proficiency and then using multiple regressional analysis to determine the dollar amount associated with it based on analysis of extensive data from individual districts. In many studies of this type, no attempt is made to control for the efficiency or inefficiency with which a district is operating. Implicitly such studies take as given the average amount of inefficiency across districts and assume that the efficiency with which a district operates is not correlated with other district characteristics that are included in the statistical model. In effect, inefficiency is assumed to be distributed randomly among school districts. Provided this assumption is reasonably valid, districts with above-average inefficiency are not rewarded when adequacy is defined or aid is distributed on the basis of results from this approach (as they would be, for example, if aid were distributed on the basis of actual expenditures). Nor, however, does this approach provide any incentive for the typical district to operate more efficiently.

The more sophisticated versions of this approach (Duncombe et al., 1996; Duncombe and Yinger 1997, 1999) enter a measure (albeit an imperfect one) of efficiency directly into the model and hence, to the extent that the model is correct, can adjust estimates of adequacy and/or state aid programs to provide incentives for districts to become more efficient.

These versions are more sophisticated in other ways as well. For example, Duncombe and Yinger used a statistical approach to determine the desired objectives of the education system as well as the costs of achieving it. Their approach determined which performance indicators are valued by voters, as indicated by their correlation with property values and school spending. The resulting "index of educational performance" for school districts in New York state includes the average share of students above the standard reference point on 3rd- and 6th-grade Pupil Evaluation Program tests for math and reading, the share of students who receive a more demanding Regents diploma (which requires passing a series of exams), and the high school graduation rate. While this approach results in a performance yardstick, Duncombe and Yinger note that it cannot determine the point on the yardstick that school districts should be expected to meet or that defines an adequate performance. The performance target must be based on the judgment of public officials, though once the target is set its costs can (at least in theory) be calculated via the statistical approach.

Because these indicators (test scores, graduation rates, and Regents diploma awards) may not accurately measure the totality of school outcomes that voters care about, Duncombe and Yinger also estimated models using "indirect" controls for school district performance. In such models, they define average performance as the level achieved by the district with average income, tax price, and voter characteristics, given its teachers' salaries and environmental factors. These (abstract) communities can then be used to observe how much per-pupil spending is necessary to achieve average educational outcomes, again while controlling for other cost or discretionary factors.

Reschovsky and Imazeki (1998) also utilized a statistical method to estimate



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