A. INTRODUCTION

In this chapter we review relatively recent AFDC participation studies. This includes a discussion of the merits and limitations of the various econometric methodologies that have been used to study AFDC participation previously, as well as an examination of the dependent and independent variables used in the studies.

The following five methodologies, or methodological groups, are considered:

• Time-series analysis of national aggregate data;
• Time-series analysis of state aggregate data;
• Pooled (cross-section time-series) analysis of state aggregate data;
• Analysis of cross-sectional and longitudinal (panel) data for individuals; and
• Analysis of pooled (cross-section time-series) data for individuals.

An important conclusion is that the pooled analyses of aggregate state data -- the method we used in this study -- holds substantial promise relative to alternative methodologies, but also has some limitations.

In Sections B through F we describe recent studies in each of the respective methodological categories. We review program participation measures used in these studies in Section G and explanatory variables in Section H. Conclusions drawn from this literature that have influenced the specification of our model appear in Section I.

B. NATIONAL TIME SERIES

In national time-series models, regression analysis is used to estimate a relationship between some measure of program participation (e.g., the average caseload) in the entire country over some time interval (usually years or quarters) to a set of explanatory variables that represent or serve as proxies for hypothesized determinants of growth.

The Congressional Budget Office model (CBO, 1993) is the most recent national time series model that we have found.⁽¹⁾ We describe this model in some detail below and use it to provide context for a discussion of the strengths and weaknesses of this methodology. We then go on to describe briefly one earlier national time-series model.

The CBO model uses quarterly data for the period from 1973.1 through 1991.3 (73 quarters). Separate models are estimated for the Basic and UP caseloads.⁽²⁾ For each model, the dependent variable is the average monthly caseload over the three months of the quarter. specifies the caseload measure as a linear function of a set of explanatory variables and a disturbance term. The disturbance term represents all determinants of the caseload that are not captured by the explanatory variables, and is assumed to change slowly over time.⁽³⁾

Explanatory variables include: a measure of female-headed families, used in the Basic equation only;⁽⁴⁾ the "employment gap" -- the difference between actual and potential employment; a measure of the real average (across states) maximum AFDC benefits for a family of three; average earnings for year-round, full-time workers aged 18 to 24 with exactly 12 years of schooling (a female series is used for the Basic model and a male series is used for the UP model); three dummy variables to capture the transitional and permanent effects of program changes enacted under OBRA81; and three quarterly dummies to capture seasonal caseload variation. The real earnings data are annual data that have been converted to quarterly data by interpolation. Both current and lagged values of the employment gap variable are included in the equations; the first through third lagged values appear in the Basic equation and the first through fifth lagged values appear in the UP equation.

Almost all of the coefficients reported are statistically significant, and the few that are not have the expected sign. The most significant variables (i.e., those with highest t-statistic for their coefficient) in the Basic equation are the measure of female-headed families, and two of the three OBRA81 dummies. The four employment gap variables (current and three lags) are collectively very significant as well. In the UP equation, the first and second quarter dummies have the most significant individual coefficients. Collectively, the six employment gap variables (current and five lags) also stand out as especially significant.

For purposes of comparison with other studies, we used the CBO results and ancillary data reported by CBO to compute the estimated effect of a one percentage point change in the unemployment rate on the Basic and UP caseloads. Because the employment gap, rather than the unemployment rate, is included in the CBO model, it was first necessary to investigate the relationship between these two variables. Based on data reported in Figure 6 of CBO (1993), it appears that a one percentage point change in the unemployment rate is approximately equivalent to a one percentage point change in the employment gap variable. Given this, and assuming that the initial unemployment rate is five percent, the CBO estimates imply that a one percentage point increase in the unemployment rate increases the Basic caseload by 1.7 percent after four quarters and increases the UP caseload by 9.7 percent after six quarters.

There are numerous advantages of the national time-series methodology relative to others, but there are also significant disadvantages. We use the CBO study to illustrate some of the advantages and disadvantages, but they apply to other national time-series studies as well. The advantages include:

It is methodologically simpler than most other approaches. This makes it relatively easy to apply, and the results are relatively easy to describe.

National data for explanatory variables are readily available from published sources. Further, national explanatory variables are available at a level of specificity and measured with a degree of accuracy that is not available for smaller geographic units such as states. Most of the data used by CBO come from readily available published sources.

The researcher can examine the dynamics of the relationship between explanatory variables and participation, as CBO does by including both current and lagged values of the unemployment rate. This cannot be done with a pure cross-section approach, which uses data from only a single point in time.

In a statistical sense, the researcher is able to "explain" a very high percentage of the variation in participation over time (high adjusted R-square) and to produce simulated participation series that closely track the actual series. This is true of CBO's models -- the adjusted R-squares are 99.6 percent and 99.5 percent for the Basic and UP equations, respectively, and sample-period simulations track the actual series well. This occurs because time-series data -- especially aggregate data over large populations -- tend to be very highly correlated with one another. With a small to moderate size sample, such as CBO's, it is fairly easy to find a small set of explanatory variables that can achieve a good fit. There is, however, a negative aspect to this advantage, which we return to later.

The disadvantages of the national time-series approach include:

It can capture state-level changes in AFDC programs only in a very crude way -- through their impact on program variables that are aggregated across states, such as CBO's AFDC average maximum benefit for a family of three. The authors of the CBO report note that this variable may be endogenous: as caseloads increase, states may cut back maximum benefits for budgetary reasons (CBO, 1993, p.39).

The method's ability to distinguish between the effects of a substantial number of variables is limited by high correlations among explanatory variables that are typical of time-series data, and by limited observations. For instance, it is difficult to be confident that the impact of an increase in unemployment ends after three quarters for Basic and five quarters for UP given the likely high correlations among the various lags of the unemployment rate and the relatively short time-series available for estimating CBO's model. The authors mention this as a specific problem with their model (CBO, 1993, p. 12).

Major national-level changes in the program are difficult to disentangle from the effects of other variables because they can usually only be modeled in a very crude way -- such as the three dummy variables that CBO used to capture the impact of OBRA81. The authors of the CBO report mention this as another specific problem with their model (CBO, 1993, p.12). This same problem would arise in modeling the impact of the 1988 Family Support Act (FSA) using more recent data. Both OBRA81 and the FSA may be viewed as changes in "regime," and it could be that coefficients of other variables also changed with the regime shift. When regime shifts occur as frequently as they perhaps have for AFDC, time-series data alone are likely to be inadequate for testing whether other coefficients did change. For instance, using national time-series data we would be unlikely to determine whether the mandating of the UP program as of October 1990, under the FSA, had an impact on the models other than a one-time shift in UP participation.

There is a serious danger that the researcher will end up with a model that fits the data well (very high adjusted R-square and the simulated series tracks the actual series closely), but that the model coefficients misrepresent the causal relationship between the explanatory variables and participation; further, out-of-sample predictions may be very poor. The reason is that the high correlations found in time-series data, combined with a fairly small sample size, make it relatively easy to get a good fit by trying a variety of different specifications or by constructing an explanatory variable that seems to track participation well but that does not have a clear theoretical rationale. CBO's measure of female-headed households (FHH) may be such a variable.

The last point merits further discussion. According to the report, the FHH variable is defined as the "number of families headed by women with their own children under age 18, multiplied by the ratio of never-married mothers to mothers who had been married." This variable was developed after an attempt to include separate variables for families headed by never-married mothers and by ever-married mothers led to nonsensical results, evidently because of multicollinearity between the two variables (CBO, 1993, p. 14, fn. 17). Although the particular functional form used to aggregate the variables does not have an apparent theoretical rationale, the path of the variable has an upturn that coincides with the upturn in Basic caseload growth.

While the variable is critical to the fit of the model, the authors find that its coefficient is implausibly large; the coefficient implies that 80 percent of all new female-headed households move onto AFDC (CBO, 1993, p. 18). The authors suggest that some of the growth attributed to this variable may be due to omitted factors, such as the Immigration Reform and Control Act of 1986 and Medicaid outreach.

Other than earlier versions of the CBO model, the next most recent national time-series model we have found comes from ASPE's last effort to model AFDC caseloads (Grossman, 1985). This effort is notable because the author also explored the development of a pooled, state-level model, and contrasted the findings to the findings from a national time-series model. We describe the national model here, and return to the state-level model in Section D, below. The national model also provides additional examples of the advantages and disadvantages of the national time-series approach.

The Grossman model is estimated with quarterly data for the period from 1974.4 through 1983.4 (37 quarters). The Basic and UP programs are modeled separately. Two equations are estimated for each program: a caseload equation and an average benefits per case equation. The primary purpose of this effort was to improve national forecasts of AFDC expenditures for each program (the product of the program's case-load and average expenditure per case).

As in the CBO model, the caseload measure for each program is the average caseload over the three months of the quarter. The average benefit variable is the average of the monthly values for benefits per case over the same three months.

Explanatory variables in the Basic caseload equation include: the number of female-headed households, the poverty rate for families, the real average hourly wage rate in retail and service industries, the average standard of need for a family of three, the lagged unemployment rate (with lags for four quarters), quarterly dummies, and three dummies to capture the transitional and final impacts of OBRA81.⁽⁵⁾ The explanatory variables in the UP caseload equation are the same except that the labor force in UP states replaces the number of female-headed households and the fifth lag of the unemployment rate is added. A first-order autoregressive disturbance is specified for each model.

