Center for Policy Research Working Paper No . 60 How Much More Does a Disadvantaged Student Cost ?

This paper provides a guide to statistically based methods for estimating the extra costs of educating disadvantaged students, shows how these methods are related, and compares state aid programs that account for these costs in different ways. We show how pupil weights, which are included in many state aid programs, can be estimated from an education cost equation, which many scholars use to obtain an education cost index. We also devise a method to estimate pupil weights directly. Using data from New York State, we show that the distribution of state aid is similar with either statistically based pupil weights or an educational cost index. Finally, we show that large, urban school districts with a high concentration of disadvantaged students would receive far more aid (and rich suburban districts would receive far less aid) if statistically based pupil weights were used instead of the ad hoc weights in existing state aid programs.


Introduction
Both scholars and policy makers have recognized that it costs more to achieve any given level of student performance when the students are disadvantaged than when they are not.
Nevertheless, scholars and policy makers tend to use different methods to account for these extra costs. This paper provides a guide to statistically based methods for estimating the extra costs of educating disadvantaged students, shows how these methods are related, and compares state aid programs that account for these costs in different ways.
Most scholars have addressed educational costs through the use of an education cost index, which operates much like a cost-of-living index. Specifically, an education cost index indicates the amount a district must spend relative to the average district to obtain the same performance target. Several scholars also have proposed that these cost indexes be used in state education aid formulas, and in particular, that higher-cost districts should receive more aid, all else equal.
Educational costs are also considered by many state aid programs. In fact, a state aid formula that incorporated a regression-based cost index was implemented for towns (including overlapping school districts) in Massachusetts in the 1980s (Bradbury et al. 1984). Cost indexes are rarely used, however. Instead, state aid formulas give extra weight to students in high-cost categories, such as poor students or students with limited English proficiency (LEP). Because state aid is based on the number of weighted students in a district, this approach, like a cost index, results in higher aid for districts with more disadvantaged students. If the extra weight for a poor student is 20 percent, for example, then a district in which half the students are poor will receive 10 percent more aid than a district with no poor students, all else equal. This paper is organized as follows. Section 1 provides background on the scholarly literature and the use of pupil weights in existing state aid formulas. Section 2 provides a guide to calculating pupil weights. This section shows how cost indexes and pupil weights are related, devises a new method for estimating pupil weights, and shows how pupil weights can be incorporated into an aid formula. Section 3 uses data from New York State to illustrate the consequences of various approaches to estimating pupil weights. In particular, this section shows which types of districts gain, and which types lose, when measures of expenditure need or associated state aid payments are based on pupil weights instead of on a cost index. The final section presents conclusions and policy implications.
Existing scholarly work on pupil weights includes Reschovsky and Imazeki (1998), Duncombe (2002), and Duncombe, Lukeymeyer, and Yinger (2003). Reschovsky and Imazeki (1998) start by estimating an education cost function. Then they use the estimated parameters to predict total spending in each district. One of the variables in their cost regression is the share of poor students (as measured by the share of students eligible for a free or reduced-price lunch).
Next they set this variable at a low value (the value below which it has no impact on costs) and predict total spending again. Finally, they obtain a weight for each district by finding the difference between these two predictions, which is the impact of actual poverty in the district on total spending, and dividing this difference by the number of poor students in the district. They find that in both the mean and median district the extra weight for a poor student is 1.59. Duncombe, Lukemeyer, and Yinger (2003) use a similar approach to calculate the cost of bringing a student with a given disadvantage up to the average performance in the state. This approach also results in a different weight in each school district. They estimate that the extra weight for a poor student is 1.10 in the upstate big three cities (Buffalo, Syracuse, and Rochester) and 0.98 in both New York City and the average suburban district. The LEP weight is 1.12 in the Big Three, 1.15 in New York City, and about 1.10 in the average suburb.
As shown in Tables 1 and 2, many state aid programs account for the higher costs of educating disadvantaged students. 1 Table 1 indicates that the weighted-pupil approach is used to adjust the main operating aid formula for poverty in 15 states, for students with limited English proficiency in 9 states, and for students with handicaps in 14 states. The legislated extra weights for students with these disadvantages vary widely across states. Among the states that adjust for poverty, 11 use weights of 0.3 or below, whereas Maryland uses a weight of 1.0 and New Hampshire's weight reaches 1.0 under some circumstances. The LEP weights vary from 0.06 to 1.2. Virtually all of these weights fall well below the values estimated by scholars. The weights for handicaps vary widely, depending on the handicap to which they apply, and no attempt is made to summarize them. Overall, this table testifies both to the intuitive appeal of the weightedpupil approach to aid and to the need for a systematic approach to determining the weights.
A legislated pupil weight may not be used in all state aid programs, and it may be subject to various restrictions. Thus, the effective weight may differ from the legislated weight. Table 2 provides information on effective or implicit poverty weights calculated in several different ways.
This table reveals wide variation in effective poverty weights across states. Alaska, Connecticut, and New Jersey, for example, provide more than twice as much aid for high-poverty districts as for low-poverty districts, whereas New Hampshire provides less aid to high-poverty districts despite a relatively high extra weight (42.6 percent) for poor pupils. Moreover, no state has an effective poverty weight as high as the estimated weight in the scholarly literature. 2 The principle of aid adjustments for student disadvantage has been endorsed by several state supreme courts. In a 1990 decision that called for a more equitable educational finance system, for example, the New Jersey State Supreme Court declared: We have decided this case on the premise that the children of poorer urban districts are as capable as all others; that their deficiencies stem from their socioeconomic status; and that through an effective education and changes in that socioeconomic status, they can perform as well as others. (Abbott v. Burke, 1990, p. 385) This type of argument has appeared in decisions by the highest courts in several other states, including Campaign for Fiscal Equity v. New York, 2003(See Lukemeyer 2004).

