5
Special Features of Formula Allocations

For many programs, a basic formula is not in itself sufficient to meet all of the objectives of the allocation process. Special features are introduced for various reasons: to promote more efficient use of program funds, to reflect fixed costs of program operations, to avoid disruptions caused by large year-to-year changes in amounts received, or to negotiate passage and authorization of the allocation program. Generally, a basic formula is a continuous function of such inputs as need, fiscal capacity, and effort; special features can introduce discontinuities. They can sometimes result in a grant system in which resources are not well matched with needs as they change over time.

Special features, such as thresholds for eligibility and upper or lower limits on total amounts (or on match rates or components of the basic allocation formula), operate on an individual year basis. Other features, such as hold-harmless provisions or caps, place constraints on the year-to-year change in the amounts allocated to each recipient jurisdiction. This chapter gives examples of these special features, discusses possible rationales for their inclusion, and examines some of their impacts and consequences.

THRESHOLDS

Thresholds on eligibility to receive funds produce discontinuities. The Title I education program, the community development block grant program, and the HIV Emergency Relief Project Grants under Title I of the



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Statistical Issues in Allocating Funds by Formula 5 Special Features of Formula Allocations For many programs, a basic formula is not in itself sufficient to meet all of the objectives of the allocation process. Special features are introduced for various reasons: to promote more efficient use of program funds, to reflect fixed costs of program operations, to avoid disruptions caused by large year-to-year changes in amounts received, or to negotiate passage and authorization of the allocation program. Generally, a basic formula is a continuous function of such inputs as need, fiscal capacity, and effort; special features can introduce discontinuities. They can sometimes result in a grant system in which resources are not well matched with needs as they change over time. Special features, such as thresholds for eligibility and upper or lower limits on total amounts (or on match rates or components of the basic allocation formula), operate on an individual year basis. Other features, such as hold-harmless provisions or caps, place constraints on the year-to-year change in the amounts allocated to each recipient jurisdiction. This chapter gives examples of these special features, discusses possible rationales for their inclusion, and examines some of their impacts and consequences. THRESHOLDS Thresholds on eligibility to receive funds produce discontinuities. The Title I education program, the community development block grant program, and the HIV Emergency Relief Project Grants under Title I of the

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Statistical Issues in Allocating Funds by Formula Ryan White Care Act impose such thresholds. In the Title I education program, individual school districts are grant recipients. To be eligible for a basic grant, a school district must have at least 10 eligible children and a poverty rate for school-age children of 2 percent or more. A likely rationale for the minimum number of eligibles is that a critical mass of funding is needed to enable a program; funding below that level would be too small to run a Title I program of sufficient size, scope, quality, and impact. A possible rationale for the minimum rate requirement is that a wealthy school district with no more than a 2 percent poverty rate would have relatively little need for Title I funding—such a district would have sufficient resources to address the needs of its at-risk students. Title I concentration grants are designed to provide funds for areas with large concentrations of poor families: a school district must have at least 6,500 eligible children or a 15 percent poverty rate. For either type of grant, a shortfall of one eligible child means that no grant funds are received. Eligibility for these two grants is based on estimates of the numbers of eligible children and poverty rates. These estimates are based on data from sample surveys, the decennial census, and administrative sources and are subject to substantial statistical variation, especially for the smaller school districts. Some school districts that would be eligible if exact counts were available receive no funds, and some that would not be eligible do receive funds. These false negatives can have a large impact on both large and small school districts. A small school district with 10 truly eligible children that failed to receive a basic grant because the estimate was 9 or fewer might be unable to serve these children, and the amount lost could be substantial for a large school district with a true poverty rate at or slightly above the 2 percent level or one whose number of eligible children was close to the 6,500 threshold for concentration grants. In a capped program, false positives take funds away from truly qualifying districts. The community development block grant program features another type of threshold. Annual appropriations for the program are divided, with 70 percent directly allocated by the U.S. Department of Housing and Urban Development to eligible metropolitan cities and urban counties (entitlement communities) and 30 percent allocated to states to be distributed to nonentitled jurisdictions that must apply to the state for funding. Entitlement communities include central cities of metropolitan statistical areas (MSAs), other cities with at least 50,000 population in MSAs, and urban counties, which are counties located in MSAs and having a population of at least 200,000, excluding entitled cities. Decennial census counts

