A. INTRODUCTION

We present and discuss the regression results in this chapter. The participation equation results for the Basic program are in Section B, those for the Unemployed Parent program are in Section C, and the average monthly benefit results for the combined programs are in Section D. We consider the estimated effects for the labor market variables in more detail in Section E, including comparisons to findings from other studies. We do the same for the estimated AFDC program parameter effects in Section F.

B. BASIC PARTICIPATION EQUATIONS

1. Determining the Final Specifications

The results reported here were arrived at after trying many alternative specifications. The number of explanatory variables that could justifiably be included in any individual model is very large, especially when multiple lags for a single variable are included. While the number of "state-quarter" observations used (51 x 60 = 3,060) makes it technically possible to include a very large number of explanatory variables in a single equation, collinearity among explanatory variables (especially multiple lags of the same variable) would have made results from specifications with many more explanatory variables both imprecise and difficult to interpret. In addition, the software we used to implement the Parks method, SAS-ETS TSCSREG, limited us to using fewer explanatory variables than quarters in any one model (no more than 60).⁽¹⁾

We conducted the specification search using the Basic caseload as the dependent variable. We first included explanatory variables that we thought most likely to be statistically significant, with a limited number of lags -- the expected participation variable, the unemployment rate, the program parameters, and a few others. We then expanded lag specifications for variables that proved to be significant, and tried alternative lags for ones that were not. Variables that had insignificant coefficients were dropped along the way, and new variables not included in the initial specification were added. We generally used absolute t-statistics in excess of 2.0 as evidence of statistical significance, but accepted a lower value when the coefficient had the anticipated sign and/or if the 2.0 standard was met in many, but not all, specifications.

As a result of the specification search, the t-statistics reported need to be interpreted cautiously. Many of the most important findings were very robust to specification changes, but others are less robust and some have not been tried in many variants. Note, in particular, that a large number of variables in the "other programs and laws" category were tried and only a few have been retained -- in some cases they have the wrong sign. In any specification search we would expect to find at least some variables with large t-statistics even if all of the "true" coefficients were zero. It is also possible that a different search strategy would have yielded a different set of explanatory variables in the final models.

The estimated effects of increases in the unemployment rate are especially robust across the many specifications tried, as well as across participation equations within each program and across estimation methodologies (Parks vs. Weighted Least Squares). The trade employment variable also had consistently strong coefficients in all specifications in which it was included, while other employment variables were consistently insignificant. For the Basic models, the maximum monthly benefit (MMB), average tax and benefit reduction rate (ATBR), and gross income limit (GIL) variable coefficients are all remarkably robust across equations, explanatory variable specifications, and estimation methodologies. UP results for these program variables were considerably weaker, but are consistent across specifications - especially for MMB. We tried many specifications of the vital statistics variables. We consistently found significant results for the out-of-wedlock births and marriage variables in the Basic equations. The divorce variable was not significant in any specification. Results for the Immigration Reform and Control Act variable in the Basic equations are consistently strong. Results for other explanatory variables were much less robust, as discussed further below.

Once we finished the specification search for the caseload equation, we estimated the total recipient and child recipient equations with the same set of explanatory variables. Coefficients proved to be very similar across participation equations, so we did not search further using the recipient dependent variables. We also estimated the same models for all three participation measures, but with the vital statistics variables omitted. As discussed previously, this was done to determine the extent to which the effect of business cycles on participation works through their effect on family characteristics.

We discuss the coefficients for the various sets of explanatory variables included in the final specification (Exhibit 5.1) below, and also discuss alternative specifications that were tried. Note that coefficients of explanatory variables that are in logarithms are elasticities -- percent change in the caseload (or other dependent variable) per one percent change in the explanatory variable (an elasticity of 0.5, for instance, means a one percent change in the explanatory variable is associated with a 0.5 percent change in the dependent variable). Coefficients of other variables can be interpreted as the percent change in the dependent variable per unit change in the explanatory variable after multiplying by 100. For dummy variables, the coefficient times 100 can be interpreted as the percent change in the dependent variable associated with a change from the "zero" category of the dummy to the "one" category. "Long-run" estimates that are reported for explanatory variables with multiple lags are sums of the coefficients over all lags and represent the effect of a permanent change in the variable after the number of quarters indicated by the maximum lag length.