How to Calculate Pupil Weights
Pupil weights are designed to indicate the extra expense associated with students in particular categories, holding student performance constant. In principle, these weights should be related to actual experience, that is, to the extra expenses that districts must actually pay to bring disadvantaged students up to a given standard. The existing literature brings in actual experience by deriving pupil weights from the estimated parameters of a standard education cost function. 3 This section begins by exploring various ways to use standard education cost functions to determine the added cost per disadvantaged student in a state, expressed as a share of the cost for a student with no disadvantages. An alternative approach is to specify an education cost function so that the pupil weights can be estimated directly. The second part of this section explores this approach. The third part shows how to incorporate pupil weights into a state aid formula.

Pupil Weights Based on a Standard Education Cost Function
Consider the following cost function, which is similar to the formulation in most of the papers cited earlier: where S j equals spending per pupil in district j; T equals a vector of student test scores and perhaps other performance measures; Z equals other control variables, such as those designed to control for district efficiency; P equals the price of the key input, namely teachers; C i equals the share of students in cost category i; and α and β indicate coefficients to be estimated. By taking logarithms and adding an error term, this equation can be estimated with standard linear regression techniques. Because they are directly influenced by district actions, T and P should be treated as endogenous (see Duncombe and Yinger, 1997, 2000Reschovsky and Imazeki, 1998.) Once Equation (1) has been estimated, a standard cost index is found in two steps. The first step is to calculate the spending required in each district to reach a given performance target, called expenditure need, assuming that districts differ only in their cost characteristics. This step is accomplished by setting the variables in T at the same performance level for all districts (T ); setting the variables in Z at the state average for all districts ( Z ); setting P at the required wage level for each district (P ), based on exogenous factors such as the regional wage level; and setting student characteristics in C at their actual value in each district. 4 Then, with the estimated values of the coefficients, a and b, substituted for the parameters in Equation (1), α and β, one obtains this expenditure need in each district, ˆj S . In symbols, The second step is to divide ˆj S by its value in a district with average required wages and average student characteristics, say * j S , which is defined as 5 * 0ˆe xp{ }.
Equations (2) and (3) lead to a cost index for each district, I j. This index equals 1.0 in a district with average characteristics, is above 1.0 in relatively high-cost districts, and is below 1.0 in relatively low-cost districts. A district with a value of 1.5, for example, has educational costs that are 50 percent above those in a district with average characteristics. The formula for a cost index is Note that the T and Z terms are the same in every district, so they cancel when the expression for I j is written out.
One complicating factor is that educational cost indexes sometimes account for economies and diseconomies of enrollment scale, as well as for teacher costs and student disadvantages. These types of adjustments are somewhat more controversial than others. There is extensive evidence, for example, that small districts have higher costs per pupil than middlesized districts (see Andrews, Duncombe, and Yinger 2002). This can be interpreted as a cost difference, but it can also be interpreted as a sign that the small districts have refused to consolidate with their neighbors and thereby to lower their costs. 6 Similarly, there is evidence that large districts have higher costs than middle-sized districts. This difference may reflect diseconomies of district scale, but it might also reflect mismanagement that arises in some large districts but not in others. Because these issues are not our primary concern in this paper, we calculate pupil weights without considering enrollment. We include enrollment variables in our cost regressions, but we treat them as Z variables. As a result, they are simply set at the average value for all districts and have no impact on the cost indexes or pupil weights.
As shown by Reschovsky and Imazeki (1998) and Duncombe (2002), district-specific pupil weights can be calculated using reasoning similar to that behind a cost index. The first step is to calculate required spending in each district, assuming now that a district has no disadvantaged students at all, that is, that every variable in C has a value of zero. In this calculation, as in a cost index calculation, T and Z are held constant and P is allowed to vary across districts. If district j had no disadvantaged students, in other words, its expenditure need would be: The second step is to find the extra spending in the district because of the presence of students with disadvantage i. This can be found by comparing required spending once disadvantage i is considered with required spending when, as above, one assumes that no students have this disadvantage, or The district-specific weight, i j W , is the extra cost per student with disadvantage i in district j expressed as a share of spending on students with no disadvantages. 7 To find this weight, Equation (6) must be divided by the share of students with this disadvantage and by 0 j S , or ( ) 0ˆe xp{ } 1 .
District-specific weights do not appear in any state aid formula. Instead, states use statelevel weights for each category of student disadvantage. The district-specific weight in Equation (7) can be translated into a statewide rate by averaging it across districts. The simulations in the next section examine statewide weights that are both simple averages and enrollment-weighted averages.
A key question for us to address is: How do measures of a district's expenditure need based on a cost index differ from those based on pupil weights? As discussed earlier, expenditure need equals the amount a district must spend to meet a given performance target, as defined by a set of values for the T variables. Using Equation (2), we know that expenditure need in district j equals the amount a district with average costs must spend to reach these performance targets multiplied by district j's cost index, or Because exp{a} ≈ (1+a) when a is small, we can also write ( ) In the case of pupil weights, the base spending concept refers to spending required to meet a given performance standard assuming no disadvantaged students but actual wages, namely, 0 j S as defined by Equation (5). Total expenditure need in district j equals 0 j S multiplied by the weighted number of students, and student need per pupil (written with a W superscript to emphasize the role of weighting, or ˆW j S ) equals 0 j S multiplied by weighted pupils relative to actual pupils, or, using Equation (7), Using the same approximation as before, we can also write which is the same as Equation (9). In other words, cost indexes and the associated districtspecific weights yield approximately the same measures of expenditure need for each district. In one special case, namely, when there is only one category of disadvantage, there is no need for approximation: according to Equations (8) and (10), these two approaches yield exactly the same measure of expenditure need.
The accuracy of the approximation used in this derivation diminishes as the magnitude of each i i j b C increases. Because this approximation is used to derive both Equations (9) and (11), however, it is not clear how this feature of the approximation affects the difference between these two equations. Switching to state-level weights adds another type of approximation to the mix, one that hurts districts with district-specific weights above the state average. In a later section we use data from New York to explore the nature of these approximations by identifying the types of districts that are put at a disadvantage by the use of various state-level weights instead of a cost index.