Note that the CBO (1993) caseload specifications are very similar to Grossman's specifications in many respects. The variables with the most significant coefficients are the OBRA81 dummies and the female-headed household variable. The only variable with a very insignificant coefficient (t-statistic less than 1.0) in both equations is the poverty rate. As with the CBO model, adjusted R-squares are very high: 97.5 percent for the Basic caseload and 99.0 percent for the UP caseload.

The implied long-run elasticities for the Basic and UP caseloads with respect to the unemployment rate are very similar to those calculated from CBO's findings: 0.1 for the Basic caseload (identical to the value we found for CBO) and 0.7 for the UP case-load (compared to 0.5 based on CBO's findings), despite the fact that only about half of CBO's sample period was used by Grossman.⁽⁶⁾ Thus, it appears from the national time-series estimates that the impact of a change in the unemployment rate on caseloads has been reasonably stable over a long period of time.

The average benefit equation for the Basic program includes the following explanatory variables: the weighted average of the maximum benefit for a family of four, an estimate of the average number of persons per family in the United States, and dummy variables for OBRA81. The average benefit equation for the UP program also includes the maximum benefit variable and an OBRA dummy, but does not include the average family size variable. The adjusted R-squares for both equations are very high: 99.4 percent for the Basic equation and 98.6 percent for the UP equation. The maximum benefit variable has a very large t-statistic in both equations, and accounts for most of the models' explanatory power.

We note that Grossman did not include the unemployment rate in the average benefit model. Holding the maximum benefit constant, increases in unemployment may be associated with reductions in earnings among existing recipients, which would increase average benefits. Earnings and other benefit determining characteristics of cases that enter the caseload due to an increase in unemployment may differ from those of existing recipients. This could offset or add to the hypothesized positive effect of unemployment on average benefits for existing cases.

C. INDIVIDUAL STATE TIME SERIES

1. Overview

Most states have developed their own AFDC caseload forecasting models, which are typically used for budget and staffing projections. A 1990 study conducted by the state of Oregon documents the variety of techniques used by the fifty states in forecasting their AFDC caseloads (Oregon State Department of Human Resources, 1990). According to this study, most states use a simple trend analysis (fitting a linear relationship to past series of caseload values) to predict future values for the AFDC caseload. Several states, however, use a multivariate regression framework, incorporating demographic, economic, and programmatic factors, to forecast their AFDC caseloads.

We reviewed state models for two reasons. First, individual state models may reveal important determinants of the AFDC caseload that have not been examined in other studies. Second, our plan called for using the model we developed to simulate caseload series in a few selected states. For this analysis, we wanted to select states that have well-developed AFDC models so that we could compare the results of our model to findings from the state models. In addition, we expect that the developers of the state models would be helpful in interpreting our findings.

The state models discussed here provide additional examples of how particular factors may be measured and incorporated in our model of caseload growth. The models also identify important state-specific policy changes that occurred during the time period we will examine. The MinnesotaCare program in Minnesota and the FIP program in Washington are examples. To the extent that such policy changes are identified, they can be included in our model of caseload growth. Given the fact that we will not be able to identify all such policy changes for all states, the state models that do provide information on policy changes will allow us to test the sensitivity of our results to the omission of state-specific policy variables.

Below, we describe selected state models that go beyond trend analysis or ARIMA models^(7), incorporating factors believed to influence AFDC caseload growth in a multivariate regression framework. The models we describe are those used by Florida, Maryland, Minnesota, Oregon, Texas, and Washington. In addition, we discuss a report by Barnow (1988) that presents a guide for states to use in developing their own AFDC caseload models, using New Jersey as an example.

2. State Models

Barnow (1988), in a study conducted for ASPE, developed a guide for states to use in constructing models to predict AFDC-Basic and UP caseloads. Models were developed using the state of New Jersey as an example, spanning the period 1978 to 1985. In these models, the quarterly AFDC caseload is regressed on: the number of divorces, the real average weekly wage in the retail trade industry, the number of persons unemployed in the state, out-of-wedlock births, the total state population, a set of dummy variables representing permanent and phase-in effects of OBRA81, a dummy variable representing the implementation of the 1984 Deficit Reduction Act (DEFRA), a set of seasonal dummy variables, and a set of variables interacting OBRA with the divorce, unemployment, wage, and out-of-wedlock birth variables. Many other variables were tested but not included in the final models. These include the need standard, the number of marriages, employment in the retail industry, earnings in the personal services industry, and the number of persons who have exhausted their Unemployment Insurance benefits.

Florida

Several governmental entities in the state of Florida employ statistical models to forecast the state's AFDC caseload. The Governor's office utilizes an OLS regression framework combined with ARIMA modeling techniques to forecast AFDC caseload and costs. Explanatory variables used in the models include the female population ages 15-45 and the unemployment rate. The Legislature bases its forecasts on a multivariate regression model using quarterly data from 1981 to the present. The dependent variable is the average monthly AFDC caseload, seasonally adjusted. Explanatory variables include the state unemployment rate (seasonally adjusted), the state female population ages 18 to 44, and a dummy variable accounting for policy changes occurring after 1987. In the past, the regression equation has included a variable representing movement in and out of unskilled labor (e.g., labor in the retail trade).

Maryland

The Maryland AFDC Net Flow Model is one component of a larger, macro model of the Maryland economy. Models are specified separately for the AFDC-Basic and AFDC-UP caseloads. The AFDC-Basic model is specified as a log linear model where the number of paid cases is a function of the unemployment rate for the at-risk population, a measure of real net income gain from work, the size of the at-risk population, an index of help wanted ads, and the rate of AFDC case closings. The at-risk population is defined as female headed households with children under age 18. The real net income gain from work is the real average monthly wages after taxes minus the real monthly combined value of AFDC, food stamp, and Medicaid benefits.⁽⁸⁾ Explanatory variables used in the AFDC-UP model include the unemployment rate, an index of leading economic indicators developed by the state, the index of help wanted ads, and a lagged value of the UP caseload.

Minnesota

The Minnesota model uses monthly time-series data to forecast AFDC-Basic and AFDC-UP caseloads, specifying a structural component and an ARMA component in the model. The AFDC-Basic structural component includes variables for unemployment (including lags), out-of-wedlock births, the real payment standard for a family of three, and the number of families with children enrolled in Minnesota's publicly-subsidized health insurance program, MinnesotaCare. MinnesotaCare is believed to have an impact on AFDC caseloads because it offers an affordable alternative to Medicaid should an AFDC recipient have an opportunity to take a job that does not offer health insurance benefits. This variable has proven to be a statistically significant predictor of Minnesota's AFDC caseload. The AFDC-UP model uses the same variables as those used in their AFDC-Basic model, except that in-wedlock births are used instead of out-of-wedlock births.

Oregon

Oregon uses a multivariate regression model to forecast caseload growth where the number of AFDC cases is a function of the fertility rate of unwed women, the female population ages 15-44, the number of divorces, the real value of the average AFDC cash grant to a family of three, the federal poverty level for a family of three, and the total number of births. The data for this model are monthly, but the model is used to produce annual forecasts. Oregon projects four years into the future. For the nine years previous to 1994, the Oregon model consistently produced R-squares of over 90 percent. In 1994, however, the State of Oregon's AFDC policy changed from one of eligibility determination to one of diversion. That is, the Department of Human Resources first seeks ways to keep individuals/families off AFDC by primarily helping parents keep their current jobs or find new adequate employment. As a result, the explanatory power of the model has dropped significantly because the model does not contain a variable that captures this important policy change.

Texas

One of several models used by the state of Texas is a multivariate regression model where the dependent variable is the number of AFDC-Basic cases. The model's explanatory variables include: the number of separated/divorced female-headed households with children under age 18; the number of never-married female-headed households with children under 18; the combined (inflation adjusted) cash value of monthly AFDC, food stamp and Medicaid benefits for the typical three-person AFDC family; the average (inflation-adjusted) wage rate in the retail sector; the gap between 'full' and actual nonagricultural employment; and a dummy variable representing the implementation of the JOBS program in the fourth quarter of 1990. Increases in the caseload have been attributed to the JOBS program; however, staff at the Department of Human Services believe there is a crossover effect due to recent expansions in Medicaid eligibility, and that new Medicaid applicants are also being found to be eligible for AFDC.

Washington

Washington uses entry and exit models to project AFDC caseloads. The state uses separate models to forecast AFDC-Basic and AFDC-UP caseloads. In both models, a month's caseload is equal to the previous month's caseload plus projected entries minus projected exits. Apparently, the models were quite successful in forecasting caseload changes until 1994, when reforms began to reduce caseloads.

The AFDC-Basic entry model regresses the entry rate (new entrants divided by the population of males less than age 19 and females less than age 49 not on AFDC) on the out-of-wedlock birth rate, seasonal dummy variables, and a reform status variable representing the impact of the Family Independence Program (FIP) initiated by Washington in 1988. The reform variable measures the weighted percentage of state community service offices operating under FIP or FSA.⁽⁹⁾ The AFDC-Basic exit rate model regresses the exit rate (exits divided by the Basic caseload) on: the ratio of the three-person grant to typical earnings in non-manufacturing employment; the gap between the employment rate and its previous maximum, where the employment rate is the ratio of total non-agricultural employment to number of Washington residents between the ages of 18 and 64); seasonal dummies; and the reform status variable.

