Estimating the cost function of connecticut public K–12 education: implications for inequity and inadequacy in school spending

ABSTRACT

This paper is the first to estimate the cost function of Connecticut public K–12 education and to evaluate the state's school spending based on regression-estimated education costs. It finds large disparities across districts in education costs and cost-adjusted spending. A large percentage of the state's public school students are enrolled in districts where spending is inadequate relative to the predicted cost of achieving a common student performance target. Thus, many school districts, especially the high-cost ones, need a large amount of additional spending to improve student performance.

KEYWORDS:

• Cost function
• education costs
• school spending
• equity in education finance
• adequacy in education finance

JEL CODES:

• H75
• I21
• I22
• I28

Acknowledgments

The author thanks Jeff Thompson, Rubén Hernández-Murillo, and participants at the Federal Reserve Bank of Boston Research Department's Brown Bag Seminar and the annual meeting of the Federal Reserve System Committee on Regional Analysis for helpful comments. Lan Ha and Eli Inkelas provided excellent research assistance.

Disclaimer

The views expressed herein are those of the author and do not indicate concurrence by the Federal Reserve Bank of Boston, the principals of the Board of Governors, or the Federal Reserve System.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 See http://schoolfunding.info/.

2 Whereas a typical student without additional learning needs receives a weight of 1 in the formula, a low-income student receives a weight of 1.3, and an English-learner student receives a weight of 1.15. Furthermore, the state gives an additional weight of 0.05 to each low-income student who pushes their school districts above the concentrated-poverty threshold, which is defined as low-income students making up 75 or higher percent of the enrolled students.

3 The U.S. Bureau of Labor Statistics defines an LMA as ‘an economically integrated area within which individuals can reside and find employment within a reasonable distance or can readily change jobs without changing their place of residence.’ (See https://www.bls.gov/lau/laufaq.htm#

4 Downes and Pogue (1994) and Duncombe and Yinger (2007) include fixed effects for school districts. Other studies without geographic-level fixed effects include Duncombe, Ruggiero, and Yinger (1996); Duncombe and Yinger (1997, 1998, 1999, 2000, 2005a, 2005b, 2006, 2011a, 2011b); Imazeki (2001, 2008); Duncombe (2002, 2006, 2007); Duncombe, Lukemeyer, and Yinger (2003, 2004); Imazeki and Reschovsky (2003, 2004, 2006); Reschovsky and Imazeki (2003); Gronberg et al. (2004); Gronberg, Jansen, and Taylor (2011).

5 See https://portal.ct.gov/-/media/SDE/Digest/2016_17/nextgenFAQ.pdf

6 See Duncombe and Yinger (2011a) for the mathematical proof. Even Imazeki (2008) cautions that ‘estimating the production function … is an approach that must be explored a great deal more before one would want to use the results to draw any conclusion about actual costs.’ (103)

7 However, based on their Texas experience, Imazeki and Reschovsky (2005) point out that ‘the courts are capable of understanding and using the results of complex statistical analysis.’ (123)

8 One potential area of inefficiency is that school districts may pay teachers higher wages if they have stronger teachers' unions with greater bargaining power. Winkler, Scull, and Zeehandelaar (2012) show that 98.8 percent of public-school teachers in Connecticut were union members, according to the National Center for Education Statistics' Schools and Staffing Survey for academic year 2007–2008. This is the highest percentage among all states in that year. There is no publicly available data on district-level teachers' union membership. But given that nearly all public-school teachers in the state are unionized, variation in union density across Connecticut school districts is likely to be small. In addition, Winkler, Scull, and Zeehandelaar (2012) indicate that collective bargaining is mandatory for public schools in Connecticut and state law permits teachers' unions to automatically collect agency fees from non-members. As such, there is no variation in the legal scope of bargaining among school districts in the state. To the extent that the strength of teachers' unions is affected by the business cycle, this is controlled for by year fixed effects in the regressions. If the strength of teachers' unions is related to the labor market competitiveness, LMA fixed effects help to account for it. If there is remaining variation in the strength of teachers' unions within an LMA, it is likely to be correlated with resources available to the school district, political leaning and socioeconomic and demographic characteristics of the population in the school district. These are accounted for through such control variables as property tax base per pupil, median household income, the percentage of registered Republican voters, the percentage of population aged 65 and older, and the percentage of adults with a bachelor's degree or higher.