Pupil Weights Estimated Directly from an Education Cost Function
The pupil weights in the previous section are approximations because the functional form of a standard education cost function differs from the algebraic form of a student-weight calculation. One way to avoid these approximations, therefore, is to re-specify the education cost function so that it estimates the pupil weights directly.
Consider a cost function of the following form: where, the γ's and the ω's are parameters to be estimated and, as before, T and P are treated as endogenous. This cost function can be estimated with nonlinear two-stage least squares. The ω's are the pupil weights we are after; with this form they can be estimated directly. Let g stand for an estimate of a γ parameter and w stand for the estimate of a ω parameter. Then, drawing on our earlier notation, with a "D" superscript to indicate direct estimation, expenditure need in district j Recall the approximation noted earlier, namely, that exp{a} ≈ (1+a) when a is small.

this approximation translates Equation (12) into Equation (1), or vice versa.
Despite this algebraic connection between the two equations, however, they are substantially different in practice. Compared to Equation (1), the nonlinear Equation (12) requires a more complicated estimating procedure but results in a dramatic simplification in the calculation of weights and student needs.
The obvious question to ask at this point is whether Equation (1) or (12) is a better specification of the cost function, that is, which one provides a better explanation for variation in school costs. 8 This is, of course, an empirical question, which we address in a later section.
However, a specification test alone cannot determine which approach is best for policy purposes.
If the two approaches lead to similar results, then one must weigh the benefits of a relatively simple estimating equation (Equation (1)) against the benefits of a relatively simple pupil-weight calculation (Equation (12)). We return to this issue in our conclusion.

Pupil Weights in State Aid Formulas
The most common type of state aid formula is a foundation formula, which is used to some degree in 43 states (Huang 2004). This type of formula is designed to bring all districts up to a minimum spending level. Another type of aid formula is a so-called "guaranteed tax base" plan, which is the main aid formula in three states and which is combined with a foundation plan in ten others. Except in the case of Missouri, which relies exclusively on a GTB formula, the weights in Table 1 refer to foundation plans. 9 Following the emphasis in existing state aid programs, we focus exclusively on the role of pupil weights in a foundation formula. 10 A foundation formula sets aid per pupil at the difference between an expenditure target, S , and the amount of money a district can raise at a standard tax rate, a rate set by state policy makers. This amount of money is the tax rate, t , multiplied by the district's tax base, V j . To be specific, A more general approach is to select an educational performance target and then to base the formula on the expenditure needed to reach this target. Suppose S is the expenditure needed to reach the desired level of student performance in a district with average costs, namely * j S , as defined by Equation (2). Then, as shown by Ladd and Yinger (1994), a cost index, I j , can be added to yield a performance-based foundation aid program: With this approach, total aid to a district obviously equals aid per pupil multiplied by number of pupils.
Pupil weights are designed to replace some, but not all, of the cost index. Specifically, pupil weights do not account for differences in teacher costs or in enrollment effects across districts. (A few states, namely, Colorado, Florida, Maryland, Massachusetts, and Texas, combine pupil weights with an adjustment for teacher costs or the cost of living.) 11 To bring in pupil weights, therefore, one needs to use a spending base that reflects teacher wages but not student characteristics, namely, 0 j S as defined by Equation (5). 12 Moreover, the number of weighted pupils is (1 ) .
Pupil weighting applies only to the expenditure target in a foundation aid formula, not to the expected local contribution. Introducing pupil weights therefore leads to the following formula for aid per (unweighted) pupil: In this formula, the ratio of weighted to unweighted pupils plays the role of the student-need component of an education cost index. The wage component of 0 j S multiplied by this ratio is equivalent to a full cost index.

