Abstract
Some education policy studies suggest that consolidation of public school districts saves resources. However, endogeneity in cost models would result in incorrect estimates of the effects of consolidation. We use a new stochastic frontier methodology to examine district expenditures while handling endogeneity. Using the data from California, we find that the effects of student achievement and education market concentration on expenditure per pupil are substantially larger when endogeneity is handled. Our findings are robust to concerns such as instrumental variable adequacy and spatial interactions. Our consolidation simulations indicate that failure to address endogeneity can result in unrealistic expectations of savings.
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Notes
For details, see the expenditure reports of the Institute of Education Sciences (IES).
Even for the use of restricted funds, there is little or no formal accountability, and as explained by Timar (2006), some districts even acknowledge that they spend the restricted categorical funds for general-purpose use.
This statement rules out the entire literature of stochastic frontier analysis, which is acknowledged by strands of studies. A detailed defense of this literature is beyond the scope of our study.
We modify the Cobb–Douglas form by adding the square of enrollment to allow testing the economies of size and the shape of the cost function.
Kumbhakar and Lovell (2003) provide an extensive survey of traditional stochastic frontier models.
See Wang and Schmidt (2002).
We programmed the analysis in this section using Stata 13 and MATLAB 2014a. The programs and the Stata ado files are available upon request. The Stata command is also available on the Statistical Software Components (SSC) at Boston College, and on the Stata repository (package st0466). For more information about this Stata module, see Karakaplan (2017) or type the following command in Stata: findit sfkk.
For statistical and financial purposes, California Department of Education began classifying County Offices of Education, California Education Authority School District, state schools for disabilities, and state charter schools as school districts in 2004. Structurally, however, these institutions are considerably different than traditional elementary, high, and unified public school districts. For example, County Offices of Education provide complementary and supplementary services, staff development, technical assistance, legal and financial advice, curriculum and instructional support to the public school districts. To give another example, Regional Occupational Centers/Programs provide career education, development, and workforce preparation by operating through Joint Powers Agreements (JPA) with traditional school districts, or through County Boards of Education. Without taking such structural differences into account, many print and online sources count these institutions as public school districts due to the change in the classification and report that the number of public school districts in California is more than 1000. Our data set, on the other hand, shows that the number of traditional public school districts in the 2010–2011 school year is 935. Our sample size drops from 935 to 913 due to the unavailability of some of the observations mostly in the student achievement data.
The CWI data are available for all states at http://bush.tamu.edu/research/faculty/Taylor_CWI/.
Even in the age of the Internet, there are Internet costs and shipping costs, which are the counterpart of transportation cost. Since the warehouses of online shops and the hubs of package delivery companies are in central locations, shipping costs for transporting the items to remote locations would be more than transporting them to central locations. In addition, a policy brief of the Rural School and Community Trust by Hobbs (2004) explains how newer education technologies are much more expensive in rural areas. Also, the report indicates that maintenance and technical assistance for these technologies are costlier in remote locations.
We do not control for the percentage middle school students since variation in this variable is low.
The source of these data is demographics from Census 2010.
The source of data is the US Geological Survey (USGS) Geographic Names Information System (GNIS).
The correlation between the proxy for material costs and the number of springs is \(-0.0879\). This outcome would indicate that our material cost/geographic isolation proxy variable is not necessarily correlated with the number of springs topography measure. While the fact that material costs measure is uncorrelated with the springs variable does not necessarily indicate that the measurement error is not correlated with this IV, such a correlation is highly unlikely.
Note that for Model EX, \(\sigma _w^2 =\sigma _v^2 \).
Expenditure variable in our paper aggregates what needs to be spent for variables that other researchers find to be effective to get more quality.
As explained by Karakaplan and Kutlu (2017), estimations with this model are done in a single stage. So, to avoid confusion, we do not use the name “first-stage statistics” and, instead, call them prediction equations.
Our simulation results confirm the possibility of such large biases.
We calculate ln(district API) corresponding to 10% above the average API as ln(1.1 \(\times \) exp(6.664)) = 6.760.
Such a change in a district’s HHI is not necessarily an outcome of a change in the enrollment of the district. The chance in a district’s HHI can solely be a result of changes in the enrollments of other districts in the district’s education market.
We can include only a single bias correction term for enrollment variable and its squared term. Intuitively, this is because the squared enrollment term is a function of enrollment, and conditional on enrollment, these two variables are exogenous. This may be seen from the decomposition of the log likelihood \(\hbox {ln}L\left( \theta \right) =\hbox {ln}L_{y|x} \left( \theta \right) +\hbox {ln}L_x \left( \theta \right) \). In the standard instrumental variable approaches such as 2SLS, a prediction equation for each function of an endogenous variable would be needed. In our setting, however, we need a prediction equation only for the original endogenous variable but not for its functions. For more details regarding this issue, see Wooldridge (2010) and Amsler et al. (2016).
These findings are available upon request.
It is important to note that Table 6 includes only two extended versions of Model EN in Table 3. Model EXs are excluded from Table 6, but available upon request. Assuming that Model EN in Table 3 is the null model, Extended Model EN-1 as an alternative model has a significantly better fit and Extended Model EN-2 has an even better fit than Extended Model EN-1 according to the likelihood ratio test.
School districts have different numbers of criteria to be met depending on the applicability of the criteria.