Exhibit 5.1

a Expected participation variable is based on national age-specific participation rates for 1990 and estimated population of the state by age in the quarter.

b T-statistics in bold are at least 2.0 in absolute value. These statistics were reduced from those calculated by SAS to make a correction for degrees of freedom that is not made by the procedure used (TSCSREG). The reduction factor used is .41, computed as [(T - K)/T].5, where T is the number of quarters (60) and K is the number of explanatory variables (50).

c All explanatory variables except quarter and year dummies are changes. Quarter and year dummies are equal to .25 in the quarters/years indicated so that coefficients can be interpreted as annualized rates of growth. For the polynomial distributed lag (PDL) variables, the coefficient of the variable lagged j periods is a₀ + a₁ j + a₂ j² for j = 0, 1, 2, ... L. Other variables are lagged the number of periods indicated.

d Indicates whether vital statistics variables are included (with v.s.) or not (without v.s.). The vital statistics variables are ln(out-of-wedlock births) and ln(marriages).

e Variables are moving averages of previous four quarters.

f This variable is the change in the state's SSDI initial allowance rate from 1977 to 1978 times the 1979 year dummy. Special dummies for three states were included due to missing initial allowance data.

g "Medicaid expansion" is the share of children in the state covered under the Medicaid expansions that began in 1988. "Share participating" is the share of children in the state who were in AFDC families in the year before the expansions began (1987 -- average monthly child recipients divided by population under 19).

The results we focus on below were estimated using the Parks method (Chapter 3). At the end of this section we compare the caseload results from the Parks method to estimates of the same model using a weighted least squares (WLS) method, with weights proportional to population size.

2. Year and Seasonal Effects

The estimated coefficients of the seasonal and calendar year dummies appear at the end of Exhibit 5.1. Recall that the calendar year dummies can be interpreted as the annualized growth rate in the first (winter) quarter holding all other variables constant. To obtain the annualized growth rate for the full year holding all other variables constant it is necessary to add the average of the seasonal coefficients (including zero for the winter quarter) to the calendar year coefficient. For the basic caseload equation, the average of the seasonal coefficients is 1.0 percent, i.e., .010 = (0 - 0.004 - 0.010 + 0.053 )/4. In interpreting these coefficients, it should be kept in mind that they may be misleading with respect to the extent of national participation growth not accounted for by the state variables because the state observations were not weighted by relative size in estimating the model. The simulations reported later (Chapter 6) do so. Nonetheless, the patterns of the year coefficients are closely related to the patterns of national growth not accounted for that are found in the simulations.

All but three of the calendar year coefficients are positive in the caseload equation with vital statistics after the seasonal adjustment. The positive values for each year from 1985 to 1991 are significant and substantial. This indicates that substantial growth in the caseload during this period is not accounted for by the variables in the model. The largest calendar year coefficient is 4.8 in 1990; after adjusting for seasonal effects, the estimate implies that the caseload grew by 5.8 percent in that year for reasons not accounted for by other variables in the model. The smallest coefficient is for 1981, the year that OBRA81 was implemented: -3.0 percent after adjustment. This represents only a portion of the possible effect of OBRA81, as we discuss further below. In all other years (1979-80, 1982-84, and 1992-94) the calendar year dummies are under 1.0 percent in absolute value after adjustment, and not statistically significant. Results are similar in other equations.

