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, bj, 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 Yst = ... b0Xst + b1Xst-1 + ... + bLXst-L ..., substitution of the quadratic equation and simplification yields the following alternative version of the model: Yst = ... a0Z0st + a1Z1st-1 + a2Z2st ..., where: Z0st = Xst + Xst-1 + ... + Xst-L; Z1st = Xst-1 + 2 Xst-2 + ... + L Xst-L; Z2st = Xst-1 + 4 Xst-2 + ... + L2 Xst-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.
12. The UP caseload series obtained from ACF reported zero UP cases for the District of Columbia in 1992.2 and 1992.3 and for Mississippi in 1993.4.
13. There are eight states in the sample in addition to the 19 full-period states and the 22 mandate states. These states had UP programs during part, but not all, of the pre-mandate subperiod. Recall that Mississippi and the District of Columbia were dropped from the sample because of evident data errors that could not be corrected. See Chapter 4.
14. The full-period estimates (Column 1) differ from the results reported previously in both specification and estimation methodology. We inadvertently did not include the ratio of the ECO to the GIL in this model. Its exclusion may account for differences in the Park and WLS results for the ATBRR, but we have not had an opportunity to confirm this.
15. The change in the log of children per family (case) equals the change in the log of children minus the change in the log of families. Because small changes in logarithms are approximately equal to percentage changes in the variable itself, the percent change in children per AFDC family is approximately the difference between the percent change in children and the percent change in families.
16. The graph suggests that the effect on both caseloads will continue to be positive for some time after the 14-month period ends; in fact, the effect for the UP caseload appears to be increasing! We specified only a 14-quarter distributed lag because in models in which we included the trade-employment variable the distributed lag coefficients for the unemployment rate became slightly negative in the 15th quarter when we specified long maximum lag lengths. We did not try longer lag lengths without the trade employment variable.
17. The overall figure assumes that five percent of the caseload is in the UP program.
18. One piece of information we have is, at least, consistent with the conjecture. As previously mentioned, we inadvertently omitted the GIL variable from the UP specification when we re-estimated the full-period model using WLS (Exhibit 5.4). Both the MMB and ATBRR coefficients in these findings are much larger than those in the Parks estimates (Exhibit 5.3). Of course, other model and estimation differences could explain these differences. In the WLS results, the long-run MMB elasticity implies that a 10 percent increase results in an UP caseload increase of 4.2 percent -- close to Shroder's 5.1 percent finding when he does not correct for simultaneity, but well short of the 16.7 percent figure he obtains when he makes the correction. The estimated long-run ATBRR effect implies that a 10 percentage point increase in that variable reduces the caseload by 1.6 percent -- in line with our finding for the Basic program (1.5 percent), but well below Moffit's estimate (5.5 percent) for the Basic program.