The AFDC-UP entry rate equation regresses the entry rate (new UP cases divided by the population under age 49) on the gap between the employment rate and its previous maximum, seasonal dummies, and the reform status variable. The AFDC-UP exit equation regresses the exit rate (UP exits divided by the UP caseload) on the gap between the employment rate and its previous maximum, seasonal dummies, and the reform status variable.

D. POOLED CROSS-SECTION TIME-SERIES: STATE AGGREGATES

1. Introduction

In this section we describe previous efforts to model AFDC caseloads that used pooled state-level data for multiple states across states and over time. These models can be viewed as individual state time-series models that have been linked by using the same explanatory variables in all states and, with some exceptions, constraining the coefficients of each variable to be the same for all states. An important aspect of these models concerns the specification of the regression disturbance.

The general specification for this class of models is:

Equation 2.1: Y_ts = a+ b'X_ts + e_ts

where:

Y_ts is the dependent variable for year "t" in state "s" (a measure of program participation);

X_ts is a vector of explanatory variables;

a is the intercept;

b is a vector of coefficients for the explanatory variables (assumed constant across states and over time); and

e_ts is the regression disturbance.

There are various subclasses of pooled models, which are defined through the specification of the disturbance. For our purposes the most important subclass is "fixed effects" models. These models assume that the disturbance, _ts, is the sum of three terms: a "state fixed effect" that is different for each state but doesn't vary over time; a "time fixed effect" that is different each year but doesn't vary across states; and a random effect. The subclass can be specified as:

Equation 2.2: Y_ts = b'X_ts + a_s + t_t + u_ts

where:

a_s is the state fixed effect;

t_t is the time fixed effect for time period t; and

u_ts is the random disturbance.

We find it useful to think of the state fixed effect as the intercept term for the state; i.e., under this specification the regression intercept varies across states. This term will "explain," in a statistical sense, all of the cross-state variation in the average of the caseload variable overtime. Another way to state this is that it captures all factors that account for cross-state variation in the dependent variable that do not change over time.

Another way to interpret the fixed state effects specification is to recognize that it is equivalent to modeling the relationship between changes in the caseload variable to changes in the explanatory variables, with no state fixed effect (assuming appropriate specification of the other parts of the disturbance). The change model can be obtained by first differencing Equation 2.2:

Equation 2.3: DY_ts = b'DX_ts + Dt_t + Du_ts

where the prefix D indicates the one period change in the variable. Because fixed state effects don't change over time, they drop out of the change specification. From this specification it is evident that we are ignoring the cross-section relationship between the levels of the variables in estimating the models' coefficients. Note that any X variables that vary only across states, and not over time, will drop out of the model, too.

The fixed time effects capture factors that change over time, but have an equal influence on the caseload variable in all states. One important example is the implementation of a new federal AFDC policy in all states at one time (example, OBRA81). To the extent that such a change has an equal impact on the participation variable in all states, its impact will be captured in the fixed time effects. The implementation of a new policy may be captured in fixed time effects over a number of periods because the impact may not be fully realized in the first period of implementation, or because it may be partly realized in anticipation of implementation. Unit impacts that are not uniform in all states will not be captured; unless captured by explanatory variables, they become part of the regression disturbance.

Fixed time effects are usually implemented by including a time dummy for each period, omitting the dummy for an arbitrarily chosen base period. These dummies "use up" all of the information that national time-series models would use to estimate the model, as is evident from the fact that a national time-series model with a dummy variable for each period could not be estimated. Assuming state fixed effects are also included, the coefficient estimates for the explanatory variables are based on the relationship between deviations of changes in the dependent variable from the national average change and deviations of changes in the explanatory variables from changes in their respective national averages. Thus, fixed effect results can be quite different than those obtained from national time series alone.

The other commonly used subclass of pooled models is known as "random effects" models. As in fixed effect models, the disturbance is usually assumed to have three components -- one that varies across states, one that varies across time periods, and a third that varies across both. The critical difference between random effects and fixed effects models is that the state and time components of the error term are assumed to be uncorrelated with the X variables in the former, but not in the latter. The uncorrelated assumption is built into estimators for random effects models. If the assumption is correct, the estimator will be more efficient than fixed effects estimators, but if it is incorrect the estimator may be biased, perhaps substantially so.

Fixed effects models are more commonly used for studying program participation at the state level than are random effects models. Fixed state effects are important because there are many time-invariant characteristics of states (e.g., geography) that could have an impact on participation and might well be correlated with explanatory variables. Fixed time effects may or may not be important, depending on whether significant national factors changed over the period under investigation, and whether those changes are associated with changes in the explanatory variables.

The two subclasses of pooled models may be mixed. The AFDC participation studies discussed below provide examples of models with fixed state effects, but not fixed time effects.

2. Previous AFDC Studies

Four studies have used pooled state data to analyze AFDC participation: Grossman (1985); Cromwell, Hurdle, and Wedig (1986); Moffitt (1986); and Shroder (1995). The first two of these use quarterly data, while the last two use annual data. In the discussion below we give more attention to the first two than the last two, for different reasons. As discussed previously, Grossman compares the findings from her analysis of pooled state data to results from a national time-series analysis. Cromwell et al. adopt an approach that is closer to the approach we recommend for this project than any other study.

Grossman (1985) estimates pooled Basic caseload and average Basic benefits models for states using specifications for each state that are very similar to the specifications she used for her corresponding national time-series models, described earlier. This is the earliest example of a pooled model that we have found. Quarterly data are pooled for 51 states (including DC) over the period from 1974.4 through 1983.4 (37 quarters) -- a total of 1,887 observations.

The caseload variable is the average monthly caseload during the quarter. Explanatory variables in the caseload equation include: a dummy for each state (i.e., a state effect); the national series for female headed households that was used in the national model, but allowing for a different coefficient for each state by interacting the variable with the state dummy; a single dummy for OBRA81, which permanently changes from zero to one in the quarter in which the state adopted a key requirement of OBRA81;⁽¹⁰⁾ four lags of the number of persons unemployed; quarterly dummies; and the state's standard of need for a family of three.

Grossman concludes that her pooled caseload model performs very poorly relative to her national model, as well as relative to time-series models estimated for individual states. This conclusion is largely based on an R-square of only 52 percent, compared to an adjusted R-square of over 97 percent in the national time-series model and similarly high R-squares in the individual state time-series models. We find the low R-square to be very surprising. Our own experience in developing similar models for SSA disability program caseloads suggests that R-squares are typically very high because the state effects (i.e., individual state intercepts) explain all of the very considerable average cross-state variation in the dependent variable. Grossman's findings for the average benefit equation are more consistent with our experience; for that equation she obtains an R-square of 93 percent in the pooled model.

The low R-square in the caseload equation suggests that there is a significant specification problem. It may be that the low R-square is due to the following problem: the coefficient of the OBRA81 dummy is constrained to be the same in all states even though one would expect that the impact of the OBRA81 change on the level of the Basic caseload (i.e., on the dependent variable) would increase with the size of the state. There is tremendous variation in size across states, and one would expect comparable variation in the size of OBRA81's impact. The same criticism applies to the specification of the quarterly dummy variables and the standard of need.⁽¹¹⁾ There may be an analogous problem in the average benefit specification, but it is presumably much less severe because relative variation in average benefits across states is much smaller than relative variation in caseload size. The lesson from this experience is that state models should be specified in such a way that it is reasonable to expect coefficients to be constant across states. Later pooled models, discussed below, solve this problem by specifying logarithmic participation variables, so that each coefficient represents (approximately) the percentage change in participation associated with a unit change in the corresponding explanatory variable.

The Cromwell et al. (1986) study is unique among the studies we have examined because the focus of the study is Medicaid enrollment; AFDC participation is examined because most AFDC participants are Medicaid eligible. The authors measure AFDC participation as Medicaid enrollees who are also AFDC recipients; no distinction is made between Basic and UP households.⁽¹²⁾

The authors pool quarterly data from 44 states for the period from 1976 to 1982 (28 quarters and 44 x 28 = 1,232 observations). They implicitly use fixed state effects by specifying their model in quarterly changes. They do not specify fixed time effects, however.

The model's dependent variable is the natural log of average monthly AFDC Medicaid enrollment per capita for the quarter. All continuous explanatory variables are also in logs: the unemployment rate (current period plus three lags); monthly manufacturing earnings per capita; the maximum monthly AFDC payment level for a family of unspecified size, deflated; and a "tax capacity" index; a political "liberalism" index. In addition to these continuous variables, they include dummy variables for: whether or not a state has an UP program; whether or not the state has a Medicaid eligibility option for independent children between the ages of 19 and 21; and an OBRA81 dummy that is equal to one in the third quarter of 1981 and thereafter. They also include an interaction between the UP indicator and each of the unemployment variables. They hypothesize that participation will be more sensitive to business cycles in UP states than in non-UP states because the UP program is designed to help two-parent families when both parents are unemployed. Their findings provide strong support for this hypothesis.

The estimates of the impact of increases in the unemployment rate obtained by Cromwell et al. are several times larger than those obtained in the national time-series models we have reviewed. Their estimates imply that a one percentage point increase in the unemployment rate increases the caseload in states without an UP program by 1.8 percent after three quarters -- almost identical to CBO's estimate for Basic programs. IN states with an UP program, the estimated effect of the same change is a 3.0 percent increase after three quarters. Assuming that five percent of the caseload in the latter states is in the UP program, which is approximately correct, and that the sensitivity of the Basic program in those states is the same as in states with no UP program, the results imply that a one percentage point increase in the unemployment rate increases the UP caseload by 25.8 percent after three quarters -- more than 2.5 times the estimate obtained by CBO for the national UP caseload. One reason for the stronger result may be that the pooled methodology is better able to separate the effects of the recession in the early 1980s from the effects of OBRA81.