9 I tried using district fixed effects. But there is not enough variation within districts in the sample period to identify the regression coefficients.

10 An alternative functional form of the cost function is a translog model (for example, Gronberg et al. 2004). While this is a more flexible specification, it adds a large number of interaction variables and quadratic terms, which makes it much more difficult to identify the coefficients with precision and interpret the results (Imazeki and Reschovsky 2005; Duncombe and Yinger 2011a).

11 I omit teacher salaries in the regression because LMA fixed effects help control for differences in teacher salaries across LMAs. Reschovsky and Imazeki (2003), Imazeki and Reschovsky (2003, 2004, 2005), and Baker et al. (2018) include the Education Comparable Wage Index (ECWI) as a proxy measure of teacher salaries, without including LMA or other geographic-level fixed effects. The National Center for Education Statistics develops this index using the salaries of college graduates in the school district's LMA who are not teachers. Because the ECWI is defined at the LMA level and does not change substantially in a short time, LMA fixed effects should already capture most (if not all) of the effects of the ECWI. Therefore, I do not concurrently include the ECWI and LMA fixed effects in the regression equation. In fact, when I include the ECWI without LMA fixed effects, the ECWI is positive but not statistically significant, which suggests that wage differences do not play an important role in cost disparities across Connecticut school districts.

12 There are six federally designated LMAs in Connecticut. I also tried using county fixed effects instead of LMA fixed effects. (There are eight counties in Connecticut, which mostly overlap with LMAs.) The regression results are similar to those using LMA fixed effects.

13 School districts in the same labor market area share the same labor pools (teachers and school managers) and compete for the same students whose families decide where to live in the area. For this reason, school districts in the same LMA are more comparable than those in different LMAs for the regression purpose.

14 See Duncombe and Yinger (1997, 2000, 2005a, 2005b, 2006, 2007, 2011a, 2011b), Duncombe (2002, 2006, 2007), Duncombe, Lukemeyer, and Yinger (2003), Imazeki and Reschovsky (2003, 2004, 2006), Imazeki (2008) and Baker et al. (2018).

15 Martorell, Stange, and McFarlin (2016) find that school facility investments have no effect on student achievement.

16 Transportation and food services expenditures are relatively small. On average, they were less than 8 percent of total current spending in Connecticut in 2013. I find that including spending on transportation and food services has virtually no impact on the cost coefficients, except to somewhat raise the coefficient on the dummy variable for regional school district. This likely reflects that regional districts tend to be rural and encompass a large area. Therefore, it costs them more to provide school transportation on a per-pupil basis.

17 Imazeki (2001) and Duncombe (2002, 2006, 2007) also use a lack of statistical significance as a reason to reject some potential cost factors.

18 The cost measure that is later calculated based on the final cost factors is an empirical, composite measure. It does not require each cost coefficient to reflect only the pure, independent impact of the factor on school spending. In fact, accounting for the correlated impact helps the cost measure to be more comprehensive, given an unavoidable omission of some potential cost factors due to data constraints.

19 First, for each test subject, I calculate a weighted average of the percentage of students who are at or above the proficiency level across grades in each district, using the number of tested students in each grade as a weight. Then, I take the simple mean of the three weighted average percentages of students who are at or above the proficiency level in math, reading, and writing.

20 Numerous studies show that school expenditures have a positive effect on students' achievement. See, for example, Elliot (1998), Guryan (2001), Card and Payne (2002), Deke (2003), Kinnucan, Zheng, and Brehmer (2006), Chaudhary (2009), Roy (2011), Nguyen-Hoang and Yinger (2014), Jackson, Johnson, and Persico (2016), Hyman (2017), Lafortune, Rothstein, and Schanzenbach (2018), and Gigliotti and Sorensen (2018).

21 See Duncombe and Yinger (2005a, 2005b, 2006, 2007, 2011a, 2011b), Duncombe, Lukemeyer, and Yinger (2003), Gronberg et al. (2004), Duncombe (2002, 2006, 2007), Gronberg, Jansen, and Taylor (2011), and Baker et al. (2018).

22 See Duncombe and Yinger (1997, 2000, 2005a, 2005b, 2006, 2007, 2011a, 2011b), Duncombe (2002, 2006, 2007), Duncombe, Lukemeyer, and Yinger (2003), Imazeki and Reschovsky (2003, 2004, 2006), Imazeki (2008) and Baker et al. (2018).