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Statistical Issues in Allocating Funds by Formula or intercensal population estimates are the basis for determining which cities and counties have attained these threshold sizes. Thresholds established for grants under Title I of the Ryan White Care Act are based on population and the number of reported AIDS cases. A metropolitan area becomes eligible for emergency relief project grants if it has a population of 500,000 or more and has reported a cumulative total of more than 2,000 cases of AIDS for the most recent 5 calendar years for which data are available from the Centers for Disease Control and Prevention. Once eligibility is established, the area remains eligible. Zaslavsky and Schirm (2002) conducted simulations to assess how an eligibility threshold in a formula interacts with sampling error in the measure of eligibility that is used. They found that as sampling error increases, the sharp cutoff that is seemingly implied by the threshold is replaced by an increasingly smooth relation between an area’s true need and its expected allocation. The reason is that, as sampling variability increases, an area with true need below the threshold is more likely to have estimated need above the threshold, while an area with true need above the threshold is more likely to have estimated need below the threshold. On average, areas with true need below the threshold get more than they deserve, while areas with true need above the threshold get less than they deserve. Allocations for small areas, which typically have smaller samples and larger sampling errors, tend to be distorted more than allocations for larger areas. Importantly, allocations depend both on formula features and on the statistical properties of estimated formula inputs, which in turn depend on the design and analysis of sample surveys and other inputs. This interaction between formula features and statistical properties of inputs has high leverage and apparently has seldom been taken into consideration by those who design formulas and surveys. Its single-year (cross-sectional) and year-to-year (longitudinal) impacts need to be evaluated in detail. LIMITS Upper and lower limits are used in various ways to constrain the outcomes that would result if an allocation were determined solely by a basic formula. In some instances, limits are imposed on the values that can be taken by selected components of the basic formula. Alternatively, limits may be placed on the allocation determined by the basic formula, whether it be expressed as an amount, a share of the total appropriation, or a federal matching proportion.

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Statistical Issues in Allocating Funds by Formula Examples of such limits can be found in the Title I education and the substance abuse block grant programs. In Title I, state per pupil expenditure (which is multiplied by the estimated number of eligible children to provide an estimate of need) is restricted to lie between 80 and 120 percent of the national average per pupil expenditure. In the block grant program, the cost of services index for a state must lie between 0.9 and 1.1. In financial terms, the most significant limits are those placed on the federal medical assistance percentage (FMAP), which is used in Medicaid and several other formula allocation programs to determine what proportion of state program expenditures will be reimbursed by the federal government. The formula for FMAP is: with the constraint that 0.50 ≤ FMAP ≤ 0.83 The lower limit of 50 percent was retained from a predecessor program that provided a flat matching rate of 50 percent to all states (U.S. General Accounting Office, 1983). At present, no states have per capita income so low that they are affected by the 0.83 upper limit. However, several states with high per capita incomes (11 in FY 2002) receive 50 percent matching funds, which is more than they would receive if there were no lower limit. An unconstrained FMAP matching percentage could be derived from a theoretical construct of how to level the playing field. Therefore, limits on the matching percentage may indicate disagreement on such a construct or other, possibly political, considerations. For some programs, limits are relatively easy to justify. For example, in the federal aid highway program, no state can receive less than 90.5 percent of its estimated contributions to the Highway Trust Fund, which funds the program. The Highway Trust Fund is financed by receipts from user taxes, and it is reasonable that each state should receive at least some minimum proportion of the taxes it provides. In the special education program, no state may receive more than an amount equal to the number of its children receiving special education services multiplied by

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Statistical Issues in Allocating Funds by Formula 40 percent of the average per pupil expenditure in U.S. public elementary and secondary schools. The long-range goal established for this program is to provide federal funding for 40 percent of the cost of special education in each state, but so far the annual appropriations have been insufficient to reach this level of support. Limits can be used to dampen the effects of outliers for formula inputs based on estimates that are highly variable. Limits can also be used to preclude attempts by recipients to increase their receipts by manipulating formula inputs. Several formula allocation programs place a lower limit on the proportion of the total funds allocated to be received by any state. The federal aid highway and EPA state capitalization grants programs each guarantee a minimum share of 0.5 percent to every state, and the substance abuse block grants program guarantees a minimum of 0.375 percent. We have identified two possible justifications for such small-state minimums, one practical and one political. Most programs require states to incur expenses to set up programs to administer the receipt and use of federal grant funds, and some of the costs may be more or less fixed regardless of a state’s population. The other consideration is that all states, regardless of population, have two senators and their votes are needed to pass authorization and appropriation legislation. HOLD-HARMLESS PROVISIONS AND CAPS Limits are imposed on year-to-year changes in the amounts received by states or other recipients. Hold-harmless provisions limit downside changes; caps limit increases. Big swings in allocations can be of great concern to legislators and administrators. Legislators for areas whose amounts or shares decline are likely to face difficult questions from their constituents. On one hand, unpredictable declines in federal program funding can cause difficulties for state and local program administrators, for example, school officials planning budgets for the coming year. On the other hand, most fund allocation programs are designed to meet specific needs and to equalize, at least in part, the fiscal capacity to meet those needs. As needs and fiscal capacities change, allocations should be responsive to those changes. Therefore, except for open-ended programs like Medicaid and foster care, there is a clear trade-off and tension between stability and addressing current needs, especially if annual program funding remains level or declines.

