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
Regression Results for Basic Models | |||||||||||||
Sample: 51 states, 1979.4 - 1994.3 | |||||||||||||
Dependent Variable is change in ln(participation/expected participation)^{a} | |||||||||||||
Coefficients | T-statistics ^{b} | ||||||||||||
Explanatory | Caseload | Recipients | Child Recipients | Caseload | Recipients | Child Recipients | |||||||
Variables^{c} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | |
ln(unemployment rate) | 10xa_{0} | 0.250 | 0.256 | 0.261 | 0.268 | 0.222 | 0.225 | 9.4 | 11.1 | 8.9 | 10.1 | 8.5 | 9.8 |
(PDL: L = 14) | 100xa_{1} | -0.343 | -0.370 | -0.477 | -0.489 | -0.334 | -0.343 | -3.7 | -4.7 | -4.5 | -5.1 | -3.9 | -4.5 |
1000xa_{2} | 0.148 | 0.167 | 0.257 | 0.264 | 0.154 | 0.161 | 2.4 | 3.1 | 3.5 | 4.0 | 2.6 | 3.1 | |
long-run elasticity | 0.165 | 0.165 | 0.152 | 0.156 | 0.138 | 0.141 | |||||||
ln(trade employment per cap.) | 10xa_{0} | -0.650 | -0.738 | -0.602 | -0.733 | -0.511 | -0.598 | -3.9 | -5.2 | -3.1 | -4.0 | -3.2 | -4.2 |
(PDL: L = 10) | 100xa_{1} | -2.427 | -2.324 | -2.290 | -2.170 | -2.252 | -2.180 | -7.5 | -8.2 | -5.8 | -5.9 | -6.5 | -7.1 |
1000xa_{2} | 2.721 | 2.690 | 2.646 | 2.663 | 2.223 | 2.244 | 6.6 | 7.6 | 5.4 | 5.8 | 5.1 | 5.8 | |
long-run elasticity | -1.002 | -1.054 | -0.903 | -0.974 | -0.944 | -0.993 | |||||||
ln(maximum monthly benefit) | current | 0.080 | 0.082 | 0.075 | 0.077 | 0.098 | 0.097 | 4.5 | 5.5 | 4.1 | 4.6 | 6.4 | 7.3 |
1st lag | 0.151 | 0.151 | 0.129 | 0.131 | 0.136 | 0.138 | 8.1 | 9.7 | 6.8 | 7.4 | 8.1 | 9.3 | |
2nd lag | 0.039 | 0.040 | 0.012 | 0.014 | 0.020 | 0.021 | 2.2 | 2.6 | 0.7 | 0.9 | 1.3 | 1.6 | |
long-run elasticity | 0.270 | 0.274 | 0.216 | 0.222 | 0.254 | 0.256 | |||||||
average tax and | current | -0.033 | -0.038 | -0.030 | -0.033 | -0.030 | -0.031 | -2.3 | -3.2 | -2.4 | -2.7 | -2.5 | -2.9 |
benefit reduction rate | 1st lag | -0.080 | -0.081 | -0.064 | -0.068 | -0.079 | -0.079 | -5.4 | -6.5 | -4.8 | -5.3 | -6.1 | -7.0 |
2nd lag | -0.041 | -0.041 | -0.016 | -0.019 | -0.035 | -0.038 | -2.8 | -3.4 | -1.2 | -1.5 | -2.8 | -3.4 | |
long-run effect | -0.153 | -0.160 | -0.111 | -0.120 | -0.144 | -0.148 | |||||||
AFDC earnings cut off | current | -0.051 | -0.052 | -0.048 | -0.048 | -0.055 | -0.055 | -8.8 | -10.2 | -7.7 | -8.1 | -10.1 | -11.1 |