23 Downes and Pogue (1994), Duncombe and Yinger (1997, 2000, 2005a, 2005b, 2006, 2007, 2011a, 2011b), Duncombe (2002, 2006, 2007), Duncombe, Lukemeyer, and Yinger (2003), and Imazeki and Reschovsky (2003) control for per-pupil property values, per-pupil income, or median household/family income in their cost regressions.

24 Downes and Pogue (1994), Duncombe and Yinger (1997, 2000, 2005a, 2005b, 2006, 2007, 2011a, 2011b), Duncombe (2002, 2006, 2007), Duncombe, Lukemeyer, and Yinger (2003), Imazeki and Reschovsky (2003),and Baker et al. (2018) include proxy variables for tax price.

25 However, some previous studies find that the impact of the elderly on school spending can be positive if the elderly are long-term residents and loyal to their communities and schools (Fletcher 2006) or if the capitalization of school spending in house prices benefits the elderly homeowners (Hilber and Mayer 2009).

26 Duncombe and Yinger (2005a, 2006, 2011a) and Duncombe (2006, 2007) control for the percentage of the population that is aged 65 and older. Duncombe and Yinger (1997, 2000, 2005a, 2011a), Imazeki and Reschovsky (2003), and Duncombe (2006, 2007) control for the percentage of adults with a bachelor's degree or higher. Duncombe and Yinger (2000, 2005a, 2011a), Imazeki and Reschovsky (2003), and Duncombe (2006, 2007) control for the percentage of owner-occupied housing units.

27 To measure market competitiveness, Hoxby (2000) develops a Herfindahl index based on enrollment distribution across school districts within the education markets. Imazeki and Reschovsky (2004, 2006), Imazeki (2008), and Baker et al. (2018) include this Herfindahl index in their regressions. However, because this index is often defined at the LMA level and does not change substantially in a short time, I do not include it and LMA fixed effects in the same regression. When I include the Herfindahl index without LMA fixed effects, it is not statistically significant.

28 During the sample period (2009–2013), the number of charter schools did not change in five out of six Connecticut LMAs, ranging from zero to eight. The remaining LMA saw the number of charter schools drop from five in 2009–2011 to four in 2012. Furthermore, the enrollment in charter schools was small and equivalent to between 0.3 and 1.8 percent of the total enrollment in traditional public schools in the same LMA. Given that charter schools play a relatively stable and small role in Connecticut public K–12 education, the LMA fixed effects should help to control for the competition pressure from charter schools (if any) on traditional school districts in each LMA. Estimating the education costs for charter schools and other non-regular local education agencies is beyond the scope of this paper and deserves a separate study.

29 I remove 19 districts that do not have their own high school and do not belong to a regional high school district. These districts instead send their high school students to designated schools in neighbor districts or to private high schools. They pay these students' tuition based on agreements with the recipient districts or private schools. Next, for districts that do not operate their own high school but belong to a regional high school district, I aggregate them to the regional district level. In doing so, I create eight pseudo regional K–12 districts. However, three of them have to be dropped from the regression analysis, because of missing data on student performance.

30 The earliest ACS 5-year estimates were collected over the 2005–2009 period. This paper treats the 2005–2009 ACS data as the 2009 data. Doing so makes the endogeneity of the ACS variables less likely, since the 2009 current spending is unlikely to affect the ACS data collected in the 2005–2008 period. I also tried treating the 2005–2009 ACS data as the 2007 data. This change has little impact on the regression results. However, the 2007 current spending could potentially affect the ACS variables collected in 2008 and 2009 if families moved across school districts in response to changes in school spending. For this reason, treating the 2005–2009 ACS data as the 2007 data are less desirable

31 If student performance improved due to teachers' and students' increased familiarity with the test over time, year fixed effects should help to control for that. In addition, in a robustness check (Column 5 in Table 3), I drop the observations from the first two years of implementing the new test when teachers and students were presumably the least familiar with the test. The results from this robustness check are essentially the same as those using the entire sample period.

32 I run a regression of the average percentage of students meeting or exceeding the achievement standard under the Smarter Balanced Test for each district in year t on a constant term and the average percentage of students reaching or exceeding the proficiency level under the CMT and the CAPT for the same district in year t−6, with t = 2015 to 2019. This regression produces an adjusted R-squared of 0.88, a slope of 1.40, and an intercept of −64.41.