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Statistical Issues in Allocating Funds by Formula Improving formula inputs may improve targeting, primarily by updating and upgrading the quality of the data used to estimate needs and fiscal capacity. Sometimes improved estimators, such as the model-based estimates that have been developed for the Title I education and WIC programs, can be developed and introduced into the formula allocation process. Although on the average they may reflect needs more accurately than the inputs previously used, they are still subject to statistical error, which can be relatively large. For example, the Title I education program requires school-district-specific estimates, and the State Children’s Health Insurance Program (SCHIP) requires state estimates for a narrowly defined subset of the total population. These estimates usually have a high relative variance. Hold-Harmless Provisions In order to maintain some degree of stability, several programs include hold-harmless provisions that guarantee that each recipient entity will receive, at a minimum, a specified proportion of the prior year’s amount1 or share. The specified proportion may be 100 percent, or it may be less than 100 percent. In some programs, hold-harmless provisions remain the same from year to year; in others they are in force for a limited period, especially at times when revised formulas were being introduced. Hold-harmless provisions, in combination with year-to-year changes in total funding, limit the extent to which allocations reflect changes in need. In an extreme case, if there is 100 percent hold harmless and no increase in funding, there will be no change from the previous year’s allocations, regardless of any changes in need or other elements of the basic allocation formula. In at least two programs, hold-harmless provisions have largely neutralized efforts to improve the targeting of allocations to current needs. In the Title I education program, a model-based estimation procedure, with estimates updated biennially, has replaced the earlier use of estimates based on the decennial census, which were updated only every 10 years. However, for FY 1998-2001 Congress enacted a 100 percent hold-harmless provision and provided only a modest increase in the annual appropriation. Therefore, the revised estimates had little effect in shifting funds to the 1   In some instances, the total amount available may be insufficient to meet the hold-harmless guarantee; in such cases, allocations to all participants are “ratably reduced” so that their sum is equal to the total funds available.

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Statistical Issues in Allocating Funds by Formula areas where needs had increased relatively the most rapidly. Indeed, there were instances in which school districts that qualified for concentration grants for the first time received no funds because there was nothing left over after the hold-harmless provision had been satisfied for school districts that had received grants in the previous year.2 In the WIC program, in which similar steps have been taken to improve estimates of need, there is a 100 percent hold-harmless provision (called a “stability grant”). Of the funds available for food grants after allowing for stability grants, 80 percent is used to cover increases in food costs due to inflation, and only the remaining 20 percent is allocated to states that are not receiving their fair shares as determined from current estimates of need. The special education program allocation rules ensure that as long as there is an increase in funds compared with the preceding fiscal year, no state can receive less than its allocation for that year (a conditional 100 percent hold harmless). Additional provisions ensure that states will receive some minimum proportion of any increase in the amount appropriated for the current fiscal year. SCHIP has a hold-harmless provision that applies to shares rather than amounts. Starting with FY 2000, no state’s share can be less than 90 percent of its share for the preceding fiscal year or less than 70 percent of its FY 1999 share. Zaslavsky and Schirm (2002) conducted simulation studies of the relations between hold-harmless provisions and variability in estimates of formula inputs. They found that when there is a hold-harmless provision in a formula that allows an area’s allocation to rise by any amount but fall by only a limited amount, sampling variability in estimates of formula inputs ratchets up allocations over time. Such ratcheting occurs because sampling variability can raise an area’s allocation—perhaps substantially— in a year, but the hold-harmless provision always prevents the allocation from falling very much the next year. The amount of ratcheting increases as sampling variability increases. Because estimates for smaller areas typically have greater sampling variability than estimates for larger areas, the upward bias in allocations from hold harmless is greater for smaller areas; thus, the smaller areas tend to benefit more from a hold-harmless provision than do larger areas. 2   In school year 2000-2001, 23 newly eligible school districts received no concentration grant allocations (see Brown, 2002).