In summary, the state-level factors in the model appear to account for most of the growth in the caseload in eight of the 16 years of the sample period. For the seven years from 1985 to 1991, substantial growth is not accounted for by these variables, and these variables do not account for some of the decline in participation in 1981. As will be demonstrated by the simulations (Chapter 6), the state-level factors do explain much of the large cyclical variation in the caseload, but leave much of the long-term trend in the caseload unaccounted for.

3. Demographic Variables

Population Growth and Aging

The expected participation variables allow us to capture the effects of both growth and aging of the at-risk population in a single variable. They do not, however, appear explicitly as explanatory variables in Exhibit 5.1. They are included, but their coefficients are fixed. As discussed in Chapter Three, we hypothesized that the true coefficients of the expected participation variables are each one (a one percent increase in expected participation due to population growth and aging leads to a one percent increase in actual participation). The expected participation variable was, by far, the most significant variable in the initial specifications. The coefficients were always less than one, but not often significantly less than one. Hence, we imposed the restriction that the coefficient is one. This is equivalent to using the logarithm of actual relative to expected participation as the dependent variable, as we report at the top of the exhibit.

Vital Statistics

Of the three vital statistics variables, out-of-wedlock births, marriages, and divorces, only the first two were significant in most specifications tried, for at least some lags. Coefficients of the divorce variable generally had the wrong sign and were, at best, marginally significant. We suspect that reported divorces are a poor predictor of participation because AFDC participation may begin in anticipation of divorce in many cases. Hence, we dropped the divorce variable from the final specification.⁽²⁾

The coefficients of the vital statistics variables are quite significant. Both a one percent increase in out-of-wedlock births and a one percent reduction in marriage are associated with a 0.1 percent increase in participation.⁽³⁾

Immigration

Two immigration variables were tried, the number of immigrants legalized under IRCA-86, and the number of other legal immigrants -- both as immigrants per capita, and both in level, rather than change, form.⁽⁴⁾ The former variable consistently had a statistically significant coefficient, and the latter never did. Waiting periods on AFDC participation for IRCA immigrants (five years) and others (three years) during this period make it unlikely that the immigrants themselves would become participants as soon as one year after immigration.

One possible reason for the significant coefficient on the IRCA variable is the "child-only" phenomenon that has been observed in California and other states with large numbers of IRCA immigrants. Many parents legalized under IRCA had children who were citizens because they had been born in the United States. These children were eligible for benefits prior to their parents' legalization, but parents evidently feared deportation should they apply. Once the parents became legal immigrants, many applied for benefits for their children. The estimates imply that each legalization per 100 population resulted in a 5.0 percent increase in the caseload. The coefficient in the child recipient equation is smaller, 3.6 percent, suggesting that the number of child recipients in these families is smaller than for the mean AFDC Basic family. The coefficient in the total recipient equation is smaller still, 2.3 percent, consistent with the "child-only" explanation.

4. Labor Market Variables

As mentioned above, the final specifications include a quadratic distributed lag in the change in the log of (age-adjusted) unemployment. We found that as many as 14 lags of the unemployment rate could be included in the models without obtaining a negative coefficient, with almost all coefficients statistically significant. This was surprising because previous studies using quarterly data had used no more than five lags. After discovering this, we imposed polynomial distributed lags (DLs) on the unemployment rate coefficients to smooth them and to conserve on the number of explanatory variables. A quadratic DL is used in the final specification.⁽⁵⁾ Three explanatory variables that are functions of the current and 14 lagged values of the change in the log of unemployment appear in the model, each corresponding to one of the parameters of the quadratic function.⁽⁶⁾ We also used polynomial DLs for other unemployment or employment variables that were tried. The models reported include quadratic DLs for the change in the log of trade employment per capita.

We also include a quadratic distributed lag in the current and first 10 lags of the log of trade employment per capita. Other labor market variables tried with distributed lags are unemployment per capita, employment per capita, and manufacturing employment per capita. None of these alone exhibited more explanatory power (i.e., results in lower mean square error) for the caseload equation than the unemployment rate variable alone, and the only variable that adds substantial explanatory power when the unemployment rate variable is included is trade employment per capita. We also experimented with several lags of average weekly earnings in trade and in manufacturing. Trade earnings are not significant, and manufacturing earnings are marginally significant, with the expected negative sign, but only in specifications without trade employment.

