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## 9 Conclusions and Recommendations

Mathematical formulas were used to allocate more than \$250 billion of federal funds in 2000 to state and local governments via approximately 180 grant-in-aid programs designed to meet a wide spectrum of economic and social objectives. Large amounts of state revenues are also distributed through formula allocation programs to counties, cities, and other jurisdictions. The essential feature of a formula allocation program is that the amounts to be allocated are determined by a formula that uses statistical information to calculate or estimate the values of its inputs, and these are processed to produce outputs. Often, the allocation process consists of a basic calculation using a mathematical formula or algorithm, followed by adjustments that place constraints on levels or shares (percentages of the total allocation) or on changes in levels or shares.

In addition to providing a mechanism for addressing changes in need and other formula components without the need for Congress to revisit the issue annually, formula-based allocations can help build consensus and credibility by:

• Creating a transparent means of allocating funds.

• Creating a relatively solid foundation on which to negotiate legislation.

• Separating the question of how to distribute aid funds from the question of why they are needed.

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Statistical Issues in Allocating Funds by Formula 9 Conclusions and Recommendations Mathematical formulas were used to allocate more than \$250 billion of federal funds in 2000 to state and local governments via approximately 180 grant-in-aid programs designed to meet a wide spectrum of economic and social objectives. Large amounts of state revenues are also distributed through formula allocation programs to counties, cities, and other jurisdictions. The essential feature of a formula allocation program is that the amounts to be allocated are determined by a formula that uses statistical information to calculate or estimate the values of its inputs, and these are processed to produce outputs. Often, the allocation process consists of a basic calculation using a mathematical formula or algorithm, followed by adjustments that place constraints on levels or shares (percentages of the total allocation) or on changes in levels or shares. In addition to providing a mechanism for addressing changes in need and other formula components without the need for Congress to revisit the issue annually, formula-based allocations can help build consensus and credibility by: Creating a transparent means of allocating funds. Creating a relatively solid foundation on which to negotiate legislation. Separating the question of how to distribute aid funds from the question of why they are needed.

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Statistical Issues in Allocating Funds by Formula Creating the appearance, if not always the reality, of a sound analytic process. Providing a starting point for the reauthorization process. ROLES OF CONGRESS AND THE PROGRAM AGENCIES Program agencies in the executive branch vary in the latitude Congress gives them to decide on the basic formula to be used, the variables (e.g., population size, tax revenue, per capita income) to be used to represent formula components, and the statistical data series to be used to calculate or estimate their values. Variants differ in the control retained by Congress and the required congressional technical expertise and monitoring:1 Congress specifies program goals and intentions, leaving the program agency to specify the allocation formula. Congress specifies the formula, and the program agency specifies which variables should be used and which statistics should be used to estimate them. Congress specifies the formula and variables, and the program agency specifies which statistics are to be used to estimate the variables. Congress specifies the formula, the variables to be used, and the statistics to be used to estimate the variables. Congress specifies the numerical allocations without the need for an explicit formula. Variant (a) allows Congress to delegate all technical issues—to build consensus on a program without resolving the fine details. If the specification of goals and intentions is sufficiently precise, relevant program agencies can use their technical expertise to develop the details. However, if goals are vague or the agency has goals different from Congress, the formula as implemented may be different from what Congress intended. Relative to variant (a), variant (b) allows Congress to reflect its goals in the specification of the formula, giving it more control over the allocations. Under variant (b), Congress must be able to assess the performance of alternative formulas. Still, if there are a variety of ways of defining and estimat- 1   Examples of the different arrangements are given in Chapter 1, in the section on “Alternative Approaches to Fund Allocation.”