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Statistical Issues in Allocating Funds by Formula Caps Caps, which limit the size of increases, are less common than hold-harmless provisions. In the special education program, no state can receive more than its allocation for the previous year increased by the percentage increase in the total amount appropriated plus 1.5 percent. In SCHIP, starting in FY 2000, no state’s share can exceed 145 percent of its share for FY 1999. MOVING AVERAGES AND HOLD-HARMLESS Stabilizing formula inputs, for example by using moving averages computed by averaging estimates from two or more consecutive years, will stabilize formula outputs. For example, the FMAP matching percentage used in Medicaid and several other matching grant programs specifies the use of moving averages of estimates of state and national per capita income for the three most recent years. As discussed before, Zaslavsky and Schirm (2002) found that sampling variability in estimates of formula inputs ratchets up allocations over time when there is a hold-harmless provision. They also found that using moving average estimates can greatly reduce the biasing effect of a hold-harmless provision and, in fact, can be as effective or more effective than a hold-harmless provision in moderating downward fluctuations in funding. Although moving average estimates will tend to be too high if there is a downward trend in need for an area and too low if there is an upward trend in need, Zaslavsky and Schirm show that using an exponentially weighted moving average that gives more weight to more current data reduces this tendency for a moving average to lag behind a trend. Although the use of moving averages in allocation formulas can eliminate or sufficiently reduce ratcheting, it may not in itself be sufficient to achieve the desired level of stability, and hold harmless may still be needed. The authorizing legislation for SCHIP required that the allocation formula use three-year moving averages of Current Population Survey state estimates of children eligible for the program. However, the year-to-year variation in these moving averages proved to be so large that, as noted earlier, it was considered necessary to introduce a hold-harmless provision ensuring that no state would receive less than 90 percent of its share for the preceding year.

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Statistical Issues in Allocating Funds by Formula STEP FUNCTIONS The statutory hold-harmless provision for basic grants under the Title I education program specifies the guaranteed proportion of the prior year’s grant as a function of the estimated poverty rate (the percentage of eligible children in the school-age population). The hold-harmless proportion is 95 percent if the estimated poverty rate exceeds 30 percent, 90 percent if the estimate is between 15 and 30 percent, and 85 percent if the estimate is below 15 percent. Thus, a difference of one person in the numerator or denominator of the estimated poverty rate could make a substantial difference in the amount received by a school district. Such discontinuities can be avoided by making the hold-harmless proportion a smooth and slowly changing function of the poverty rate.3 BONUSES AND PENALTIES Several formula allocation programs have provisions for bonuses and penalties. Although these provisions usually do not affect the initial allocations for the current fiscal year, they can lead to subsequent additions to or subtractions from a state’s allocation as determined by the basic formula. The Temporary Assistance for Needy Families (TANF) program includes an annual appropriation of $1 billion through FY 2003 for bonuses to the states that do best in moving aid recipients into jobs and an appropriation of $100 million to reward states that are most successful in reducing the number of out-of-wedlock births and abortions. The program also has contingency and loan funds that are used to assist states experiencing economic downturns. Penalty provisions are more common than bonuses. TANF has several penalties that are assessed on states failing to satisfy work requirements, failing to comply with paternity establishment and child support enforcement requirements, and failing to meet state maintenance of effort requirements. Penalties are assessed as a varying percentage of state allocations and states that are penalized must expend additional state funds to replace the amounts lost. The total of penalties assessed in a fiscal year may not exceed 25 percent of a state’s block grant. 3   In recent years, the statutory provision has been overridden by insertion at the appropriation stage of a 100 percent hold-harmless provision for both basic and concentration grants.

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Statistical Issues in Allocating Funds by Formula The federal aid highway program, which includes highway planning and construction and several other subprograms, has numerous penalty provisions. State allotments for various subprograms can be reduced by specified percentages for failure to enforce vehicle size and weight laws, to control outdoor advertising, to comply with the 1990 Clean Air Act Amendments, to have a law that prohibits purchase or public possession of any alcoholic beverage by a person under age 21, and to comply with other requirements. The national school lunch program includes maintenance of effort requirements. States are required to appropriate or use for program purposes state funds amounting to up to 30 percent of the federal funds received. The 30 percent requirement is reduced for states whose per capita income is less than the national average. If a state fails to meet its requirement, certain federal funds are subject to recall by or repayment to the U.S. Food and Nutrition Service. These examples illustrate the carrot and stick aspect of federal grant programs, whereby payments to states are made contingent on behavior considered by Congress and executive branch agencies to be in the national interest.4 SUMMARY Special features, such as thresholds, limits, hold-harmless provisions, and caps, are used in many formula allocation programs to serve various purposes. They can lead to allocations quite different from those that would result if only the basic formula is used. Any attempt to evaluate the performance of a formula allocation program must, of necessity, take account of the effects of these special features, both on initial allocations and on changes over time. Generally, policy analysts and congressional staff simulate one year’s allocations under different formula provisions, and sometimes longitudinal assessments are conducted to explore how allocations vary over time under different hold-harmless levels (including no hold harmless) and other provisions. Similarly, longitudinal analyses are needed to assess how program operations in an area might be affected by, for example, a 10 percent reduction in funding, especially if need had really decreased by 10 percent. 4   For more on this topic, see the series of articles on welfare quality control programs (estimation of penalties), from the Journal of the American Statistical Association, vol. 85, no. 411 (1990): articles by Kramer (pages 850-855), Hansen and Tepping (856-863), Fairley et al. (874-890), Puma and Hoaglin (891-899).