The effect of business cycles is jointly captured by the unemployment rate and trade employment variables in these models. The estimated long-run elasticity for the unemployment rate -- the effect of a permanent one-percent increase in the unemployment rate on the caseload after 14 quarters, in percent -- is the sum of the 15 implied coefficients on the current and 14 lags of the unemployment rate variable. The sum is 0.165; i.e., a one percent increase in unemployment eventually leads to an 0.165 percent increase in the caseload. This is not as small as it first appears, as is evident in the following illustration. A one percentage point increase in the unemployment rate from 5 percent to 6 percent is a 20 percent increase, and applying the long-run elasticity of 0.165 to 20 percent yields an increase in the caseload of over three percent.⁽⁷⁾ The increase in the national unemployment rate associated with the most recent recession was more than twice as large as the hypothetical one percentage point increase of our illustration: 2.4 percentage points.⁽⁸⁾

The reported long-run unemployment elasticity alone substantially understates the possible effect of a recession on the Basic caseload because it does not include estimated effects that work through the trade employment variable. When the trade employment variable is not included in the caseload equation, the long-run unemployment rate elasticity is substantially higher, 0.313. Using this figure, the hypothetical one percentage point increase in the unemployment rate described above results in a 5.7 percent increase in the caseload after 14 quarters.⁽⁹⁾

The long-run unemployment elasticities in the recipient and child recipient equations are somewhat smaller, indicating that families induced to obtain welfare benefits by a recession have fewer children than the average AFDC family.

The estimated long-run elasticity for trade employment per capita is -1.00; i.e., a permanent one percent increase in trade employment reduces the caseload by one percent. The corresponding values in the recipient and child recipient equations are slightly smaller, but all above 0.90 in absolute value. It would probably be a mistake to conclude that increasing employment in trade, specifically, would substantially reduce the caseload. The estimated elasticity likely reflects the effects of business cycles, and captures a feature of business cycles that is particularly important to potential AFDC families. The full estimated effects of changes in the unemployment rate and trade employment variables during business cycles are illustrated in the simulations presented in the next chapter.

Dropping the vital statistics variables from the specification increases the magnitude of the long-run elasticities for the two labor market variables, as expected, but only slightly. There are also minor increases in the magnitudes of the AFDC program parameter coefficients. These results suggest that very little of the impact of recessions or of changes in program parameters works through changes in family characteristics; alternatively, we may have missed a more substantial effect because the vital statistics variables are inadequate proxies for family characteristics.

Further discussion of the estimated employment effects appears in Section E, below.

5. AFDC Program Variables

Program Parameters

We used four program parameters in some early specifications of the model: the maximum monthly benefit (MMB), the marginal tax and benefit reduction rate (MTBRR), the average tax and benefit reduction rate (ATBRR), and the AFDC earnings cut off (ECO) relative to the gross income limit (GIL). The MTBRR coefficient was never significant, even marginally, which we attribute to relatively little variation in changes across states in any given year during the sample period. Coefficients of the current and first lag of the other three parameters were very significant in virtually all specifications tried, and the second lag of each was sometimes as significant as well. Hence, the current and first two lags of each of MMB, ATBRR and the ratio of the ECO to the GIL are included in the specifications reported.

The estimated effects of increases in the MMB, the ATBRR, and the ratio of the ECO to the GIL are all statistically significant and consistent in sign with the predictions of the static participation model (Chapter 1). We estimate that a one-percent increase in the MMB increases the caseload by about 0.27 percent after two quarters. An increase in the ATBRR of 10 percentage points (e.g., 70 percent to 80 percent ) is estimated to reduce the caseload by about 1.5 percent after two quarters as some families receiving small benefits because of earnings or other income leave the caseload. An increase in the ratio of the ECO to the GIL of 10 percentage points is estimated to reduce the caseload by 1.0 percent. Estimated effects on recipients and child recipients are quite similar. We discuss the results for the program parameters further in Section F, below.