Moffitt (1986) uses the pooled state-level methodology to investigate whether there was a positive shift in AFDC participation among female-headed households from 1967 through 1982 that cannot be explained by labor market or programmatic changes, as many have alleged. Moffitt's analysis uses annual data for nine of the 16 years during the period of interest. The reason that seven years are excluded is that the participation variable in his model is estimated using data from the March Current Population Survey (CPS) and the biennial AFDC Characteristic Surveys (AS). Many states are excluded from the analysis because of missing data. The number of states included varies across years; the average number is just over 27; the total sample size (average number of states times years) is 245.

The dependent variable in Moffitt's model is the rate of AFDC participation among female-headed households; the numerator of the rate is based on AS tabulations and the denominator is based on CPS tabulations. Explanatory variables include the AFDC guarantee level, the benefit reduction rate (BRR), a dummy variable for southern states, the unemployment rate, and each of the following for female-headed households: mean age of head, mean education of head, percent of heads who are white, mean number of children, mean hourly wage rate, and mean unearned income.

Moffitt estimates equations for each year (i.e., using the cross-state data for the year alone), and also estimates three versions of pooled models: a model with fixed time effects but random state effects, a model with fixed time and state effects, and a "between" estimator.⁽¹³⁾ The between estimator incorporates fixed time effects also, but ignores the possible existence of an error component that varies across states but not over time -- either random or fixed. It is obtained by jointly estimating cross-state regressions for each time period, constraining all coefficients to be the same except the intercept. As a result, the coefficients depend solely on cross-state relationships between the levels of the model's variables.

Moffitt's findings, taken alone, would be discouraging for those interested in examining AFDC participation through pooled analysis of state data. Very few of his explanatory variables, except the time dummies, are statistically significant using any of the three pooled estimators he considers, and those that are significant are not significant for all three estimators. The model with fixed state and time effects is especially disappointing, with no statistically significant coefficients other than for the fixed effects themselves. Further, the unemployment rate has an insignificant coefficient in all three specifications. The most significant findings are for the guarantee and BRR variables, in both the model with random state effects and the between model. The fact that these variables are not significant in the model with fixed state effects means that their significance in the other two models relies heavily on the cross-state relationship between these variables and participation, rather than on the relationships between changes in these variables and changes in participation over time. It is possible that the significant coefficients are substantially biased because important factors that both vary across states and are correlated with the explanatory variables, are not included in the equation.

There are many possible reasons why there are so few statistically significant coefficients in Moffitt's results, especially when compared to the findings of Grossman and Cromwell et al. The use of biannual data and the relatively limited number of observations are obvious ones. Moffitt also documents a number of problems with the construction of the dependent variable, as well as with some explanatory variables. Note, too, that Moffitt controls for a key demographic variable -- the number of female-headed households -- by using it as the denominator for the dependent variable. Hence, his estimates implicitly control for the effect that any of the explanatory variables would have on AFDC participation via their effect on the number of such households.

Of course one reason the estimates are insignificant may be that the explanatory variables are not very important determinants of participation. Moffitt explores this further with analysis of individual data for three of the seven years, and does find more significant effects. Nonetheless, these estimates, like the estimates using state-level data, imply that most of the increase in AFDC participation rates for female-headed households during the 1967-82 period was due to factors not accounted for by the models.

Shroder (1995) estimates a pooled state model using annual data from 1982 to 1988 for all 50 states and the District of Columbia (357 observations). His research focuses on the issue of endogeneity of AFDC benefits in participation equations. Specifically, he investigates the hypothesis that increases in AFDC participation reduce benefit levels because they increase the cost per taxpayer of each marginal dollar spent on benefits; as a larger share of the population participates in AFDC, there are fewer taxpayers per recipient to fund the program. For this reason he develops a two-equation model in which AFDC participation and benefits are jointly determined. Another unique feature of Shroder's model is the use of variables representing benefits and economic conditions in neighboring states.

The dependent variable in his participation equation is the log of the recipiency ratio (the ratio of AFDC recipients to non-recipients). The dependent variable in his AFDC benefit equation is the log of the maximum benefit for a 3-person household, including both the AFDC income guarantee and the value of Food Stamps.

Explanatory variables in the benefit equation include: the log of the recipiency ratio; an index of "Republican power;" the log of the share of AFDC recipients who are non-Hispanic whites; the log of the share of recipient households in which the mother of the youngest child is not married; the log of per capita disposable income; and the log of the state's share of AFDC benefit payments. Explanatory variables in the participation equation are (all in logs): the AFDC benefit variable, the average annual wages for laundry, cleaning and garment services (SIC 721), the unemployment rate, the ratio of women age 15-65 to employed men (F/EM; "it proxies for the availability of the marriage option"), a wage variable for the state's "composite neighbor" (see below), the composite neighbor's unemployment rate, F/EM for the composite neighbor, and the composite neighbors AFDC benefit.

Economic conditions and AFDC benefits in neighboring states are potentially important determinants of AFDC participation in one's own state because state residents may work in other states or migrate to other states to get work or obtain better AFDC benefits. The neighbor variables are weighted averages of variables in other states, with weights based on migration patterns from the 1980 census. Technically, the weight for state "k" as a neighbor for state "I" is defined as W_ik = M_ik/å_kM_ik, where M_ik is sum of migration from i to k and k to i.

Shroder uses instrumental variables for the benefit variable in the participation equation and the participation variable in the benefit equation to avoid simultaneity bias. He also estimates each model in two ways. In the first, he specifies fixed state effects (but not time effects), and in the second he averages the state data over seven sample years and estimate a cross-section model using the average state data. As discussed previously, the explanatory variable coefficients from the first estimator are based on the cross-state relationship between changes in the variables, and do not depend on the cross-state relationship between the levels of the variables. The second estimator is the antithesis of the first, relying on just the cross-state relationships in the average levels. Shroder's rationale for considering these estimators is worth examining:

"The fixed-effects model is particularly appealing in analyzing the recipiency ratio. Welfare recipiency across states may be affected by differential rates of divorce, abortion, teen pregnancy, size and social isolation of minorities, level of stigma attached to recipiency, structure of the economy, school quality, and so on. Many factors are difficult to measure well; the inclusion or exclusion of numerous imperfect measures will be controversial.

Assume these factors are time-invariant characteristics of the state. With the fixed-effects model, the choice problem of the representative agent can be conceived in terms of variables that do change over time, like the AFDC benefit paid in that state or in a neighboring state, and independent of the invariant factors.

However, two problems may arise with the fixed-effects model. First, the response to a change in the explanatory variables might occur with a lag of some unknown form. Second, the change in the explanatory variables from one period to another may consist of two shocks, one permanent and one transitory; the agents may be able to distinguish between them even if the econometrician cannot, and may respond only to the permanent component. In either case, the fixed-effects estimator will then be biased toward zero (Griliches and Hausman, 1986)." (Shroder, 1995, p. 186)

He goes on to argue that the estimator based on state averages, although possibly biased due to fixed effects, will mitigate problems arising from delayed responses and permanent versus transitory "shocks."

We concur with Shroder's rationale for using state fixed effects. The rationale for the state average estimator deserves closer examination. First, although the state-average estimator is likely to mitigate problems associated with delayed responses, another way to accomplish this and still include fixed state effects is to include lagged explanatory variables. In modeling SSA disability program participation using annual data, we have successfully used lags of as long as three years for unemployment with as few as seven years of data. The lagged variable strategy is likely to be even more successful with quarterly data, as the results reported by Cromwell et al. seem to indicate.

Second, use of lagged values will also help mitigate the problem of permanent versus transitory shocks to the extent that "agent" expectations are based on past experience rather than on other information that might portend a different future.

There is a third problem that Shroder does not mention, but which is mathematically identical to the permanent versus transitory shock problem: measurement error associated with the explanatory variables. As is well known to econometricians, random measurement error on an explanatory variable generally biases the variable's coefficient toward zero (i.e., the estimate understates the magnitude of the true coefficient), and the size of the bias is positively related to the share of the variation in the variable that is due to measurement error. Measurement errors in state average data tend to cancel each other out, implying that the share of variation in state averages for a variable that is due to measurement error is lower than the corresponding share for levels of the same variable in any given year. When year-to-year changes in a variable are examined, however, the share of variation due to random measurement error is higher than the share for the level of the same variable in a given year because some permanent variation across states has been removed and the variance of the random change in the measurement error is twice as large as the variance of the measurement error itself.

While we concur that the problems of delayed behavioral responses and permanent versus transitory shocks (including the measurement error problem) are problematic for the fixed state effects estimator and are mitigated by the state average estimator, it is preferable to deal with these problems as directly as can be done in the context of the fixed state effects estimator (e.g., by using lagged explanatory variables) than to resort to the state average estimator. Even though the latter mitigates these problems, it does so at the expense of accepting bias due to the state fixed effects. There are compelling reasons to believe that state fixed effects are important, and it seems likely that the bias resulting from ignoring them would be very large.