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Statistical Issues in Allocating Funds by Formula ing the components of the formula (e.g., need, capacity, effort), the implementation of the formula may be different from that envisioned by Congress. Relative to variant (b), variant (c) allows Congress to be sure that implementation will be closer to what it envisioned. However, variant (c) requires more technical capability and understanding by Congress to ensure satisfactory performance. Under variant (c), Congress still delegates to the program agency the choice of data sources for estimating the inputs. With variant (d), Congress retains full control over the allocation formula, the definitions of inputs, and the statistical data series used to calculate the allocations. It needs some expertise in statistical matters to evaluate and select candidate statistical data series. Variant (e) does not require the use of statistical data in the allocation process; Congress specifies the amounts or shares for each recipient. This variant does not allow the allocations to reflect changes in social conditions over time, except through new legislation. Recommendation 1. For each formula allocation program, an effective trade-off between congressional control and locus of expertise and monitoring must be found. As they have already done for many formula allocation programs, legislators should consider giving some flexibility to program agencies, especially in determining what data sources and procedures should be used to produce estimates of the components of allocation formulas. PERIODIC EVALUATIONS In the initial design stage and in subsequent revisions, formula designers are faced with a large array of choices, including: Which of the basic components—need, fiscal capacity, and effort— should be included in the formula? What conceptual variables should be used to represent those components? What data sources and procedures should be used to estimate those variables? How should the separate components be combined in the basic formula? What limits, if any, should be applied to the values of individual formula components or to formula outputs (recipient amounts or shares)?

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Statistical Issues in Allocating Funds by Formula Formula allocation programs are designed either to operate for an indefinite period of time (with annual allocation of funds appropriated for the program) or to operate for a specified number of years (usually four to six), at which time reauthorization of the program is considered. Formula designers and evaluators need to be aware of this longitudinal nature. Inevitably, there will be changes in the total funds available for the program, the distribution of need among aid recipients, and the nature and quality of the data sources available for use in estimating formula components. Trade-offs between stability of funding and adjustments to meet shifting needs should be considered. Modification of the formula or other program features may be indicated. Changes can be effected either by amending the initial authorization legislation or as part of the annual appropriations process. Changes effected by the latter can be made only for one year at a time; permanent changes require amending the authorizing legislation. Recommendation 2. At reauthorization and possibly at other times, policy makers should evaluate whether a formula allocation program is performing as intended. Evaluations should include a study of how the relations between formula outputs (allocations) and inputs (measures of need, fiscal capacity, and effort) are affected by special provisions, such as hold harmless and small-state minimums. Evaluations should attempt to identify misallocations due to statistical variation or inherent bias in formula inputs and the degree of improvement in the accuracy of allocations that would be achieved by using improved inputs. For some programs, a more comprehensive evaluation may be warranted to consider the quality of services delivered to program beneficiaries, the impact of those services, and program efficiency. Such assessments could be used to determine the net value to society of improving the accuracy of formula inputs or of revising the formula. For example, comparing the potential benefits of conducting a new survey or expanding a current survey to obtain more accurate estimates of need to the costs of data collection would show whether the overall improvement in performance of the formula allocation program justifies the investment in the survey or whether justification must depend on benefits outside the program. The ability to conduct such assessments successfully will depend to a considerable extent on how explicitly the goals of the formula allocation program have been defined in its authorizing legislation.

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Statistical Issues in Allocating Funds by Formula CHOICE OF VARIABLES, ESTIMATION PROCEDURES, AND DATA SOURCES Many aid formulas include components that are intended to represent variation in need among recipient jurisdictions. However, some variables used as proxies for need are not closely linked to the resources needed to provide particular services, and developing good proxies is quite difficult. While there is general agreement that aid formulas should take account of differences in need, efforts to do this are often inadequate. Furthermore, statistical data series used to calculate the formula allocations can be updated and in many cases are revised as additional data become available or refinements are identified. Recommendation 3. The formula design process should include evaluation of the trade-offs in timeliness, quality, costs, and other factors that are inherent in the selection of variables, the methods used to estimate them, and the data sources used to provide inputs. Specific considerations include: —Whether to use direct estimates of a program’s target population or need or to use proxies that may be more current, reliable, or available at less cost. For example, current estimates of total population by state (or population in a specific age range) might be an adequate proxy for some more difficult-to-measure population group. —Whether to use estimates from a single source or model-based estimates that combine data from several sources. For example, the small-area estimates of income and poverty used in the Title I education program combine information from the decennial census with more timely data sources, such as the Current Population Survey and the federal income tax and Food Stamp programs. —Whether to use the latest available data to compute each year’s allocations, while recognizing that updating some series but not others may prove counterproductive. —Whether, when modifications in data series or statistical estimation procedures are expected to produce a substantial change in allocations, to smooth the transition from the old to the new allocations.