Federal Legislation

The only dummies for federal legislation that have statistically significant coefficients in any models are the OBRA81 and DEFRA84 dummies. The current and first two lags of the OBRA81 dummy are significant (the last only marginally so). It should be noted that the dummy coefficients do not capture the full effect of OBRA81. OBRA81 increased the ATBRR in most states and also introduced the GIL. The effects of these specific OBRA81 changes are presumably captured by the program parameter variables themselves. Further, the dummy variables for calendar years 1981 and 1982 imply annual caseload reductions of about 3.0 (i.e., -0.03 = -0.04 + (0 - 0.004 - 0.010 + 0.053)/4) and 1.2 percent, respectively, after controlling for other factors and adjusting for seasonal effects, which might also be attributable to OBRA81. The full estimated effects of OBRA81 on the Basic caseload are more apparent in the simulations (Chapter 6).

The coefficient for the DEFRA84 dummy lagged one period is negative, but not significant in the equations reported; it was negative and significant in many specifications we tried. Other lags had much smaller coefficients. We expected this coefficient to be positive if significant, because DEFRA84 partially reversed the changes of OBRA81. The full estimated effect of DEFRA84 includes the effects of resulting changes in program parameters, including the increase in the GIL from 150 percent of the state's need standard to 185 percent. Further, the calendar year coefficients in the caseload equation indicate a 1.4 percent unexplained increase in the caseload in 1984 (after adjusting for seasonality, i.e., 0.014 = 0.004 + (0 - 0.004 - 0.010 + 0.053)/4), and a 4.4 percent increase in the following year, which might at least partly be attributable to implementation of DEFRA84. It may also be that coefficients for other federal legislation dummies (for OBRA87, the provisions of FSA88, OBRA90, and OBRA93) are insignificant because the effects of the legislation are captured by the program parameters and the year dummies. These changes are one of several possible explanations of the substantial caseload growth not accounted for by state-level variables from 1985 to 1991.

We were somewhat surprised to find that the federally mandated introduction of UP programs in 1990 in states without existing UP programs did not have an identifiable impact on the Basic caseload. We had expected to find some shift from the Basic caseload to the UP caseload in these states, especially those with 12 month programs, but did not find any statistically significant shift.

1115 Waivers

Only one of the 1115 waiver dummies we tried had a statistically significant coefficient, the "family cap" dummy for restrictions on benefits for children born while the mother is an AFDC recipient. According to the estimate, such restrictions reduce the caseload by 2.3 percent after one quarter, but have no further effect. We did not expect such restrictions to have an impact on caseloads, at least so quickly. The most likely effect would be a reduction in child recipients, and perhaps only after several quarters (allowing nine months for gestation). The estimated effect on child recipients, however, is slightly smaller than the effect on the caseload.

While it could be that some one-parent families are deterred from welfare dependency by such restrictions, there are three more likely explanations. First, it may be that this waiver dummy is proxying for other administrative efforts in the waiver states to reduce caseloads. The three states that instituted family caps during the sample period are New Jersey (1992.4), Georgia (1994.1), and Wisconsin (1994.3), and the family caps are part of broader efforts to reduce welfare dependency in each state. Second, some AFDC families who have children subject to the cap may have migrated to other states without caps. Third, the finding may be due to random error; given the number of other variables we tried, it would be surprising if we didn't include at least one or two that really were not important in our final specification.

We had not expected to find strong effects for the waivers because each is implemented in only a small number of states and only for a short period before the end of the sample period. Hence, it would be premature to conclude that the requirements implemented under the waivers have little effect on caseloads or recipients.