Results for New York State
To examine the implications of different approaches to estimating the cost of disadvantaged students, we now use data from New York State for the 2000-2001 school year to compare the distribution of state aid using Equation (15) with the aid using Equation (17) and various forms of pupil weights. As shown in Table 3, we have data for 678 school districts and have classified these districts into eight categories ranging from New York City to small rural districts upstate. 13 These districts differ substantially in terms of enrollment, wages, and the share of students with various disadvantages. This table shows, for example, that the share of students who applied for a free or reduced-price lunch, a commonly used measure of poverty, ranges from 74.9 percent in New York City to 11.2 percent in downstate suburbs. In addition, districts vary widely in their child poverty rates and in their concentrations of students with limited English proficiency or in special education. The special education variable, which provides one way to measure the share of students with disabilities, is discussed in more detail below.
The last column of Table 3 presents a student performance index, which we will use in our cost estimation. This index combines the passing rates on elementary and secondary math and reading tests. The elementary tests cover both fourth and eighth grades, and the secondary exams, called Regents exams, are given twice as much weight because students must pass them to graduate from high school. 14 The resulting index can range from 0 (no students pass any test) to 200 (all students pass every test).

Cost Indexes
We begin by estimating standard education cost models. These models use the functional form given in Equation (1), with operating spending per pupil as the dependent variable.
Following Duncombe, Lukemeyer, and Yinger (2003), the regressions also control for schooldistrict efficiency by including variables that have a conceptual link to efficiency, namely, property value, income, and state aid-all on a per-pupil basis.
We estimate four versions of this model. These models are distinguished by (a) the variable used to measure economic disadvantage and (b) whether special education students are included. We use two different variables to measure economic disadvantage: the child poverty rate in the school district, which is provided by the Census every two years, and the number of students in grades K through 6 who sign up for a free lunch or for a reduced-price lunch. 15 The latter variable fluctuates significantly from year to year, so we use a two-year average in all of our estimations.
Although these two variables are correlated, they are by no means identical. 16 As shown in Table 3, for example, the subsidized lunch variable tends to have a substantially larger value than the child poverty variable. Moreover, the two variables have different strengths and weaknesses. The Census poverty variable has the desirable feature that it cannot be manipulated by school officials, but it is not available every year, it is often excluded from data bases maintained by state education departments, and we have no evidence about its accuracy in years not covered by a decennial census. The subsidized lunch variable has the advantages that it is available every year, is included in many state data bases, and covers a broader population than does the poverty variable. This variable has the disadvantage, however, that it reflects parental participation decisions, and perhaps even school management policies. Given these contrasting strengths and weaknesses, we do not believe that either variable dominates the other and we present results using both of them.
One final difference between the two variables arises when another measure of student disadvantage, the share of students with limited English proficiency (LEP), is added to the cost model. As shown below, this LEP variable is highly significant in cost models that include the census poverty variable. In contrast, this variable is not close to significant in models that include the subsidized lunch variable. Thus, in case of New York, the subsidized lunch variable appears to capture the cost effects both of poverty and of LEP, and the LEP variable is dropped from the models in which the subsidized lunch variable appears.
The second distinction is whether the model includes a third measure of student disadvantage, namely, the share of students in a special education program. We focus on a measure of students with relatively severe disabilities, because these students have a relatively large impact on educational costs and because the identification of these students is largely insulated from district discretion. To be specific, this variable indicates the share of students who require placement for 60 percent or more of the school day in a special class, or who require special services or programs for 60 percent or more of the school day, or who require home or hospital instruction for a period of more than 60 days. As we will see, this variable is highly significant when it is included in a cost regression. However, this variable does not provide a full analysis of the extra costs imposed by student disabilities. It does not include students with relatively minor disabilities, for example, and it does not recognize the wide variations in spending required for different students in the special education category. Moreover, some states prefer to treat special education with categorical grants, instead of incorporating them into basic measures of expenditure need and operating aid. As a result, we present all of our results with and without special education students in the analysis.
These cost models include several cost variables in addition to student characteristics, namely, teacher salaries (treated as endogenous) 17 and student enrollment categories. The omitted enrollment category is districts with enrollment below 1,000 students. As explained earlier, we do not include enrollment effects in our analyses of education costs, expenditure need, or state aid.
Selected parameter estimates from the cost models are presented in Table 4. (Full results for two of these cost models are presented in an appendix. 18 ) The performance index is highly significant in all cases, and the teacher wage variable has an elasticity close to unity. The student characteristics also have large, statistically significant impacts on costs. A school district's costs increase with the share of students in poverty (whether measured by census poverty or subsidized lunches), with limited English proficiency, or with a severe handicap. As noted earlier, the LEP variable is not close to significant in models that use the subsidized lunch variable so it has been dropped from these models.
The second panel of Table 4 presents results for Equation (12), which provides direct estimates of the pupil weights. This equation also performs well, and the results in this panel are similar to those in the first panel. We conducted specification tests to determine whether Equation (1) (the first two panels) or Equation (12) (the last panel) provides a better fit for any given column. 19 We find that neither one of these models can be rejected in favor of the other; that is, there is no statistical basis for selecting one of them. This choice must be made on other grounds.
We then use the cost models in Table 4 to calculate cost indexes, using the approach presented earlier. Our cost indexes reflect teacher wage costs (based on exogenous factors only) and student characteristics. Not surprisingly, the resulting cost indexes vary widely by district category. The first panel of Table 5 presents cost indexes based on the census poverty and LEP variables. As shown in

