(relative to the average district) to meet a selected value of the performance variables, given its own input prices and environmental factors.
By controlling for performance directly, this approach avoids the complexity of the tax-price effect, which was discussed earlier. No fancy algebra is needed to obtain the correct cost index. However, this approach runs into another problem, namely, that unobserved school district characteristics might affect both school spending (the dependent variable) and measures of performance (explanatory variables). This simultaneity problem, like the one associated with teachers' salaries, requires the use of two-stage least squares. Fortunately, instruments for this procedure are readily available and are, in fact, defined by the indirect method just discussed. The indirect method is based on the well-established result that school performance depends on income and tax price, so income and tax price are natural instruments.7
As first pointed out by Duncombe and Yinger (1997), this approach also leads to a school district performance index. This index is a weighted average of the performance measures included in the regression, where the weights are the regression coefficients. Duncombe and Yinger also explain that these weights can be interpreted as demand weights; that is, they reflect the weight households place on each of the performance measures. With this approach, the performance measures used in a final regression are those that prove to be statistically significant; under this demand interpretation, this approach allows a researcher to identify the performance measures that play a significant role in households' demand for educational performance.
In addition, this performance index, with its statistically determined weights, makes it possible to set a single performance standard that covers a wide range of performance indicators. Instead of setting performance standards separately for several different indicators, such as test scores and a dropout rate, policymakers could set a standard based on this index. They could, for example, set the standard at the median of the index's current distribution.
One might criticize the approaches in the two previous sections because they could confuse high cost and inefficiency: large districts may not have higher costs, for example, but may instead just be inefficient. This problem is analogous to the problem of controlling for household choices, about furnace maintenance for example, in the case of home comfort. In technical terms, ignoring inefficiency could lead to ''omitted variable bias" in estimating the effects of environmental factors on costs. In this section we will discuss a method for measuring district inefficiency and including it in an expenditure regression. Inefficiency is, of course, difficult to measure, and the approach we will present is not the final word on the subject, but it does provide one tractable method to control for