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Statistical Issues in Allocating Funds by Formula ANTICIPATING BEHAVIORAL RESPONSES OF AID RECIPIENTS In the absence of legislated controls, there are many program-allowed strategies that recipient jurisdictions can adopt to maximize their benefits in a formula allocation program, and sometimes these strategies run counter to program goals. The most obvious example is substitution, whereby funds provided for a specific program are used to replace existing expenditures for that purpose, leaving total expenditures on the program at or near their original level. Other examples of unintended behavior include: When the measure of need depends on data reported by the recipient jurisdictions and reporting requirements are not clearly defined, some recipients may adopt definitions that work in their favor. When the formula includes current expenditure in the program area as an input, the amount shown in the books could be manipulated to maximize the jurisdiction’s benefit without changing the amount actually spent. When the aid formula includes a representative tax system-style measure of fiscal capacity and a single jurisdiction accounts for almost all of a particular source of revenue (e.g., some varieties of oil in Saskatchewan in the Canadian Equalization Program), there can be a disincentive for the jurisdiction to tax that source, since any increase in the tax rate may be directly offset by a loss of equalization funds. When one jurisdiction is dominant in the state or the nation in which an aid program operates (e.g., the city of New York in the formulas for New York state aid to education), there may be incentives for that jurisdiction to influence overall state or national averages to its advantage. Recommendation 4. Designers of formula allocation programs should evaluate the potential for unintended behavioral responses by recipient jurisdictions. Particular attention should be given to the possibility of substitution, to the ability of recipient jurisdictions to influence input data, and to the effect on calculated amounts that might result from particular actions by dominant jurisdictions. When formulas use data collected and provided by the recipient units, Congress and the relevant agencies should require use of standard definitions and should establish procedures to monitor and control quality.

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Statistical Issues in Allocating Funds by Formula SPECIAL FEATURES: HOLD-HARMLESS PROVISIONS Many formula allocation programs have special features that modify the initial allocations determined by application of the basic formula. Hold-harmless provisions limit the extent to which recipients’ aid amounts can decline from year to year, either in absolute terms or as a share of the total amount appropriated. Their effects on allocations depend on the amount of change permitted (a 100 percent hold harmless on shares guarantees no change in shares; a 100 percent hold harmless on amounts guarantees no reduction in amounts) and on the annual changes in total appropriations available for allocation (increase, decrease, level). Hold-harmless provisions can limit disruptions in program administration and service delivery at state and local levels. However, they can also delay response to changing patterns of need. In the extreme case, a 100 percent hold-harmless provision coupled with no increase in program funding results in no changes from the previous year’s allocations, regardless of changes in the indicators of need, fiscal capacity, and effort. In the presence of sampling variability in the estimation of formula components, hold-harmless provisions can cause a ratcheting-up effect that tends to favor smaller jurisdictions with less stable estimates. Moving averages, used alone or in conjunction with hold-harmless provisions, can reduce this ratcheting effect, but they also may delay responses to significant changes in the distribution of need. Recommendation 5. The probable effects of hold-harmless provisions on allocations should be evaluated before including them in a formula, and their effects should be part of subsequent review. If undesired effects are identified, policy makers should consider such changes as loosening the hold-harmless constraint or introducing the use of moving averages—as an alternative means for reducing the volatility of allocations—to estimate formula components. SPECIAL FEATURES: THRESHOLDS Some allocation formulas include thresholds to ensure that certain designated funds are allocated only to those jurisdictions in which need is relatively great. For example, the concentration grant formula for the Title I education program includes a threshold on the number of poor children and the child poverty rate. With a threshold, a small change in estimated