Pupil Weights
Our next step is to calculate statewide pupil weights and to extract the pupil weights estimated using Equation (12). The results are in Table 6. All the weights in this table are above 1.0, indicating that the cost of educating a student with any one of the three disadvantages we observe is more than twice as high as the cost of educating a student with none of these disadvantages. These weights are therefore higher than the weights used by any state except Maryland (see Tables 1 and 2). Moreover, the weights for special education students are all above 1.8.
In every case, the pupil weight goes up as one moves from column 1 to column 2 or from column 2 to column 3. In other words, enrollment-weighted weights are larger than weights for the average district, and directly estimated weights are larger than the weights calculated from a standard education cost function. In addition, the poverty weights in the first and third models, which are based on the census child poverty variable, decline by a small amount when students requiring special education are added to the analysis, whereas the LEP weight increases slightly when this change is made. Overall, this poverty weight ranges from 1.22 to 1.67, the LEP weight ranges from 1.01 to 1.42, and the special education weight varies from 2.05 to 2.64. Table 6 also presents estimated weights using the number of students applying for a subsidized school lunch. Without either the special education variable or a direct estimating procedure, the extra weight for an economically disadvantaged student is higher with the child poverty variable than with the subsidized lunch variable. If the pupil weights are estimated directly or if the special education variable is included in the estimation, the weight based on subsidized lunch is larger, sometimes considerably larger, than the weight based on census poverty.

Expenditure Need
Tables 7 and 8 compare expenditure need calculations using various approaches to the cost of disadvantaged students. Table 7 is based on the census child poverty variable; Table 8 uses the subsidized lunch variable. The baseline in all cases is expenditure need with a full cost index, which we regard as the most direct approach with the clearest conceptual foundation. Our objective is to determine how much expenditure need diverges from this baseline when pupil weights are used. As explained earlier, pupil weights approximate a cost index approach, so our objective is equivalent to calculating which categories of districts are placed at a disadvantage by this type of approximation. All our calculations include an adjustment for teacher wages.
The first row in each panel of Tables 7 and 8 compares aggregate expenditure need using the weights identified in each column with aggregate expenditure need using a standard cost index. A value below 1 indicates that aggregate expenditure need falls below the baseline value and a value above 1 indicates that aggregate expenditure need is higher with those weights than with the baseline cost index.
The first column in Table 7 shows how much expenditure need diverges from the baseline when student characteristics are not accounted for at all. In the first panel, without special education, this approach lowers aggregate expenditure need substantially, namely, by almost 30 percent, compared to the baseline and places large cities at a significant disadvantage.
To be specific, the expenditure-need numbers for New York City, Yonkers, and the Big Three fall about 40 percent below the baseline. In contrast, this approach leads to expenditure needs that are only about 10 percent below the baseline in suburbs, both upstate and downstate.
The introduction of pupil weights brings the expenditure need calculations much closer to the baseline for all types of districts. As shown in the second and third columns of the first panel in Table 7, expenditure need falls no more then 8 percent below the baseline for big cities, and no more than 1 percent below the baseline for suburbs (on average), when estimated statewide pupil weights are used. Because the enrollment-weighted average weights tend to be larger than the simple average weights, the use of an enrollment-weighted average boosts expenditure need and narrows the divergence from the baseline. Indeed, the results in the third column of Table 7 reveal almost no divergence from the baseline outside the large cities. The divergence in the large cities is about 6 percent.