While the education market areas are larger for each district with the HHI-R15 measure, HHI-R15 of a district can be larger or smaller than, or equal to the HHI measure of that district depending on how much the enrollment and the number of districts in the district’s education market change with the increase in the education market area. In our sample, HHI of 723 districts is larger than their HHI-R15 and 166 districts have smaller HHI than their HHI-R15. HHI of 24 districts is equal to their HHI-R15.
The regression results with 20-mile and 30-mile radii approaches are in line with the results in Table 8, so we omitted them.
CBSAs are identified by the Office of Management and Budget. A metropolitan area contains a core urban population of 50,000 or more, and a micropolitan area contains an urban core population of more than 10,000 but less than 50,000. Each metropolitan or micropolitan area contains counties with the core urban areas and any neighboring counties with a high degree of social and economic integration with the urban core. If a county does not belong to a metropolitan or micropolitan area, then we take that county as a separate educational market.
CDE assigns each cross-county school district to a single county. To determine the locations of public school districts, we use that designation. We get the location and enrollment data of private schools from the National Center for Education Statistics (NCES) Private School Universe Survey (PSS).
California State is known to be carrying out various consolidation projects in the past that the number of traditional public school districts decreased from more than 2,000 to less than 1,000 in fifty years.
Using only the regression subsample is an alternative economic simulation approach. However, our findings indicate that excluding the 22 school districts with missing observations from the analysis does not change the results qualitatively.
There is not any strong factual evidence against this assumption, and families do not have many incentives to move solely due to a consolidation decision of a district, at least not within the first year of consolidation during when the overall consequences of consolidation would be ambiguous to the public. If we relax this assumption, and if families expect that the consolidated district will be worse than their current district, they would move to other districts, or if they are already in other districts, they would not move to the consolidated district. That would decrease the education market share of the consolidated district compared to the sum of pre-consolidation market shares of the consolidated districts and decrease the resources that were available to them to spend. In this case, education market concentration would not increase as much as it would if parents do not move. So, the cost inefficiency of the consolidated district would not be as big as it would be if parents do not move.
In 2010, California has 90 “very small” districts which has less than 100 students. The decrease in the total number of districts is less than 90 because some very small districts have another very small district in the neighborhood so that they merge and form a district with more than 100 students by eliminating two very small districts at the same time.
This assumption is in line with the assumption that parents do not move as an immediate reaction to the consolidation decision of districts. As explained in Footnote 39, we do not have any substantial real-life evidence for students’ changing schools as a response to accommodate the consolidation of traditional school districts.
Median HHI before consolidation is 0.194, and the average HHI of the districts more than or equal to 100 students is 0.240 before consolidation. These numbers are smaller than 0.249 as expected.
Since price index is a proxy based on linear remoteness of a district from the nearest major metropolitan area, the linear average of the price indices from the two consolidation districts is a suitable proxy for the new district.
By efficient cost, we mean the cost when there is no inefficiency.
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Appendix: Baseline prediction equation estimates for endogenous variables
Appendix: Baseline prediction equation estimates for endogenous variables
Dependent variable: ln(district API) | |||
|---|---|---|---|
Constant | 6.7887*** | (0.0341) | [198.82] |
ln(Enrollment) | 0.0253*** | (0.0066) | [3.83] |
\(\hbox {ln(Enrollment)}^{2}\) | \(-\) 0.0013** | (0.0004) | [\(-\) 2.80] |
Percent limited English proficiency | \(-\) 0.0545*** | (0.0150) | [\(-\) 3.62] |
Percent special education students | \(-\) 0.1226** | (0.0463) | [\(-\) 2.65] |
Percent high school students | \(-\) 0.1070*** | (0.0073) | [\(-\) 14.67] |
Percent low-income students | \(-\) 0.2068*** | (0.0114) | [\(-\) 18.14] |
Price index | \(-\) 0.0412 | (0.0313) | [\(-\) 1.32] |
Wage index | \(-\) 0.1137* | (0.0486) | [\(-\) 2.34] |
Unemployment rate | \(-\) 0.0031*** | (0.0007) | [– 4.51] |
Number of springs | \(-\) 0.0001*** | (0.0000) | [– 4.28] |
Observations | 913 | ||
Dependent variable: HHI | |||
|---|---|---|---|
Constant | 0.7466*** | (0.1113) | [6.71] |
ln(Enrollment) | \(-\) 0.1194*** | (0.0215) | [\(-\) 5.56] |
\(\hbox {ln(Enrollment)}^{2}\) | 0.0035* | (0.0015) | [2.42] |
Percent limited English proficiency | \(-\) 0.0061 | (0.0493) | [\(-\) 0.12] |
Percent special education students | \(-\) 0.2189 | (0.1507) | [\(-\) 1.45] |
Percent high school students | 0.0915*** | (0.0238) | [3.85] |
Percent low-income students | 0.1427*** | (0.0372) | [3.84] |
Price index | \(-\) 0.0252 | (0.1020) | [\(-\) 0.25] |
Wage index | 0.4270** | (0.1587) | [2.69] |
Unemployment rate | \(-\) 0.0120*** | (0.0025) | [– 4.88] |
Number of springs | 0.0005*** | (0.0001) | [5.78] |
Observations | 913 | ||
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Karakaplan, M.U., Kutlu, L. School district consolidation policies: endogenous cost inefficiency and saving reversals. Empir Econ 56, 1729–1768 (2019). https://doi.org/10.1007/s00181-017-1398-z
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DOI: https://doi.org/10.1007/s00181-017-1398-z