Based on Shroder's fixed effects estimates, the relationship between the recipiency ratio and the level of AFDC benefits is dominated by the effect of the former on the latter, although a significant reverse effect is identified. The fixed effect estimate of the coefficient of the benefit variable in the recipiency equation is both significant and large; a 10 percent increase in the benefit measure is associated with an increase in participation of almost 17 percent. The fixed effect estimate of the recipiency ratio coefficient is negative and significant, but not very large; a 10 percent increase in the recipiency ratio is associated with just a one percent reduction in benefits.

The fixed effects estimates of the own wage and unemployment coefficients in the participation equation are both statistically significant. A 10 percent increase in the benefit is associated with a 17 percent increase in the recipiency ratio, while a one percentage point increase in the unemployment rate is associated with a 3.5 percent increase in the ratio. Although his dependent variable differs from those used by CBO and Cromwell, his estimated effect is of the same magnitude for states with combined programs.

Only one of the four neighbor variables in Shroder's model has a statistically significant coefficient in the fixed effect estimates of the participation equation: a 10 percent increase in the neighbor wage index is associated with a 10 percent decline in the recipiency ratio. The neighbor AFDC benefit, unemployment and marriage variables are not statistically significant.

The state average estimates differ in many ways from the fixed effects estimates, indicating that the two sets of relationships captured by the two estimators are quite different from one another. Hence, it is important to recognize that these two sets of relationships are not the same.

3. Strengths and Weaknesses of the Pooled Approach

Strengths of the pooled approach include the following:

• For a given time period and data periodicity there are up to 51 times as many observations as in the national time series or the time series for a single state. This helps solve the multicollinearity problems that are common to national time-series models and allows for a richer specification of the model. Subject to availability, more variables can be included in the model;
• As with national or individual state time-series data, the dynamic impacts of explanatory variables can be explored, and the relatively large number of observations allows for richer dynamic specifications than are possible with a single time series;
• In comparison to national or state time-series estimates, it is feasible to control for all changes in national program policy or other national factors, at least to the extent that they have the same effects on participation in all states, by using time fixed effects. In the time-series approach, sample-size restrictions severely limit the number of dummy variables that can be used to estimate the effects of such changes without obscuring the effects of other variables;
• In comparison to national time-series analysis, the prospects for obtaining strong estimates of the impacts of program variables are much better because there is much more variation in individual state program variables than in weighted national averages of these variables. This applies to AFDC and other state programs, such as Medicaid and general assistance. The studies discussed above provide examples of multiple AFDC program variables used in the same model (Moffitt (1986) uses a maximum benefit variable and the earned income reduction rate, and Cromwell et al. (1986) use a maximum benefit variable and an UP dummy);
• The pooled methodology creates opportunities to test the validity of the model that are not possible with individual time series. Perhaps most importantly, we can test a set of constraints that is implied by the methodology itself -- identical coefficients for every state. Failure to reject the constraints would bolster confidence in the validity of the model. We could also test whether some or all coefficients are the same across two sample subperiods -- something which is difficult with a single time series because of limited observations. It is also possible to learn from comparing findings for various pooled specifications (e.g., fixed effects versus random effects), and to test whether the coefficients based on, say, cross-section relationships in the levels are the same as those based on cross-section relationships in changes of the variables.

The pooled methodology does have its limitations, however:

• It is methodologically more complex than the national time-series approach, which makes it more difficult both to implement and to describe the findings.
• Cross-state relationships between participation measures and explanatory variables in the model may in part reflect substantial cross-state variation in variables that have not been included, thereby biasing estimated coefficients for the included variables. State fixed-effects may need to be used to control for such factors;
• Collecting state level data requires more effort than collecting national level data. Data for some variables may not be available and the quality of other data may be poor. Data quality is particularly problematic when state fixed effects are used because measurement error bias is exacerbated by the use of the fixed effects;
• The estimated model is likely to track the participation measure for each state less well than a state time-series model, and the sum of the simulated series (i.e., across states) may track the national series less well than a simulation from a national time-series model. The reason for this is that the pooled methodology constrains explanatory variable coefficients to be the same in all states. If the constraints are valid, however, the pooled methodology should perform relatively well.

E. CROSS-SECTIONAL AND PANEL STUDIES OF MICRO DATA

Numerous studies have examined issues related to AFDC participation using cross-sectional data on individuals, usually obtained from large national surveys such as the Survey of Income and Program Participation (SIPP), the Panel Survey of Income Dynamics (PSID), and the Current Population Survey (CPS). The focus of many of the early studies has been the impact of AFDC benefit levels and the benefit reduction rate on the labor supply of female heads of households. These studies rely on the cross-state variation in AFDC benefits to estimate the impact on labor supply and find that the AFDC program does generate work disincentives; however, the magnitudes of the effects vary considerably across studies. For example, a review by Danziger et al. (1981) found the reduction in work effort to range from one to ten hours per week (10 to 50 percent of non-transfer labor supply levels).

Other studies have involved static models of AFDC participation, that is, the researchers estimate the likelihood of AFDC participation at a point in time as a function of demographic, economic, and AFDC program variables. These studies also find that the level of AFDC benefits and the benefit reduction rate significantly affect AFDC participation. The impact of wages on participation is also found to be important in some studies. Other factors significantly and positively associated with participation include age, having less education, poor health, and having greater numbers of children.⁽¹⁴⁾

Recent cross-sectional studies of AFDC participation have examined the importance of Medicaid, private health insurance, and medical need on AFDC participation (Blank, 1988; Moffitt and Wolfe, 1992). The findings from these studies have been mixed, mainly due to the manner in which the Medicaid benefit variable is specified. When the Medicaid variable (the value of Medicaid benefits) is specified at the individual level, based on the individual family's health status and medical care utilization, the results indicate that Medicaid availability has a significant positive effect on AFDC program participation. When a cruder measure of the value of Medicaid benefits is used, specified using a state-level average cost estimate, the relationship between Medicaid availability and AFDC participation is not significant.

Other recent literature on AFDC program participation has focused on the estimation of dynamic models of welfare participation using panel data on individuals, typically from the PSID or the National Longitudinal Survey of Youth (NLSY). These studies examine the determinants of program entry and exit, and time spent on the AFDC rolls. Using data from the PSID, Bane and Ellwood (1994) find race, education, marital status, work experience, and disability status to be important determinants of first-spell duration and recidivism. In examining the reasons for the first-spell of AFDC receipt, Bane and Ellwood find that changes in family structure account for about 80 percent of first-spells: over 40 percent begin when a wife becomes a female head, and 39 percent begin when an unmarried woman without a child becomes a female head with child. Only 7 percent begin due to a fall in the female head's own earnings, and about 5 percent due to a fall in other sources of income. The study does not address the factors that influence family structure. While their findings suggest that economic factors are unimportant, the effect of economic factors on family structure is not considered.

In studying welfare exits, Bane and Ellwood found that only about 25 percent of exits are due to an increase in the female head's earnings, while about 30 percent are due to marriage. The low rate of exits due to earnings in the Bane and Ellwood study is partially due to their hierarchical classification scheme that attributes an exit to marriage rather than earnings if the woman both married and increased her earnings. Other studies have found the proportion of welfare exits due to increased earnings to be in the range of 30 to 50 percent (Blank, 1988; Gritz and McCurdy, 1991; Pavetti, 1993). When Bane and Ellwood examine earnings in the first year off welfare regardless of marital status, they find that 41 percent of former recipients earned over $6,000 (1992 dollars) in that year.

Not surprisingly, the factors found to be associated with welfare exits differ between exits due to earnings and exits due to marriage or other factors. Greater education and previous work experience significantly increase the likelihood of earnings exits, but are not strongly associated with other types of exits. Race and marital status (never married versus widowed or divorced) have a much stronger influence on exits due to other reasons than on earnings exits. Never-married women and blacks are significantly less likely to leave welfare for marriage or other reasons (Bane and Ellwood, 1994).

The cross-sectional and panel studies of the determinants of AFDC participation provide important information about specific factors found to affect the likelihood of welfare entries and exits, and the duration of time spent on the rolls. The dynamic studies of welfare participation indicate that changes in family structure play a more important role than changes in a female head's earnings. This suggests that the direct effect of changes in the economy (i.e., the effect on the female head's earnings) on AFDC caseloads may be minimal. What one cannot determine from these studies is the indirect effect of changes in the economy on the AFDC caseload, as it impacts and causes changes in family structure. If poor labor market conditions for males affect the probability of marriage and/or divorce, then AFDC participation among females is also likely to be affected.

Fitzgerald (1991) touches on this issue in his study of determinants of AFDC exits by including a variable representing the quality of the "marriage market", that is, the ratio of single employed males to single males. This variable has a significant positive effect on the likelihood of welfare exits. His estimates indicates that a ten percent increase in the ratio of single employed males to single males decreases the likelihood of being on AFDC after 24 months by 8 percent (from 0.50 to 0.46).⁽¹⁵⁾ In models where the effects were estimated separately for blacks and whites, however, the availability of single employed males had a significant positive effect on exits for whites only (a 10 percent increase reduces the likelihood of AFDC participation from 0.46 to 0.35 at 24 months). Another interesting finding from this study is that the unemployment rate had a significant negative effect on exits for blacks but was insignificant for whites. Taken together, these results indicate that the marriage market is more important for whites, and the labor market is more important for blacks in contributing to welfare exits.

While cross-sectional and panel studies are useful in identifying the important determinants of AFDC participation, they are much less useful in estimating the impacts of specific factors on caseload growth over time. The primary reason is that the marginal effects estimated in a cross-section (which rely on the variation across individuals at a point in time) are likely to differ from those estimated using aggregate changes over time. This is partly because the relationships among factors may change over time, and partly because the idiosyncratic behavior of individuals may introduce sufficient "noise" in the data to mask the effects of aggregate variables, unless very large sample sizes are available.

