The Adequacy of Educational Cost Functions: Lessons From Texas

Abstract

Adequacy studies based on cost functions have come under attack. A recent Texas court battle featured two cost function studies that reached markedly different conclusions about the additional funding needed to meet designated performance goals. Some critics see such disparities as indicators of a general futility in the whole education cost function enterprise. We argue that the more appropriate conclusion is that it is critically important to demand best-practice techniques from any analyst of educational costs. This article uses the Texas litigation studies as a lens through which to explore best practices in the estimation of educational cost functions. The analysis highlights five key decisions that researchers must make when using the cost function methodology in an educational setting and explores the implications of the various possible choices using recent data on public schools in Texas. As the analysis demonstrates, some common practices in cost function analyses of education are not best practices, and these deviations from best practice can have a significant impact on the estimated cost of an adequate education.

Acknowledgments

We thank John Yinger for helpful comments and Maria Granda for research assistance on this project. Professors Gronberg and Jansen also thank the Private Enterprise Research Center at Texas A&M University for research support. All views expressed are those of the authors alone.

Notes

See Gronberg, Jansen, Taylor, and Booker (2004, 2005) and Taylor (2004b).

See Imazeki and Reschovsky (2004) and Smith and Seder (2004), respectively.

The GJTB study used data through the 2001–2002 school year. In her extension of the GJTB model, Taylor (2004b) updated the GJTB estimates including data for 2002–03. Because spending increased in most plaintiff districts between 2002 and 2003, the estimate of the total needed to bring all plaintiff districts up to the performance standard, holding all other districts harmless, fell to $68,000. We focus on the original GJTB analysis to remove any differences in model predictions attributable to differences in the time frame for the cost projections.

These estimates adjust for geographic variations in cost using the same index as in the GJTB and I&R models. Using other geographic deflators, the MAP estimates are as low as $504 million and as high as $990 million.

Current operating expenditures per pupil declined 0.5% between 2004 and 2007. As in the Texas studies under evaluation, current operating expenditures per pupil have been adjusted for inflation using the Employment Cost Index.

GJTB used a variant of the translog adapted to incorporate percentage changes in place of natural logs. GJTB also include a cubic term for enrollment.

Technically, the translog model nests the Cobb-Douglas specification.

Other flexible functional forms for the cost function, such as the Generalized Leontief and the Generalized McFadden, also share this property.

The standard theoretical regularity condition requires that the isocost frontiers are boundaries of a convex set in outputs. To illustrate the problem, consider the simple case of two outputs and one input price. The Cobb-Douglas specification would take the form: C(q₁, q₂, w) = Aq₁ ^α,q₂ ^βw^γ. A cost frontier for cost level C⁰ and input price w⁰ would then take the implicit form: q₁ ^αq₂ ^β = K⁰, and (for positive output levels) the upper contour set, rather than the relevant lower contour set of feasible outputs with respect to C⁰ is then convex.

Unfortunately, Taylor (2004b) showed that the I&R model has heteroskedastic residuals even after pupil weighting, so pupil weighting need not address the heteroskedasticity concerns.

Wooldridge (2002) wrote, “It has become popular to estimate β by OLS [ordinary least squares] even when heteroskedasticity is suspected but to adjust the standard errors and test statistics so that they are valid in the presence of arbitrary heteroskedasticity” (p. 56).

An alternative test statistic, Sargan's statistic, can also be used if the disturbance is homoskedastic.

The four school outcome measures in their model were the TAKS-equivalent passing rate, the SDAA passing rate, the retention rate, and the share of students scoring above criterion on the SAT or ACT. Their five instruments, which were designed to reflect the determinants of educational demand, were the median household income, property values per pupil (which I&R labeled “tax price”), the educational attainment of households, the share of households with children, and the share of households that own their homes.

The Shea's partial R ²s are .02, .02, .03, and .07 for the TAKS-equivalent passing rate, the SDAA passing rate, the retention rate, and the SAT/ACT performance measure, respectively, whereas the Stock and Yogo weak identification test statistic (i.e., the Kleibergen-Paap rk Wald F statistic) is only 1.294. The probabilities of a greater Hansen's J statistic or Sargan's statistic are 0.0292 and 0.0038, respectively, rejecting the null hypothesis of exogeneity. We note that Imazeki and Reschovsky (2006) substantially altered the IV specification when their Texas cost analysis was published in a peer-reviewed journal. In particular, Imazeki and Reschovsky dropped one endogenous outcome variable (the annual retention rate) and modified their list of instruments to exclude median household income and include other demographic characteristics.