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Statistical Issues in Allocating Funds by Formula need, whether caused by statistical variation or a change in true need, can substantially affect the funding received by a jurisdiction. Another aspect of thresholds worth noting is that they may, given their all-or-nothing nature, raise the incentive for manipulation by potential fund recipients. Recommendation 6. The effects of a threshold on allocations should be evaluated both before it is included in a formula and subsequently whenever the formula is being evaluated. Evaluation should focus on how errors in estimated need cause jurisdictions to gain or lose funding erroneously and how fluctuations in estimated need affect the stability of allocations over time. If undesired effects of a threshold are discovered, policy makers should consider revising the formula to allow for a more gradual transition from no funding to above-threshold funding. SPECIAL FEATURES: SMALL-STATE MINIMUMS Many formula allocation programs have provisions for small-state minimums, which ensure that no state will receive less than a specified dollar amount or a specified share of the total allocation. A wide range of values have been used for small-state minimums. For example, the State Children’s Health Insurance Program requires that each state receive at least \$2 million annually, and the Substance Abuse Prevention and Treatment Block Grants program requires that no state receive less than 0.375 percent of the total amount allocated to the states. Recommendation 7. Formula designers should examine the degree to which proposed minimum amounts or shares produce departure from allocations based on estimated needs. SIMULATIONS OF FORMULA ALLOCATIONS When Congress develops or reauthorizes a formula allocation program, it is customary to run one or more currently available data series through alternative versions of the allocation procedure. Legislators and staff examine such “simulations” to see whether the results are satisfactory or whether the formula or data require modification. Sometimes program agencies, the General Accounting Office, or the Congressional Research Service provides assistance. For example, Brown (2002) and Riddle (2000) have

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Statistical Issues in Allocating Funds by Formula presented results of simulations designed to show the effects of selected features of the allocation procedure for the Title I education program. To assist program recipients in their fiscal and budgetary planning, some agencies and organizations have published the results of simulations based on advance estimates of the formula inputs or on alternative formula features being considered by Congress. The panel endorses simulation as an important tool to assess cross-sectional and longitudinal effects of alternative formulas, data inputs, and special features and to explore more general issues, such as the effects of alternative methods of combining formula components representing need, fiscal capacity, and effort. Simulations can shed light on the trade-offs faced in designing formulas, such as the trade-off between stability and responsiveness to need. Simulations have usually been cross-sectional, dealing with inputs and outputs for a single year; sometimes two consecutive years are considered, for example, to study the effects of a hold-harmless provision. Recommendation 8. Use of simulations when programs are being developed and when they are being reviewed prior to reauthorization should be expanded. Particular attention should be given to analyses of the effects of special features, such as hold-harmless provisions, caps, thresholds, and limits. Expanded analyses should include longitudinal studies to explore how statistical error in the input data series affects allocation patterns over time. Changes in funding levels, need distributions, and input data sources also indicate a need for evaluations of allocation processes, using simulation techniques, that are longitudinal rather than just cross-sectional. Simulations should be used to explore cross-cutting issues, such as choices among alternative measures of fiscal capacity and the relative merits of using hold-harmless provisions or moving averages to dampen the effects of large changes in the formula inputs. INFORMATION ABOUT FEDERAL FORMULA ALLOCATION PROGRAMS Formula allocation programs and procedures are often complex, and it is often necessary to consult several sources to achieve a full understanding of the history and current status of a particular program. The Catalog of Federal Domestic Assistance, maintained by the General Services Adminis-

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