A few panel studies have a feature that makes them especially useful for examining the impact of labor market conditions and other area factors on AFDC participation of individuals. These studies link area labor market and other variables to individual observations, permitting the researcher to examine the relationship over time between changes in participation and changes in the area variables. Some studies have used major longitudinal survey databases and state-level area variables, but have found little evidence that changes in these variables have an impact on participation measures.⁽¹⁶⁾ There are several possible reasons for the lack of findings, in addition to the possibility of no real effects. First, the number of observations in these surveys is small given the level of idiosyncratic variation in participation measures. Second, the studies typically control for demographic variables such as marital status, so don't recognize the potential impacts of area variables on household structure. Third, it may be that area variables for smaller areas than states are more relevant to participation than state variables, and changes in the variables for smaller areas may not be highly correlated with state-level changes.

Hoynes (1995) is the only study we know of to estimate a panel model of AFDC participation with area factors that uses a very large micro database and sub-state area variables. Her data were constructed from administrative records for AFDC recipients who are California Medicaid (Medi-Cal) enrollees -- nearly all AFDC recipients in the state -- for the period from 1987 to 1992. She estimates models for duration on AFDC using data for households that entered AFDC during the period. Her explanatory variables include household demographic variables, several county labor market variables, and community (Census tract or zip code area) demographic and economic variables based on the 1990 Census.

Hoynes found that the county market variables had substantial, statistically significant effects on duration in a variety of specifications. Her estimates imply that a three percentage point increase in the unemployment rate -- comparable to the average state-wide increase observed in California during the 1990-91 recession -- increases the chance that AFDC household that has been on the roles for less than 6 months will continue on the roles for at least one more month by 10 percent. Smaller effects were found for longer stayers. Similar results were found for other labor market variables. The findings are especially strong given that Hoynes controls for household characteristics that could themselves be influenced by labor market conditions.

F. POOLED CROSS-SECTION TIME SERIES: MICRO DATA

Several studies have used cross-sectional data on individuals, pooled over a number of years, to estimate the impact of various factors on AFDC participation. This approach offers the advantage of being able to capture the effects of and control for detailed demographic characteristics of the household while also estimating the impact of changes in state-level factors such as programmatic and labor market variables. Thus, for instance, the researcher might specify a binary choice (logit, probit, or linear probability) model for AFDC participation of families, using some explanatory variables that are specific to the family and others that are specific to the family's state, which may vary over time, but not across families within a state and time period (e.g., the state's unemployment rate).

Another advantage of the approach is that it can use variation in variables across families within a state and time period to estimate coefficients for variables that vary across families within each state and time period -- variation that is lost when state aggregate data are used. In fact, the researcher can use or not use a variety of sources of variation in the data, depending on how the model is specified. Just as in pooled analysis of aggregate data, the researcher can include dummy variables for each state to capture and control for all effects of factors that vary across states but not over time or across individual's within a state. Symmetrically, time dummies can be included to capture and control for all effects of factors that vary over time, but not across states. In addition, state and time dummies can be interacted to capture and control for all factors that vary both across states and over time, but not across families within a state and time period. When this is done, coefficients of other explanatory variables reflect only variation and covariation of variables across families within both time periods and states -- i.e., all of the variation that is lost when state aggregate data are used. As with the pooled analysis of state data, results will depend on which specification is used and differences in findings across various specifications may provide information that is useful in interpreting the results.

Yelowitz (1993) pooled individual-level data from the CPS for the years 1989 to 1992 to estimate the impact of Medicaid availability on the labor force and AFDC participation of single mothers. He tests the hypothesis that Medicaid expansions for children mandated by OBRA89 and OBRA90, that severed the link between AFDC and Medicaid eligibility for some groups, had the effect of reducing AFDC participation and increasing labor force participation. This is because the expansions allowed single mothers, in some cases, to earn more and still retain Medicaid benefits.

The Medicaid eligibility changes made it possible to study the effects of Medicaid coverage on AFDC and labor force participation in a number of ways: analysis of within-state variation based on the age of children (mothers with children of different ages either were or were not subject to the new expansions); analysis within states over time, as the expansions occurred; and analysis across states and/or time, as states adopted the optional expansions. Yelowitz estimates a probit model in which the Medicaid expansions are captured in a single variable, "GAIN%," which represents the increase in the Medicaid need standard as a result of the eligibility expansion, measured as a percent of the family's poverty line. The value of this variable for a family depends on both the family's state of residence and the age of the family's children.

While Yelowitz controls for individual demographic characteristics such as age, education, marital status, and the age of children in the family, he does not include any state-specific economic variables, such as the average wage or unemployment rate, in his analysis. He assumes the effect of these factors are captured by his state, time, and state-time interaction variables. This assumption is clearly correct when state-time interactions are included, because they control for all factors that don't vary across individual's within a state and time period. Evidence from previous analyses of pooled state data suggest that the assumption is incorrect when the state-time interactions are not included.

Yelowitz finds that Medicaid coverage independent of AFDC eligibility has a significant effect on both AFDC and labor force participation. Using the models with both state dummies and state-time interactions, he estimates that increasing the Medicaid need standard by 25 percent of the poverty line decreases the proportion of single mothers between the ages of 18 and 55 who receive AFDC by 4.6 percent, and increases their labor force participation rate by 3.3 percent. Estimated AFDC participation effects are somewhat smaller when state dummies and time dummies alone are used, with no interactions, suggesting that smaller effects would have been found if he had used aggregate data alone. This finding may have changed, however, had he controlled for the unemployment rate and other state-level factors. While variation in the Medicaid variable across families within states and time periods clearly made a substantial contribution to the strength of Yelowitz' Medicaid findings, the fact that the findings are still significant when state-time interactions are omitted provides some reason for optimism that Medicaid eligibility expansion effects can be identified using aggregate level data.

Gabe (1992) uses data from the March 1988 and March 1992 CPS to examine the effect of demographic changes on AFDC participation over the 1987 to 1991 period; intermediate year data are ignored. Gabe's analysis decomposes the recent growth in the AFDC caseload into growth due to change in the number and type of mother-only families and that due to change in the rate of AFDC recipiency. Gabe examines the growth in AFDC participation within subsamples of mother-only families with specific marital status and living arrangement characteristics. This is equivalent to controlling for those characteristics by estimating a linear probability model of AFDC recipiency using dummy variables for all possible marital status/living arrangement combinations as explanatory variables.⁽¹⁷⁾

Gabe attributes most of the caseload growth over the period to the growing number and changing composition of mother-only families, rather than to a change in the rate at which mother-only families receive AFDC. Changes in living arrangements contribute little to the growth experienced over the period, while changes in the number of mother-only families by marital status account for a substantial share (93 percent) of the growth in the AFDC caseload between 1987 and 1991.

Gabe's analysis does not take into account economic factors and their potential effect both on the rate of AFDC recipiency within the various marital status/living arrangement subgroups, and the share of the total population within each subgroup. His results indicate, however, that the effect of economic factors on recipiency rates would have to be strong in order to account for much of the growth up to 1991. They do not provide any indication of the extent to which changes in marital status and living arrangements can be attributed to economic factors.

A case could be made for developing a pooled cross-section time series model with individual data rather than a state aggregate model. For instance, a probit model for family AFDC participation could be estimated, limiting observations to those families with children. Separate models for single and two parent households could be developed. If duration data exist, duration could also be modeled. The explanatory variables would be a combination of household demographic variables, state economic variables, and variables that combine state program and household demographics (e.g., the AFDC payment for a household with the demographic characteristics of the observation; the value of Medicaid benefit for average household of same size). We might, for instance, build a SIPP database for this.

As discussed previously, this approach has the advantage of being able to control for household demographic characteristics while estimating the impact of aggregate changes over time. This approach does, however, have a number of important limitations:

• It would require considerably more resources to develop than would a state-level model, and would not span as long a time period, particularly if SIPP data were used;
• The demographic variables that this approach allows us to include are mostly ones that may themselves be influenced by the economic factors we are considering. In fact, the existence of the family itself -- the basic unit of analysis -- may, in part, be determined by economic factors. Gabe's failure to find evidence of the impact of economic factors may be due this problem;
• Individual-level survey data are often annual in nature, and therefore would not capture important dynamics associated with AFDC participation; and

As with cross-sectional studies of individual data, the idiosyncratic behavior of individuals may obscure the effects of aggregate variables in the analysis unless sample sizes are extraordinarily large.

G. PARTICIPATION MEASURES

In this section we describe the measures of AFDC participation that have been used in previous studies of caseload growth and discuss issues relevant to the choice of a dependent variable for our analysis.

The most common measure of AFDC participation used in previous studies of caseload growth is the number of AFDC Basic or UP cases, typically measured as the number of families receiving benefits in a given month, or the average number of families for a given quarter or year (Exhibit 2.1). Quarterly data are used in the national studies of caseload growth, while the use of monthly data is more common among the individual state forecasting models. The pooled studies of state date use either quarterly or annual data.