6. Other Programs and Laws

Very few of the variables in the category of new programs and laws had statistically significant coefficients in the specifications we tried, and the signs of several of these coefficients are opposite that expected. In contrast to the several strong findings for some of the demographic, labor market, and program variables reported above, none of the findings reported below stand out as particularly strong or convincing. It is important to keep in mind that they are a product of a specification search over many variables.

Medicaid

The final specification of the Basic participation models includes two Medicaid variables. "Medicaid expansion" is the share of children under 19 in the state eligible for Medicaid under the Medicaid expansions that began in 1988. The variable is zero before 1988. The second variable is the Medicaid expansion variable multiplied by the share of children under 19 in the state who were AFDC recipients in 1987 (average monthly child recipients), the pre-expansion year. Both variables are entered as changes in the current quarter.

We expected the Medicaid expansion variable when included alone to have a negative coefficient, especially given the strong findings reported by Yelowitz (1994) from his analysis of the same expansions using CPS data for individuals linked to state eligibility and expenditure data.⁽¹⁰⁾ Instead, however, we found a marginally significant, positive coefficient. We added the interaction term on the hypothesis that the expansion would have a larger negative effect (or smaller positive effect) in states in which a large share of children were already on AFDC and, therefore, covered by Medicaid. The coefficient of the interaction variable is consistent with this hypothesis. In fact, the combined coefficients imply that the estimated effect of the expansion is positive in states where the share of children on AFDC in 1987 is below 14.6 percent, and negative in states where the share is larger.⁽¹¹⁾ Most states are in the former category, but a few are in the latter -- including California (15.9 percent).

One other notable feature of the findings is that it is only the current value of the variables that is statistically significant. We expected some lag in the effects of the expansions, but found that lagged values of the expansion variables did not have significant effects even when the current values were omitted from the equation.

One hypothesis about why the estimated effect of the expansion is positive for most states is that efforts to enroll newly eligible individuals into Medicaid also encouraged enrollment in AFDC. This does not explain, however, why the findings are at odds with those of Yelowtiz.

We also estimated models including a measure of the value of Medicaid benefits as an explanatory variables, but never found this variable to be statistically significant.

SSI children

The coefficient of the log of current SSI child beneficiaries is positive in all participation equations and significant in most. We were unsure about inclusion of this variable; on the one hand, it might capture the effects of shifts in AFDC children onto SSI after Sullivan v. Zebley, implying a negative coefficient, but on the other hand it might proxy for unobserved factors affecting participation in both programs in the same direction, which would imply a positive coefficient.

Unemployment Insurance

The log of the percent of unemployed persons who are insured has statistically significant, positive coefficients in all equations. As with the SSI child variable, there are conflicting sign expectations: increases in the share of insured unemployed should reduce the share of unemployed persons who meet the AFDC means test, suggesting a negative coefficient, but the variable might also proxy for unobserved factors that have the same effect on both the share insured and AFDC caseloads.

Abortion Restrictions

Two abortion dummies are included in the reported models (parental notification or consent requirements, and Medicaid funding restrictions), both lagged one period. The coefficient on the dummy for restrictions on Medicaid payments for abortions was more consistently significant than that on the parental consent/notification dummy. Both are negative, suggesting that the negative hypothesized effect of such restrictions on conceptions exceeds the positive effect on births for babies already conceived. The estimated coefficients imply that parental notification or consent requirements reduce caseloads by 0.2 percent and Medicaid funding restrictions reduce them by 0.3 percent. The estimated effects on the number of child recipients are larger -- reductions of 0.3 and 0.5 percent, respectively -- presumably because fertility reductions occur among AFDC mothers as well as potential AFDC mothers.

As with the findings for the family cap, these estimates are surprisingly strong, especially because the estimated effects occur after just two quarters. It may be that these effects are due to other efforts in the states that adopted these restrictions to reduce fertility and AFDC participation.