relative to gross income limit | 1st lag | -0.038 | -0.038 | -0.027 | -0.028 | -0.040 | -0.042 | -6.2 | -7.1 | -4.4 | -4.7 | -6.9 | -8.0 |
2nd lag | -0.011 | -0.009 | -0.004 | -0.004 | -0.014 | -0.015 | -1.9 | -1.8 | -0.6 | -0.7 | -2.6 | -3.1 | |
long-run effect | -0.100 | -0.099 | -0.079 | -0.079 | -0.109 | -0.112 | |||||||
OBRA81 | current | -0.039 | -0.037 | -0.025 | -0.026 | -0.033 | -0.033 | -5.3 | -6.2 | -3.2 | -3.6 | -4.9 | -5.5 |
1st lag | -0.020 | -0.021 | -0.002 | -0.002 | -0.022 | -0.021 | -2.5 | -3.1 | -0.2 | -0.3 | -3.0 | -3.3 | |
2nd lag | -0.010 | -0.012 | -0.022 | -0.022 | -0.010 | -0.008 | -1.4 | -1.9 | -3.0 | -3.3 | -1.4 | -1.4 | |
long-run effect | -0.069 | -0.070 | -0.048 | -0.050 | -0.064 | -0.063 | |||||||
DEFRA84 | current | -0.006 | -0.006 | -0.006 | -0.005 | -0.002 | -0.002 | -1.3 | -1.8 | -1.6 | -1.4 | -0.6 | -0.9 |
Coefficients | T-statistics^{b} | ||||||||||||
Explanatory | Caseload | Recipients | Child Recipients | Caseload | Recipients | Child Recipients | |||||||
Variables^{c} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | w/ v.s.^{d} | w/o v.s.^{d} | |
family cap | 1st lag | -0.023 | -0.024 | -0.017 | -0.018 | -0.020 | -0.019 | -3.2 | -4.0 | -2.1 | -2.3 | -2.6 | -2.8 |
IRCA immigrants per 100 | 1st lag | 0.050 | 0.044 | 0.023 | 0.020 | 0.036 | 0.034 | 5.0 | 5.6 | 1.5 | 1.5 | 4.8 | 5.3 |
Medicaid expansion^{g} | current | 0.179 | 0.190 | 0.058 | 0.045 | 0.160 | 0.153 | 2.5 | 2.9 | 0.8 | 0.6 | 2.9 | 3.1 |
Med. exp. x share participating^{g} | current | -1.230 | -1.327 | -0.216 | -0.120 | -1.206 | -1.148 | -2.1 | -2.5 | -0.3 | -0.2 | -3.1 | -3.2 |
ln(out-of-wedlock births)^{e} | 0.101 | 0.111 | 0.073 | 3.8 | 3.3 | 3.0 | |||||||
ln(marriages)^{e} | -0.097 | -0.139 | -0.094 | -3.3 | -4.2 | -3.2 | |||||||
ln(SSI child beneficiaries) | current | 0.009 | 0.009 | 0.007 | 0.007 | 0.009 | 0.009 | 2.3 | 2.7 | 1.5 | 1.8 | 2.6 | 2.9 |
ln(% insured unemployed) | 1st lag | 0.013 | 0.014 | 0.009 | 0.009 | 0.008 | 0.008 | 3.7 | 4.3 | 2.2 | 2.5 | 2.0 | 2.5 |
abortion: parental consent/notice | 1st lag | -0.002 | -0.002 | 0.000 | 0.001 | -0.003 | -0.003 | -1.0 | -1.6 | 0.3 | 0.4 | -2.2 | -2.5 |
Medicaid restricted | 1st lag | -0.003 | -0.003 | 0.002 | 0.002 | -0.005 | -0.005 | -1.8 | -2.4 | 1.1 | 1.2 | -3.1 | -3.5 |