Models that use case openings and closings for the dependent variables are rare. The state of Washington uses entries and exits to forecast its AFDC-Basic and UP caseloads in separate models. CBO includes an analysis of case openings and closing (Basic and UP cases combined) in the appendix to their main caseload analysis (CBO, 1993). The CBO study cites inconsistencies in administrative reporting of openings and closings as a major problem in using these data to analyze caseload growth. By comparing the estimated caseload, computed using changes in openings and closings, to the actual caseload, they illustrate how the estimated and actual series diverge due to the reporting problems associated with case openings and closings.

According to the CBO report, the problem arises because only families who enter the caseload through the formal application process are counted as case openings. Families who were on AFDC and return after a short period off the program are often not required to file a formal application. For these families, a closing would have been initially reported, but the subsequent opening would not be counted. It is also frequently the case that, for families who go on and off the rolls during a short period, neither the subsequent openings nor closings are documented.⁽¹⁸⁾ This poses a major problem in using openings and closings to analyze the dynamics of caseload growth, as a portion of the caseload that is in transition will be lost to the analysis. This is especially unfortunate because those moving on and off the rolls during shorter periods are likely to be most affected by changes in the economic factors that determine AFDC participation.

Another important measure used as the dependent variable in a few models is AFDC expenditures or average payments. A primary reason for analyzing caseload growth is to determine the implications of changing factors on federal and state AFDC expenditures. Using expenditures or average payments as the dependent variable allows the effects of economic, demographic, and programmatic factors on AFDC expenditures to be estimated directly, rather than inferred from analyses that model caseloads. It is possible that expenditures are more sensitive to changes in the economy than are caseloads. They may be more sensitive if economic downturns not only induce additional individuals to apply for benefits, but also cause those already on the rolls to experience a fall in earned income and a corresponding increase in AFDC benefits. This effect may be negated, however, if those coming on the rolls receive lower than average benefits.

Exhibit 2.1

Measures of AFDC Participation used as Dependent Variables in Previous Studies

H. EXPLANATORY VARIABLES

In this section we describe the main explanatory variables that have been used in previous studies. We focus on the variables included in time-series and pooled models using aggregate data, which are the most relevant for the purposes of this project. We organize the section by the different types of variables -- demographic, economic, AFDC program, and other programs and laws.

1. Demographic Factors

All the models of AFDC caseload growth we reviewed include a number of demographic explanatory variables, the most common being the size of the population of female-headed households; however, a number of studies have simply used total population or total female population measures. Other commonly used variables are births (out-of-wedlock, in-wedlock, or fertility rates), and the number of divorces (Exhibit 2.2).

Several issues arise in considering which demographic variables to include in a model of AFDC caseload growth. First, should overall population measures be used or population measures that also incorporate family structure (e.g., total female population ages 15 to 45 versus the number of female-headed households)? Because economic factors are likely to determine family structure and therefore, their importance may be understated in models that use family structure as an explanatory variable.

Second, there is an issue surrounding the changing age distribution of the female population. As the population ages, presumably the propensity to participate in the AFDC program among female-headed households also changes. None of the studies we reviewed addressed the changing age distribution of the population or the potential impact on AFDC caseloads.

Third, there is the issue of stock versus flow measures when using births, marriages, or divorces as explanatory variables. Births, marriages, and divorces all are flow variables that each have marginal impacts on the size of the population at risk for AFDC participation (single mothers). Such variables would be appropriate for models that focus on openings and closings because these, too, are flow variables, but are inappropriate for modeling the size of the caseload directly. For the latter, explanatory variables reflecting the stock of individuals at risk for AFDC participation seem more appropriate.

Fourth, while detailed family structure data are available from surveys at the national level, they are not generally available at the state level. One pooled model (Grossman, 1985) used a national series for female-headed households in the equation for each state, allowing the coefficient to vary across states. Two state models, for Maryland and Texas, use monthly state female-headed household series. In the case of Maryland, however this series is interpolated between Census observations. We have not received information on how the series for Texas was constructed, but note that Texas is large enough so that estimates based on the Current Population Survey would be reasonably accurate.

Exhibit 2.2

Demographic Variables used in Previous Analyses of AFDC Caseload Growth

2. Labor Market and Economic Factors

Nearly all models of AFDC participation include measures of labor market conditions as explanatory variables (Exhibit 2.3). The most commonly used measure is the unemployment rate. Many models include both current and lagged values of the unemployment rate in their specifications. Other variables intended to capture labor market conditions that were used instead of the unemployment rate include: the number of unemployment insurance claims filed, employment rates of females or in female-dominated industries, and employment gap measures (the difference between current and full employment).

In addition to the unemployment rate, models frequently include a measure of earnings. These range from overall average earnings for females within a specified age/education group (males, in UP models), to average wages in specific industries. Wage data for the retail trade industry or for "predominantly female" industries were used in some Basic models, while manufacturing wages were used in some UP models as well as in one Basic model (Cromwell et al., 1986). Many researchers have found that both the unemployment rate, or some other employment measure, and an earnings variable have statistically significant coefficients -- experience that suggests there is room for both types of measures in our models.

Availability of accurate data at the state level and by quarter is a key issue. The unemployment rate is available at this level and is generally considered to be of high quality. It is our understanding that the Bureau of Labor Statistics (BLS) collects state-level earnings data by industry on a monthly basis, but does not tabulate it at the state level other than annually. The state-level monthly or quarterly earnings data used in some models evidently come from special tabulations of BLS data. We are investigating the feasibility of having BLS prepare special tabulations for this project.

Another issue related to the use of an earnings variable is whether or not earnings should be adjusted to reflect taxes. Only one model has used an after-tax measure of earnings, and in this case, it is used in combination with the AFDC benefit to construct a variable representing the net gain to participating in the AFDC program.⁽¹⁹⁾ Federal tax adjustments, including potentially important Earned Income Tax Credits, can be made fairly simply based on a set of standard assumption about a family. State adjustments are probably not feasible because they vary by state.

A final issue concerns the inclusion of lagged values. CBO (1993) uses the longest lag specification on a labor market variable of any study reviewed -- six quarters in the UP equation only (three quarters in the Basic equation). No other specification uses a lag length of more than one year. Use of pooled data for states makes it feasible to explore longer lag lengths in comparison to what might be feasible using time-series for a single geographic area. This could be important because national time-series data suggest that economic recovery from a recession is followed by very slow declines in program participation. The pooled studies to date, however, have not considered longer lag lengths.

Exhibit 2.3

Economic Variables Used in Previous Studies of AFDC Caseload Growth

Exhibit 2.3 (Continued)

Economic Variables Used in Previous Studies of AFDC Caseload Growth

3. AFDC Program Variables

Variables reflecting changes in AFDC benefit levels and the implementation of new policies that either affect eligibility or the conditions under which individuals may receive benefits are typically incorporated in models of AFDC caseload growth (Exhibit 2.4).

The AFDC benefit variable is expressed in a variety of ways. It is most commonly expressed as the real dollar value of the maximum benefits for a family of a given size, typically a three person household. In some studies, the value of Food Stamp benefits combined with the AFDC benefit is used instead of the AFDC benefit alone. The combined variable more accurately reflects the benefits to a household that participates in the AFDC program than the AFDC benefit alone because Food Stamp benefits are reduced by 30 cents for each additional dollar of income, including income from AFDC. As a result, Food Stamp benefits are relatively high in states with relatively low AFDC benefit levels. This leads to less variation across states when the combined benefit measure is used than when the AFDC benefit alone is used as the explanatory variable.

A few studies have used measures of AFDC benefit levels that express benefits relative to some measure of earnings, either as a ratio of benefits to earnings, or as the difference between after-tax earnings and AFDC benefits. Incorporating earnings and benefits in this manner imposes a restriction on the independent effects of earnings and benefits on participation. An alternative strategy is to specify the two variables separately and then test the implied restriction. We think this strategy is especially important in a study that seeks to identify causal relationships.

One study treats the AFDC benefit level as an endogenous variable, jointly determined with the rate of recipiency (Shroder, 1995). The logic is that welfare benefits are a public good and their provision is, in part, determined by the preferences of voters (taxpayers). In states with a high proportion of AFDC recipients, AFDC benefits will be lower because the cost of an additional dollar of benefits to taxpayers is higher. An instrumental variable is used for the AFDC benefit level in the recipiency model. This study finds a very large estimated effect for increases in the maximum monthly benefits -- about twice as large as when no instruments are used.

Variables reflecting changes in national AFDC policies over time are also included in a number of models. Examples of these are OBRA81, the Deficit Reduction Act of 1984, and the JOBS program in 1990. In many of the individual state models, variables reflecting state AFDC policy changes are also included. None of the national studies cover the most recent years when a variety of waivers have been granted to states to experiment with their AFDC programs, and therefore do not address these policy changes in their models. Such variables are included in some of the individual state models we reviewed.

Exhibit 2.4

AFDC Program Variables Used in Previous Studies of AFDC Caseload Growth

Exhibit 2.4 (Continued)

AFDC Program Variables Used in Previous Studies of AFDC Caseload Growth

4. Other Variables

Other than demographic, economic, and AFDC program variables, there are few examples of additional explanatory variables used in studies of AFDC caseload growth (Exhibit 2.5). Two studies have used indicators of a state's political climate, and several of the individual state models include indicators for policy changes indirectly related to the AFDC program.

One potentially important factor, included in only one caseload model we reviewed, is the Medicaid eligibility expansions that took place during the late 1980s and early 1990s. These changes expanded eligibility for Medicaid beyond AFDC eligibility for certain groups, including pregnant women and young children. Yelowitz (1993) finds that the expansions had a significant negative impact on AFDC participation.

