1.279) | (0.202) | (302.0) | |

Sample size | 9665 | 9663 | 2665 | 9829 |

Notes: The specification is the ITT regression described in eq. (16). Clustered standard errors by school district are shown in parentheses. *, ** and *** indicate the statistical significance at 10%, 5% and 1% levels, respectively.

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**Published Online:**2017-9-29

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