33 The percentage of the property tax base from businesses (which affects the tax price) and the percentage of adults without a high school degree influence the demand for student performance (Duncombe and Yinger 2011b). Baker et al. (2018) suggest that ‘one could use indicators of the education level of the adult population in surrounding districts…, which might indicate competitive pressures to improve educational outcomes at any given spending level,’ as valid IVs to instrument the measure of student performance in the home district (41).

34 When I use the percentage of total revenue from federal sources and the percentage of total revenue from state sources separately in a regression, instead of grouping them together, it has virtually no impact on the cost coefficients.

35 I do not directly include the percentage of special-education students because it is likely to be endogenous. School districts have a financial motive and discretion in determining students' special-education status, given that special education is expensive but the state does not consider the number of special-education students in distributing the Education Cost Sharing grants to school districts. Local news publications report that some Connecticut districts have been accused of intentionally and systematically denying special education to students with disabilities (Thomas 2011, 2014; McCready 2018). In addition, the percentage of special-education students is highly correlated with the percentage of school-age children from families living in poverty and with the percentage of public school students living in single-parent or non-family households. Therefore, these two cost factors should capture most (if not all) of the effect of special-education students. Furthermore, I explore using disability data that were collected from the ACS. However, the ACS changed the disability questions in 2008, resulting in new questions that are incomparable to the previous years' disability questions. Since I use the 2008–2012 ACS 5-year estimates (the first 5-year data with the new disability questions) as the 2012 data, it means that I have only two years of new ACS disability data (2012 and 2013) within this paper's sample period, which is insufficient for regression analysis.

36 In theory, the state could mandate small districts to consolidate. However, until such state-level policy occurs, the size of district enrollment remains outside the direct control of local officials at any given point in time.

37 The quality of the homeless data is poor. Federal and state laws require schools to provide transportation and other costly services to homeless students; therefore, districts may have a financial incentive to under-identify homeless students. Many districts simply rely on self-identifying by homeless families and students, who may be reluctant to do so for fear of stigma. In my data, nearly 56 percent of district-year observations reported zero homeless students. The highest percentage of students who were identified as homeless was 3.9 percent. Housing advocates believe that student homelessness is more widespread and severe than the official data suggest.

38 I tried replacing the percentage of school-age children from families living in poverty with the percentage of the population receiving Temporary Assistance for Needy Families (TANF). In that regression, the TANF variable is positive and significant. However, the TANF variable significantly underestimates the number of low-income students, since only about 1 percent of Connecticut's population received TANF in the 2009–2013 period. In addition, I also tried using the number of children under age 19 and enrolled in Husky A (Connecticut's Medicaid for children) per pupil as an alternative measure of low-income students. This variable is positive but not significant, likely because it reflects not only poverty, but also policy changes. The state significantly expanded the Medicaid program during this period, which resulted in a continuous increase in the Husky A enrollment, despite economic fluctuations.

39 The CEP allows all students, not just low-income students, to receive free meals in the qualified participating districts or schools. To qualify for the CEP, districts and schools must have at least 40 percent of their students directly certified by the state for free meals, without the use of a household application. The state can use administrative data to directly certify (1) students whose households participate in public benefits programs, such as the Supplemental Nutritional Assistance Program (SNAP), TANF, and Medicaid for children, and (2) students in other categorically eligible programs, such as homeless, runaway, migrant, foster care, and Head Start programs. In districts and schools that qualify for and participate in the CEP, parents of students no longer need to submit an application for FRPL. However, the Connecticut State Department of Education requires the CEP districts and schools to continue reporting each student's hypothetical eligibility for FRPL by using the following protocol (see https://portal.ct.gov/-/media/SDE/Digest/cep_memo_and_alt_inc_survey_08092014_2.pdf?la=en). These districts and schools should report (1) the FRPL status of directly certified students as eligible for FRPL, (2) the FRPL status of returning students who are not directly certified the same as they were in the previous year, and (3) the FRPL status of new students who are not directly certified based on the ‘alternative income survey’ that their parents are supposed to complete and return to schools. However, parents of new students have no personal incentive to complete the survey, since their children are already guaranteed to receive free meals in these CEP districts and schools. The resulting student-level hypothetical FRPL data are likely to be inaccurate, drawing serious concerns from state officials. When testifying before the Connecticut General Assembly's Appropriations Committee on 6 March 2019, the Connecticut State Department of Education officials highlighted ‘data integrity’ issues in the student-level FRPL data (Connecticut School Finance Project 2019).