No studies we have examined have looked at the potential impacts of changes in SSI (including state supplements), Social Security Disability Insurance, general assistance, unemployment insurance, or any other program on participation in AFDC. Further, none have included variables for changes in laws concerning child support, abortion policy, or others that might have an impact on participation.

Exhibit 2.5

Other Variables Used in Previous Studies of AFDC Caseload Growth

I. CONCLUSIONS

In this section we discuss how our findings from the literature review have influenced how we have specified our models. We begin with a discussion of substantive issues concerning the specification of econometric models, then turn to a discussion of more technical specification issues.

1. Substantive Issues

Basic and UP Models

Although some previous researchers have examined the combination of Basic and UP participation, the predominant approach is to consider these programs separately. There are sound methodological reasons for considering them separately, especially given the fact that the UP program became mandatory in 1990, and because data for doing so are readily available -- although potential interactions between the programs should be recognized. Hence, we have developed separate participation models for these programs. It should be recognized, however, that cases in one program may shift, over time, into the other program. We have allowed for potential interactions in only a very simple way -- by including dummy variables for the presence and type (6-month or 12-month) of UP program in the Basic participation equations.

Household Demographic Variables

Past efforts have commonly used the number of female-headed households with children under the age of 18 or similar measures as explanatory variables, rather than more general population measures such as the size of the female or male population in a specific age group. There is reason to believe, however, that such variables themselves are sensitive to economic factors, and controlling for such variables may result in an understatement of the impact of changes in economic variables on AFDC participation. We examine this issue by estimating models with and without such variables included.

Age Distribution of the Population

We did not find any studies using aggregate data at either the national or state level that directly examined the impact of changes in the age distribution of the population on AFDC participation, although there is evidence from studies using individual data that participation varies with the age of the household head. During the time period we are examining, there was a substantial change in the age distribution of the population that is most at risk for AFDC -- those between 16 and 45 -- as the Post World War II "baby boom" generation completed its entrance into this age group earlier in the period and began to age out of the age group by the end of the period. In our previous work in modeling participation in the Social Security Administration's disability programs over the same period, we have taken this factor into account and found it to be important (Lewin-VHI, 1995b and 1995c).

Labor Market Variables

The specifications and findings in the literature we have reviewed suggest that the unemployment rate and some measure of average hourly earnings will be key explanatory variables and that the full impact of changes in these variables on AFDC participation will be realized only after four to six quarters have passed, especially for the unemployment rate. As discussed above, even longer lags may be important. State-level data for such variables are available although only at the annual level. Earnings adjustments to reflect federal taxes and earned income tax credits are feasible, although they have rarely been done in previous studies, and we have followed suit. Previous studies have not considered industry level measures of employment, but such measures may more accurately reflect the effects of business cycles on the segment of the labor market that is most relevant to potential AFDC recipients.

Measuring AFDC Benefits

Most studies use a maximum monthly benefit variable for a household of specified size, and some adjust this variable for Food Stamps. Failure to adjust for Food Stamps results in an overstatement of the effect of an increase in the benefit of the family's income. The AFDC benefit variable is sometimes combined with the earnings variable into a single variable, but we think this is a mistake if it is feasible to start with separate variables and test the restriction that combining the variables implies. One study that is methodologically similar to this study also uses a benefit reduction rate and obtains strong results (Moffitt, 1986). Hence, we were encouraged to develop a similar measure for this study.

Simultaneous Determination of AFDC Benefits

While most studies have assumed that the AFDC benefit variable is exogenous to participation, there are theoretical reasons and empirical evidence to support the hypothesis that increases in participation create down-ward fiscal and political pressure on the level of benefits. If so, the coefficient of the benefit variable may understate the effect of an increase in benefits on participation. Correction of this problem requires use of an instrumental variables technique, but it may be difficult to find the right instruments. This is complicated when more than one benefit variable is used, as in our models. While this issue may be an important one, we have neglected it here in favor of addressing other issues that seemed more important and easily dealt with.

Average Benefits per Case

We originally had planned to focus on participation in our analysis, ignoring the issue of average benefits per case. Some previous studies have, however, examined average benefits per case and found some evidence of sensitivity to economic factors -- even though average benefits are largely determined by maximum benefit levels. We decided to develop a model of average benefits per case in order to more fully understand the impacts of the business cycle and other economic factors on AFDC expenditures.

Other Programs

The literature we reviewed has not examined the impact of changes in other state programs on AFDC participation, with the exception of Medicaid. The findings from the Yelowitz (1993) study on Medicaid expansions encouraged us to think we would find a similar Medicaid impact. Based on our previous analysis of SSI participation, it seemed likely that impacts of changes in other programs on AFDC might be identified using the pooled state methodology.

2. Technical Issues

Pooled analysis of state data versus alternative methods

Based on our prior experiences and the findings of this review, we concluded that pooled analysis of state-level data shows greater promise than other approaches to studying the determinants of program participation. In comparison to time-series approaches for the nation as a whole, this approach allows examination of state-level programmatic changes, provides a much richer way to study the impact of other factors that vary at the state level, and provides many more opportunities for testing the validity of the model. Individual state time-series models offer some of these advantages, but don't take advantage of the efficiency gains and testing opportunities that are afforded by pooled analysis.

Pooled analysis of individual-level data also has considerable appeal, but would require a substantially greater effort. An advantage of that approach is its ability to capture the influence of household demographic variables; this becomes a disadvantage, however, when the possibility that economic factors have a significant influence on household demographic variables is considered. Further, idiosyncratic variation in participation for individuals may mask the effects of state-level variables on participation; such variation is averaged out in aggregate state data.

Caseloads versus Openings and Closings

While some investigators have developed models of openings and closings rather than caseloads, difficulties in the administrative measures of openings and closings led us to conclude that it would be better to follow the direction taken by most investigators and model the size of caseloads directly. Despite these difficulties, it may well be worth developing an openings/closings model in the future.

Logs vs. Levels

While some researchers have used caseload levels for their dependent variable, many others have used logarithms. There is an important reason to use logarithms rather than levels in the pooled analysis: in a levels specification the coefficient of an explanatory variable estimates the effect of a unit change in the variable on the level of participation, while in a log specification it represent the percentage effect of a unit change. Looking across states that vary greatly in population size, the latter is more likely to be constant than the former. A reasonable alternative would be to use a caseload rate -- caseloads divided by the number of women in the relevant age group or some other measure of the at-risk population.

Mixing data with differing periodicities

One problem in specifying a quarterly or monthly model using state-level data is that quarterly or monthly data may not be available for some key variables. Other studies have addressed this problem by interpolating between annual data points. We have made substantial efforts to obtain quarterly data for many variables, but have had to resort to interpolation in a number of cases. This is especially unfortunate for variables that have lagged impacts on the caseload because the timing of major changes in the variable will not be captured accurately by interpolation methods. We necessarily are more skeptical about our findings for these variables than for others.

1. The model discussed is an update to the model described in CBO (1991). CBO staff have informed us that the CBO (1993) study has not been updated and that there are no immediate plans to update it in the future.

2. An analysis of Basic openings and closings appears in the appendix to CBO (1993). We discuss this analysis further in Section III.

3. Specifically, the disturbance term is assumed to have a "second-order autoregressive regressive" structure -- the current quarter's error is a linear function of the errors in the previous two quarters, plus a random error.

4. This variable is defined as the number of families headed by women with their own children under age 18, multiplied by the ratio of never-married mothers to ever-married mothers.

5. This discussion refers to the equation reported by Grossman (1985) in Table V.4. A latter specification omits some of the explanatory variables because forecasts of the omitted variables are not available.

6. The CBO sample period includes the entire sample period used by Grossman, plus an additional period of almost the same length (73 quarters for CBO versus 37 for Grossman).

7. An ARIMA model is an autoregressive integrated moving average specification--such data series are not stationary and differencing is required to convert them into a stationary series (an ARMA model) that can be modeled.

8. The Maryland model defines the real average monthly wages after taxes for the at-risk population as the average monthly wages of females in low-wage, female dominated industries (i.e., industries in which females represent at least 40% of the total work force) adjusted for the EITC, FICA, state income tax, and food stamp benefits.

9. Washington's FIP is virtually identical to the 1990 Family Support Act which created the JOBS program.

10. The requirement referenced is the stipulation that a household must have an income that is less than or equal to 150 percent of the state's standard of need before deductions -- known as the "150 percent rule."

11. We note that Grossman uses the level of unemployment in place of the unemployment rate in order to solve this scale problem for the unemployment variable.

12. Additional equations for Medicaid enrollees who are SSI recipients and "non-cash" Medicaid enrollees are also estimated.

13. The South dummy is omitted from the model with state fixed effects because that model already includes a dummy variable for each state.

14. See Moffitt (1992) for a review.

15. Fitzgerald estimates a hazard model of exits from AFDC. Using the results of his model, he simulates average survivor functions at 6, 12, and 24 months for particular variables, holding all other variables constant at their mean values.

16. See, for example, Blank and Ruggles (1996) and Hoynes and McCurdy (1994).

17. The categories of marital status used in this study are never married, divorced, widowed, and separated/other. Living arrangements are categorized as independent families, extended families, cohabiting with a single, unrelated adult male, and unrelated families.

18. Based on a phone conversation with Herbert Lieberman of the Office of Family Assistance in the Agency for Children and Families.

19. This type of variable, one that expresses earnings relative to AFDC benefits, is discussed further in the next section.