One simple approximation to estimated weights that is similar to the program passed in
Maryland is to use a weight of 1.0 for both poverty and LEP. The fourth column of the first panel in Table 7 indicates that this approach provides a reasonable approximation to estimated weights in the suburbs, where expenditure need is about 3 percent below the baseline, but only a rough approximation in the big cities, where expenditure need falls about 15 percent below the baseline.
Finally, as shown in the last column of this panel, a calculation using weights that are directly estimated comes very close to matching the results of a cost-index calculation. Indeed, with this approach, New York City and the Big Three are only 1 percent below the baseline and no group of districts falls above or below the baseline by as much as 3 percent. This result is not surprising; as shown earlier, cost indexes and directly estimated pupil weights are approximately the same thing.
The second panel of Table 7 provides comparable results based on a cost model with special education students included. The results from this model are similar to those in the first panel, although the first two models (with no weights and with simple average weights) and the last model (with directly estimated weights) diverge from the baseline somewhat more than the comparable models in the first panel. With enrollment-weighted weights, for example, the big cities now fall about 10 percent below the baseline.  Table 7, particularly those based on directly estimated pupil weights. Recall that with a single cost variable, as in the first panel of Table 8, a district-specific weight is identical to a cost index. Hence, the only source of deviations from the baseline in the second and third columns of this panel is the averaging procedure. The results in these two columns therefore prove that moving from district-specific weights to statewide weights is unfair to highcost districts, particularly large cities, and that an enrollment-weighted average is preferable to a simple average.
One contrast between Tables 7 and 8 can be found in the fourth column of the panel with special education. In this case, the use of rounded weights (1.0 for subsidized lunch, 1.0 for LEP, and 2.0 for special education) leads to a much larger underestimate of expenditure need, particularly in the big cities, in Table 8 than in Table 7. This understatement is implicitly predicted by the relevant directly estimated weights in Table 6, which are 2.1 for subsidized lunch and 3.0 for special education.

Foundation Aid
As explained earlier, expenditure-need calculations feed into foundation aid formulas.
Thus, baseline state aid is the aid a district would receive with a foundation aid formula that incorporates a full cost index. Our simulations define a baseline aid program by setting the student performance index at 160, which is the current state average. Tables 9 and 10 show how switching to pupil weights alters state aid for each category of district compared to this baseline.
To make the columns comparable, we hold the total budget constant (that is, equal to the baseline amount) in all cases by raising or lowering the foundation level. 20 Results for a baseline aid program defined by a student performance index of 140 are very similar to those in Tables 9 and   10.
As in Tables 7 and 8, the first column of these two tables indicates the impact of ignoring student characteristics. In Table 9, which examines aid programs based on the census poverty and LEP variables, this step would cut the aid of the big-city districts by 20 percent or more (compared to the baseline) and would greatly boost the aid of all other categories of districts.
Indeed, both the upstate and downstate suburbs would receive at least 46 percent more aid, on average, with this approach than with the baseline approach.
The next four columns show that introducing pupil weights would bring all categories of districts much closer to their baseline aid. Indeed, regardless of which pupil weights are used, the big cities would all be within 8 percent of their baseline aid. In all cases, both the upstate and the downstate suburbs receive more aid with pupil weights than with the baseline cost index. In columns two through four, the aid in these districts is between 6 and 20 percent above the baseline. Not surprisingly, the divergence from the baseline is smallest with directly estimated weights (the last column). Indeed, in this case, aid to large cities and suburbs is always within 4.5 percent of the baseline amount.
Note that use of rounded weights is less disadvantageous to large cities in Table 9 than in   Table 7. This result reflects the fact that Table 9 holds the state aid budget constant and thereby, in effect, eliminates the absolute decline in expenditure need in the earlier table. Finally, a comparison of the two panels of Table 9 indicates that deviations from baseline aid are somewhat larger when special education is included in the analysis. However, the difference between a result in the second panel and the comparable result in the first panel is rarely above 2 percentage points.
As shown in Table 10, the patterns across districts are similar when the subsidized lunch variable is used instead of the census poverty and LEP variables. In most cases, the divergence from baseline is somewhat larger in Table 10 than for the comparable result in Table 9, particularly when special education is included. With rounded weights and special education, for example, the Big Three fall 8 percentage points below the baseline when child poverty is used but 16 points below the baseline with subsidized lunch. Most of the other differences are considerably smaller than this.