40 I do not include all three measures for the individual subjects in one regression because they are highly correlated with each other; the correlations among them are greater than 0.9. As a result, I am unable to find three or more IVs that would separately identify each of the three performance measures if they were included in one regression.

41 I also tried using log of average scale score, either in individual test subjects or averaged over three test subjects. These log variables are positive and significant. Using them does not affect the cost coefficients.

42 I also tried five-year and six-year high school graduation rates, which are available only from 2011 onward. The results using five-year and six-year rates are similar to those using the four-year rates.

43 Imazeki (2001) also finds the graduation rate for high school students never statistically significant in her regressions for Illinois, likely due to a high amount of measurement error.

44 In the CCJEF v. Rell case, the superintendent of the Bridgeport School District acknowledged that the district could award a high school degree to a functionally illiterate student; the presiding state superior court judge called Connecticut's high school graduation requirements and rising graduation rates meaningless (Harris 2016).

45 Its sample period is 2009 through 2012. Year 2013 is dropped because as of that year, the state stopped counting participation in non-degree postsecondary programs as pursuing higher education.

46 In each year and for each subject, there are five measures of growth: (1) growth from grade 3 in the previous year to grade 4 in the current year, (2) growth from grade 4 in the previous year to grade 5 in the current year, (3) growth from grade 5 in the previous year to grade 6 in the current year, (4) growth from grade 6 in the previous year to grade 7 in the current year, and (5) growth from grade 7 in the previous year to grade 8 in the current year. I first calculate a weighted average of these growth measures for each subject, using the number of matched students in the comparison grades as the weight. Then, I take a simple average of the weighted averages between math and reading.

47 This growth measure has several disadvantages over the level measure of student performance. First, Connecticut did not develop vertical scales for the writing tests and therefore does not measure student growth in writing. Second, growth in high school students' performance was not measured. Third, policymakers find it harder to interpret the change in vertical scales between grades than to make year-over-year comparisons in the percentage of students reaching or exceeding the proficiency level. Therefore, the state published the growth measures mainly for informational purposes and did not use them for school accountability. In contrast, it heavily relied on the percentage of students reaching or exceeding the proficiency level to identify low-performing schools and districts.

48 Appendix Table A2 shows that districts with the largest enrollments, the highest school-age-child-poverty rates, or the least amount of property wealth, on average, have the highest or second-highest dollar amounts of current spending on instruction. They spend similar dollar amounts per pupil on support services and other non-instruction programs as districts in other quintiles do.

49 The state could potentially use efficiency-related variables as a policy level to influence the predicted cost measure and then ultimately affect state funding responsibility for schools. Instead of setting $O_{it}$ at the statewide average values in Equation (5(5) ${\mathrm{S}\mathrm{C}}_{it}=\frac{E_{it}}{\widehat{E_{it}}\left(S_t\right)}=\frac{E_{it}}{e^{(\alpha C_{it}+\text{β}{S}_t+\gamma O_t)}},$(5) ), the state could first calculate each district's actual $r{O}_{it}$, rank them from low to high, choose a value from this distribution, and then assign this chosen value to every district in calculating its predicted costs. For example, if policymakers choose the value at the 25th percentile of the distribution of $r{O}_{it}$, it means that the state assumes that every district should operate at the top 25th percentile efficiency level, which will result in a lower predicted cost measure than that calculated at the average efficiency level. The value of r multiplying the statewide average values of $O_{it}$ in FY 2013 is close to both the mean and the median of the statewide distribution of $r{O}_{it}$ that year.

50 I also tried three other performance target levels: the statewide average for students reaching or exceeding the proficiency level, 95 percent, and 100 percent of students reaching or exceeding the proficiency level. The results using these other performance target levels are available upon request.

51 When a district has a negative spending gap, which means that it spends more money than the predicted cost, its needed additional spending is equal to zero. This paper does not allow the needed additional spending to be negative, because it is unlikely that the state will want to force districts to reduce spending. Some districts have legitimate reasons for spending above the predicted cost. For example, they may aim at a performance level that is higher than the common target, or they may spend more on untested subjects in response to parents' demands.