SSA Allowance Rates

Another variable in this category with a significant coefficient is our measure of SSA's administrative tightening of initial allowances from 1977 to 1978. According to our estimate, reductions in the allowance rate during this period resulted in increases in AFDC participation in 1979. This finding would be substantially strengthened if the sample period was extended back through 1978 and a strong effect were found in that year as well.

General Assistance

The final variable in this category is the measure we developed for cuts in state general assistance (GA) programs. Given the strong findings we obtained for the impact of these cuts on SSI participation in earlier research, we had expected to find some effect for AFDC even though the connection between AFDC and GA is more tenuous than that between SSI and GA. The coefficient of this variable, including lagged values, was not significant in any of the equations we tried. Given the size of the cuts that occurred and the success we had in using this variable in SSI models, we conclude that the GA cuts during this period had at most very small impacts on AFDC participation. States that cut their GA benefits may have been successful in assuring that AFDC-eligible families who sought GA, which is financed entirely from state and local revenues, obtained AFDC benefits, which are partially funded by the federal government, even before the GA cuts occurred. From our earlier work, it appears that the effect of the GA cuts on SSI applications and awards was high because the effort required to apply and the uncertain result discouraged them GA recipients from applying earlier. AFDC eligibility is easier to determine than SSI eligibility and, consequently, the determination process is much simpler. It might also be that those states which cut their GA programs also tightened their AFDC eligibility requirements or screens.

7. Weighted Least Squares Results

We conclude this section with a comparison of the Basic caseload results from the Parks method (Column 1 of both Exhibits 5.1 and 5.2), to estimates of the same model using a weighted least squares (WLS) method, with weights proportional to population size (Column 2 of Exhibit 5.3). Given the Parks specification, the latter method is presumably less efficient (higher standard errors) than the former. If the model specification is correct, both methods produce unbiased estimates. If the correct specification varies across states in a manner not captured by the specified model, then the coefficients could be substantially different. In that case the WLS coefficients would more accurately reflect the national caseload experience.

In general, the results from the two methods are very comparable. The long-run unemployment elasticity is 25 percent lower in the WLS results, but the trade employment elasticity is 13 percent higher. The MMB elasticity is slightly higher, and the ATBRR elasticity is almost unchanged. The change in the coefficient of the ratio of the ECO to the GIL is more substantial; it drops to about 60 percent of the value from the Parks model. The family cap coefficient changes very little. The IRCA immigrant coefficient increases somewhat, but the standard error is much larger so the t-statistic falls. The coefficients of the Medicaid expansion variable and its interaction with the share of children participating in AFDC both increase substantially in magnitude, as do their t-statistics, but these are offsetting changes when considering the actual effect on a specific state; the effect is negative if the share of children participating in AFDC is greater than 14.3 percent, compared to 14.6 in the Parks results. The magnitudes of the coefficients on both vital statistics variables are substantially reduced and no longer significant. The coefficients of the SSI child beneficiary variable, the percent insured unemployed variable, and the two abortion variables change very little. The coefficient of the SSDI initial allowance rate changes sign, however, and is essentially zero.

Exhibit 5.2

Numbers in parentheses are t-statistics.

a Expected participation variable is based on national age-specific participation rates for 1990 and estimated population of the state by age in the quarter.

b. All explanatory variables except quarter and year dummies are changes. Quarter and year dummies are equal to .25 in the quarters/years indicated so that coefficients can be interpreted as annualized rates of growth. For the polynomial distributed lag (PDL) variables, the coefficient of the variable lagged j periods is a0 + a1 j + a2 j2 for j = 0, 1, 2, ... L. Other variables are lagged the number of periods indicated.

c Variables are moving averages of previous four quarters.

d. This variable is the change in the state's SSDI initial allowance rate from 1977 to 1978 times the 1979 year dummy. Special dummies for three states were included due to missing initial allowance data.

e. "Medicaid expansion" is the share of children in the state covered under the Medicaid expansions that began in 1988. "Share participating" is the share of children in the state who were in AFDC families in the year before the expansions began (1987 -- average monthly child recipients divided by population under 19).

f. Autocorrelation coefficients vary across states in this specification.