Conclusions and Policy Implications
There is widespread agreement among scholars, policy makers, and state courts that school districts with relatively high concentrations of disadvantaged students should receive relatively more state aid per pupil, all else equal. In the academic literature, the state-of-the-art approach is to estimate an education cost function that includes measures of student disadvantage, to calculate an education cost index on the basis of this estimation, and then to introduce this education cost index into a foundation aid formula. Although a few state aid formulas contain elements of the cost-index approach, most state aid formulas adjust for the presence of disadvantaged students using pupil weights. Pupil weights appear to be more appealing to policy makers than the more abstract notion of an education cost index. The key problem is that, in almost every case, the weights that appear in state aid formulas are determined on an ad hoc basis and are far below the weights estimated by scholars.
We show that a state aid formula using pupil weights can be thought of as an approximation for a state aid formula using a cost index. The closeness of this approximation cannot be determined a priori, but it can readily be calculated on the basis of an estimated education cost function. We show that a state aid formula combining pupil weights and teacher wage cost adjustments derived from a standard cost function distributes aid in a way that is approximately the same as an aid formula based on a cost index. For the large, urban school districts where most disadvantaged students are concentrated, aid based on statistically based pupil weights provides a reasonable approximation for aid based on the preferred cost-index approach. These two approaches differ somewhat more in their treatment of suburban and rural school districts, which receive almost 20 percent more aid with some types of weights than with a cost index. Finally, switching to a nonlinear cost function that estimates pupil weights directly yields an aid formula that closely approximates the baseline approach in almost every case.
Indeed, with directly estimated weights, aid to big cities never falls more than 4 percent below baseline aid (and aid to suburban districts falls within 5 percent of the baseline), unless special education is included in the formula and subsidized lunch is the measure of poverty.
The pupil weights we estimate are much larger than the weights that appear in any state aid formula except for Maryland's. In a typical aid formula, the extra weight for a pupil from a poor family or with limited English proficiency is about 25 percent. We estimate that these extra weights should be between 111 and 215 percent. The use of pupil weights obviously results in a much poorer approximation of our preferred aid formula when these lower weights are used. At the extreme, defined by extra weights of zero, the aid received by large urban districts falls at least 20 percent below the baseline level, and the aid received by suburban districts may exceed the baseline by over 100 percent. The low weights used in most state aid programs yield results not too different from this extreme case. The key problem, therefore, is not the use of pupil weights per se; it is the use of pupil weights that are far below the levels supported by the evidence.
We estimate similar weights for the census poverty and subsidized lunch variables. We also conclude that in New York the LEP variable need not be included in an aid formula based on the subsidized lunch variable, at least not when the weight on the subsidized lunch variable is high enough, but we find that rounded weights of 1.0 for both subsidized lunch and LEP provide a reasonable approximation to a cost-index approach. We also estimate an extra weight of at least 185 percent for a student in special education, but this weight obviously is linked to our special education variable and may not apply to the special education variables that appear in state aid formulas.
Overall, public officials who design state aid formulas face two key choices regarding disadvantaged students. 21 The first choice is whether to account for the extra cost of educating these students using a cost index or pupil weights. Judging from the choices states have made so far, the use of pupil weights appears to be a more appealing approach, and we show that, for most districts, it can result in aid amounts that closely approximately the aid amounts from a formula based on a cost index, which is the approach many scholars prefer.
The second choice is how to select pupil weights. The ad hoc process used in most states is not up to the task. Indeed, the weights used by most states are far below the weights estimated in this paper and by other scholars. These low weights result in aid payments that support far lower levels of student performance in school districts with more disadvantaged students than in other school districts. This outcome violates the key objective of a foundation aid formula, namely, to bring all districts up to the same minimal performance standard. Finally, we find that any pupil weights based on an estimated cost function provide a reasonable approximation to the use of a full education cost index, but that an even better approximation can be obtained using pupil weights estimated directly from a nonlinear cost function. We find no statistical basis for preferring a standard cost function to this nonlinear version, so the choice of method depends on whether policy makers prefer the complexity of the weight calculation with a standard cost function to the complexity of a nonlinear estimating procedure.
A state aid program is not consistent with student performance objectives unless it accounts for the higher cost of reaching a performance target in districts with a relatively large share of disadvantaged students. The use of state aid formulas with extra weight for disadvantaged students is a reasonable approach to this problem, but the fairness of this approach can be greatly enhanced through the use of statistically based pupil weights.

The information in this table is based on legislative language in various published sources
and web sites, so it may not be complete or include all the latest aid revisions. We are grateful to Yao Huang for compiling this information. Table 2 predate the new aid program in Maryland; this new program may be an exception to this claim.

The figures in
3. Some scholars (e.g., Guthrie and Rothstein 1999) have criticized the cost-function approach and have proposed alternatives, such as the use of professional educators to identify the programs necessary to reach a given performance standard in a school with many disadvantaged students. (The high pupil weights in Maryland are based on a study that uses this so-called "professional judgment" approach. See Maryland Commission on Education Finance, Equity and Excellence 2002.) In our judgment, however, a cost function makes the best use of available information and is the preferred approach. For a detailed discussion of the strengths and weaknesses of the various methods, see Duncombe, Lukemeyer, and Yinger (2004).