Assuming the usual symmetry restrictions (d_ij = d_ji, f_ij = f_ji, and k_ij = k_j), the modified translog specification becomes:

It is a modified translog specification in two respects. First, because some of the variables (such as the share of low income students) can take on values of zero, we do not take the log of all variables. Second, for enhanced flexibility the model also includes the cube of the first output measure—student quantity.

We also estimated our baseline model under two alternative one-sided error specifications, the gamma and the truncated normal. The marginal effect and efficiency estimates were quite similar across specifications, so we do not report these alternative results in the article.

Small (less than 1,600 students) and midsize (1,600–5,000 students) school districts receive additional funding under the Texas school finance formula.

I&R included expenditures in these categories in their estimate of current operating expenditures.

We also estimated models that included a measure of quality at the left-tail of the performance distribution—the 4-year longitudinal high school completion rate. The marginal effect of this variable was never significant, and we do not include the completion rate as a regressor in our reported regression results.

I&R and GJTB were able to divide the special education population into two groups based on the severity of the need. Such data are not available to us during the time frame under analysis.

We calculate the marginal effect for a representative variable, say output q₁ as the coefficient on q₁, plus the coefficient on q₁-squared times the mean of q₁, plus the coefficient on q₁q₂ times the mean of q₂, and so on, for all of the interaction terms.

This finding is not sensitive to excluding the two largest districts, Dallas and Houston, from the estimation.

The point estimate on the cubic term is –.00299, with a standard error of 0.0007.

The Farrell measure of efficiency is the inverse of the expected value of the exponent of the one sided error, Eexp(u)|e.

In this context, the Herfindahl index is the sum of the squared enrollment shares in an education market.

The chi-squared test statistic is 266.77 with 65 degrees of freedom, and a probability of a greater chi-squared of 0.0000.

The pupil-weighted model would not converge when the two-sided error was adjusted for heteroskedasticity using both the small and midsized indicators. Therefore only the small district indicator was used. As in the baseline, both the small and midsized indicators were used to adjust for heteroskedasticity with respect to the one-sided error.

If the model is estimated without modeling the heteroskedasticity in the one- and two-sided errors, a test on the residuals suggests that the heteroskedasticity problem remains. That is, pupil weighting alone does not solve the problem.

For the nonteacher wage, we predicted the full-time-equivalent monthly salary for a nonprofessional, noninstructional staffer with average characteristics. For the beginning teacher salary, we predicted the full-time-equivalent monthly salary for a 1st-year teacher who had average characteristics with respect to gender and ethnicity, educational attainment, teaching assignment, certification status, and coaching status. Salary predictions below the state's minimum salary for beginning teachers ($2,732 per month in 2006–07) were set equal to that minimum.

Because the surrounding district choices are, in an urban general equilibrium sense, endogenous, potential alternative sets of candidate instruments are the, arguably, exogenous demand factors that influence the pattern of outcomes in the comparison districts.

The Hansen's J test statistic is 60.819, which is distributed as a chi-squared with 58 degrees of freedom. The probability of a larger test statistic is 0.3748

The DWH test statistics are F tests of the joint significance of the residuals from the first stage of 2SLS in an OLS estimate of the structural equation. The test statistic for the change in passing rate and its interactions is 1.47 with 12 and 819 degrees of freedom, and the probability of a greater test statistic is 0.1291. The test statistic for the percentage taking advanced courses is 3.30, with 12 and 819 degrees of freedom, and the probability of a greater test statistic is 0.0001.

One form of the DWH test uses the residuals from the first stage of 2SLS as additional regressors in an OLS estimation of the structural equation (including the potentially endogenous ones). If the coefficients on the first-stage residuals are jointly significant, then the variables in question are endogenous. Using the same methodology, but estimating the structural equation using stochastic frontier analysis, we reject the hypothesis that the first-stage residuals are jointly insignificant.