Except in the first three years of the sample, the year coefficients change little. The two exceptions are for the first two years, in which they are much larger in the WLS results. We do not understand the reason for the changes in the initial years. Note, however, that the first two of these years are the only two that provide information about the effect of the SSDI allowance rate change, so the change in the year coefficients may be related to the change in the SSDI coefficient.

End Notes

1. This constraint could be eliminated through modification of the software or use of an alternative estimation methodology. We estimated some models with SAS-ETS PROC MODEL, a seemingly unrelated regression procedure for estimating linear and non-linear multivariate models. We found, however, that PROC MODEL ran very slowly for our 51 equation model, even though the specification is linear.

2. Like most of the explanatory variables, the vital statistics variables are specified as changes. Initially, however, we included these variables as the current value, based on the argument that they measured flows of families into the pool of families that might participate in AFDC. Multiple lags were included, and we consistently found that the coefficients of the first and second lags were both significant, of opposite signs, and approximately equal in magnitude. Hence, we converted to the change specification.

3. Some have suggested that growth in out-of-wedlock births and declines in marriages have been a more important contributor to caseload growth in the latter part of this period than in the former (see, for instance, CBO, 1993). If so, the coefficients of the vital statistics variables might be larger if estimated using data for the latter part of the period only. To test the idea, we estimated a variant of the final specification in which we included interactions between each of the two vital statistics variables and two dummy variables -- one for the middle third of the sample and one for the last third. The only coefficient that was significant was on the interaction for the marriage variable with the dummy for the last third of the period, and its sign was opposite that expected (coefficient: 0.20; t-statistic: 2.0).

4. We use the level rather than change because legal immigration represents a flow of families into the pool of families that may be eligible for the program.

5. We started with fourth-order polynomials, but found that the third and fourth order coefficients were not significant and could be dropped with little reduction in the fit.

6. In a quadratic DL for a variable, X, the coefficient of the jth lag of the variable, b_j, equals a₀ + a₁ j + ₂ j² for j = 0, 1, ..., L, where the alphas are parameters to be estimated and L is the maximum lag length. If the original model is Y_st = ... b₀X_st + b₁X_st-1 + ... + b_LX_st-L ..., substitution of the quadratic equation and simplification yields the following alternative version of the model: Y_st = ... a₀Z0_st + a₁Z1_st-1 + a₂Z2_st ..., where: Z0_st = X_st + X_st-1 + ... + X_st-L; Z1_st = X_st-1 + 2 X_st-2 + ... + L X_st-L; Z2_st = X_st-1 + 4 X_st-2 + ... + L² X_st-L. Thus, the alphas can be estimated by replacing the Xs in the model with the Zs. Once the alphas are estimated, the betas can be recovered from the quadratic equation. See Greene (1990).

7. The calculation described in the text yields 3.3 percent, but this somewhat overstates the estimated effect of the assumed change in the unemployment rate because the method used to calculate the effect is only accurate for small percentage changes in the unemployment rate. The exact method--see equation 3.7--yields an estimated increase of 3.0 percent, obtained from .165(ln(.06) - ln(.05)) =.165ln(.06/.05 ) = .030 .

8. From 5.2 percent in 1989.3 to 7.6 percent in 1992.3.

9. The 5.7 percent figure was computed by the exact method, described in the previous footnote, i.e., 0.057 = .313ln(.06/.05).

10. As discussed in more detail in Chapter 2, Yelowitz estimates that increasing the Medicaid need standard by 25 percent of the poverty line reduces the AFDC participation rate of single mothers by 4.6 percent.

11. For a state in which the share of children on AFDC is P, the estimated effect of a change in the share of children eligible for Medicaid under the expanded benefit is .179 - 1.23 x P, which is negative for P >.145.

Continue to Part 2 of Chapter 5