4.
A note on notation: A "^" indicates a spending level "required" to reach a performance target under some specified set of conditions (or a wage level required to attract teachers), a "~" indicates a policy parameter, and a "-" indicates a mean value. In a few cases the first and last symbols both appear, indicating the mean of a predicted value.
5. An alternative base in this type of calculation is the value of ˆj S in the average district.
This alternative base leads to similar results, but we find it less appealing because it shifts the focus away from the average values of the student characteristics on which the weights are based.

6.
In spite of these problems, about one-third of the states give more aid to small or sparsely settled districts. See Huang (2004).

7.
If the data made it possible to identify students with multiple disadvantages (which ours do not), then each combination of disadvantages could be treated as a separate cost category.

8.
The specifications in (1) and (12) are not the only possible ones. In fact, some scholars, such as Gyimah-Brempong and Gyapong (1991), have used a more general specification.

9.
Texas is one of the ten states with a first-tier foundation formula and a second-tier GTP formula. It uses pupil weights in both tiers. See Huang (2004).

10.
The widespread use of foundation plans reflects, among other things, a widespread emphasis on an adequacy objective in recent state supreme court decisions concerning education finance. See Lukemeyer (2002Lukemeyer ( , 2004.
11. This information was provided by Yao Huang.

12.
In principle, one could also include enrollment effects in this baseline spending number. 14.
For more information on this index, see Duncombe, Lukemeyer and Yinger (2003). This index is treated as endogenous. We used geographic proximity to identify instruments.
Specifically, our list of potential instruments consists of averages, minimum and maximum values for adjacent school districts for various measures of fiscal capacity (income, school aid, and property wealth), student need (poverty, LEP, subsidized lunch, and special education), physical conditions (pupil density, population density, enrollment), and student performance (test scores). To select instruments from this list, we used three standard rules. The instruments must (1) make conceptual sense, (2) help to explain the endogenous explanatory variables, and (3) not have a significant direct impact on the dependent variable. We also implemented an over-identification test (Woolridge 2003) to check the exogeneity of our final set of instruments and the Bound, Jaeger, and Baker (1995) procedure to check for weak instruments. The latter procedure is not formally specified for a model like ours, so we examined various combinations of the instruments and used the set that produced the highest F-test for most endogenous variables. In most cases, the F-statistic was 5.0 or above, indicating reasonably strong instruments for that endogenous variable. When estimating equation (12) we use the same instruments selected for estimating (1).

15.
The free lunch and reduced-price lunch programs are separate, but we combine them in all our analyses. Eligibility rules and funding for these lunch programs is provided by the federal government. Subsidized lunches are also offered after sixth grade but many eligible students do not sign up for them, so a subsidized lunch variable for nonelementary grades does not appear to be useful.

16.
For the year 2000, for example, the correlation between the share of K-6 students who sign up for a subsidized lunch and the child poverty rate is 0.773. Correlations in other years are similar.

17.
The teacher wage variable was first limited to teachers with five years or less of experience. Teacher wages for individual teachers were then regressed on teacher experience and whether the teacher had a graduate degree. The results of this regression were used to construct a predicted teacher salary for each district for a teacher with statewide average experience (among those with no more than 5 years of experience) and average probability of a graduate degree. The potential instruments for this variable are pupil density in the district, private wages in professional occupations, unemployment rate, concentration of area teachers in the district, and the average (maximum and minimum) salaries of adjacent districts. The final list was selected using the rules presented in an earlier footnote.

18.
The two regressions in this appendix table, along with comparable regressions for other models, which are not presented, indicate that the performance index always has the expected positive impact on costs and is statistically significant. The three efficiency variables also have the expected signs and are significant in most cases, and all districts in all enrollment classes except the largest have significantly lower costs per pupil than districts in the smallest enrollment class.

19.
We use the specification tests in Davidson and MacKinnon (2004, Chapter 15).

20.
These simulations set the required local property tax rate, t , at 1.5 percent, which is lower than the rate in most districts. Alternative tables that hold the foundation level constant and allow the state aid budget to change are available from the authors upon request, as are tables with a performance standard of 140 instead of 160.

21.
Another key choice, which is not examined in this paper, is whether to use a teacher wage index. Even with accurate pupil weights, an aid formula would not be fair to high-wage locations unless in it included a wage index or a cost of living index. Only about a dozen states have this type of index now (Huang 2004).  (2001); updated with information from various sources as cited in Huang (2004). a. Weights for students with handicaps vary widely depending on the nature of the handicap. b. These states also provide categorical grants for students in this category. c. These states adjust aid per teacher unit for weighted pupils, which is similar to standard pupil weights.  The share of students who require placement for 60 percent or more of the school day in a special class, or require special services or programs for 60 percent or more of the school day, or require home or hospital instruction for a period of more than 60 days. The share of students who require placement for 60 percent or more of the school day in a special class, or require special services or programs for 60 percent or more of the school day, or require home or hospital instruction for a period of more than 60 days.       The base enrollment is 0 to 1000 students. The coefficients can be interpreted as the percent change in costs from being in this enrollment class compared to the base enrollment class.