SSDI initial allowance rate^{f} | -0.053 | -0.046 | -0.072 | -0.071 | -0.157 | -0.153 | -2.1 | -2.0 | -2.6 | -2.9 | -5.5 | -5.9 | |
1979 dummies for: | Alaska | 0.018 | 0.005 | 0.053 | 0.056 | 0.183 | 0.176 | 0.1 | 0.0 | 0.3 | 0.3 | 0.9 | 1.0 |
Hawaii | 0.060 | 0.041 | 0.074 | 0.048 | 0.191 | 0.175 | 0.7 | 0.5 | 0.8 | 0.5 | 2.3 | 2.4 | |
D.C. | -0.057 | -0.059 | -0.013 | -0.011 | 0.017 | 0.008 | -1.3 | -1.5 | -0.3 | -0.2 | 0.4 | 0.2 | |
Seasonal Dummies | Spring | -0.004 | -0.002 | -0.004 | -0.002 | -0.006 | -0.006 | -0.6 | -0.4 | -0.6 | -0.3 | -1.4 | -1.4 |
Summer | -0.010 | -0.007 | -0.008 | -0.005 | -0.013 | -0.013 | -1.4 | -1.2 | -1.1 | -0.7 | -2.1 | -2.3 | |
Fall | 0.053 | 0.054 | 0.046 | 0.046 | 0.040 | 0.041 | 7.1 | 8.9 | 5.8 | 6.2 | 6.5 | 7.4 | |
Calendar Year Dummies | 1979 | 0.010 | 0.021 | 0.006 | 0.014 | -0.012 | -0.005 | 0.6 | 1.5 | 0.4 | 1.1 | -0.9 | -0.4 |
1980 | 0.009 | 0.014 | -0.007 | -0.002 | 0.016 | 0.021 | 0.8 | 1.4 | -0.6 | -0.2 | 1.8 | 2.7 | |
1981 | -0.040 | -0.039 | -0.061 | -0.059 | -0.033 | -0.032 | -2.9 | -3.3 | -5.0 | -5.0 | -3.1 | -3.6 | |
1982 | -0.022 | -0.019 | -0.006 | -0.003 | -0.016 | -0.016 | -1.9 | -1.8 | -0.5 | -0.3 | -1.7 | -2.1 | |
1983 | 0.004 | 0.007 | 0.013 | 0.018 | 0.016 | 0.019 | 0.3 | 0.7 | 1.4 | 2.0 | 2.0 | 2.7 | |
1984 | 0.004 | 0.011 | 0.027 | 0.033 | 0.024 | 0.030 | 0.3 | 1.1 | 2.8 | 3.6 | 2.9 | 4.5 | |
1985 | 0.034 | 0.044 | 0.028 | 0.041 | 0.042 | 0.051 | 3.1 | 4.8 | 2.7 | 4.3 | 4.9 | 7.2 | |
1986 | 0.034 | 0.039 | 0.024 | 0.032 | 0.031 | 0.037 | 3.2 | 4.3 | 2.5 | 3.4 | 3.7 | 5.3 | |
1987 | 0.016 | 0.021 | 0.010 | 0.018 | 0.021 | 0.028 | 1.5 | 2.4 | 1.1 | 2.1 | 2.6 | 4.2 | |
1988 | 0.031 | 0.037 | 0.022 | 0.028 | 0.030 | 0.035 | 2.9 | 4.1 | 2.4 | 3.1 | 3.7 | 5.3 | |
1989 | 0.042 | 0.048 | 0.038 | 0.047 | 0.040 | 0.048 | 3.7 | 5.2 | 4.0 | 5.0 | 4.5 | 6.5 | |
1990 | 0.048 | 0.055 | 0.047 | 0.055 | 0.059 | 0.063 | 4.5 | 6.1 | 4.7 | 5.7 | 7.0 | 9.0 | |
1991 | 0.038 | 0.045 | 0.034 | 0.042 | 0.032 | 0.038 | 3.5 | 5.1 | 3.5 | 4.5 | 4.0 | 5.7 | |
1992 | 0.005 | 0.005 | -0.004 | -0.003 | -0.006 | -0.003 | 0.4 | 0.5 | -0.5 | -0.3 | -0.7 | -0.5 | |
1993 | -0.006 | -0.003 | -0.022 | -0.019 | -0.016 | -0.013 | -0.6 | -0.3 | -2.4 | -2.0 | -1.9 | -2.0 | |
1994 | -0.018 | -0.016 | -0.022 | -0.021 | -0.013 | -0.011 | -1.1 | -1.2 | -1.7 | -1.6 | -1.1 | -1.2 |
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 a_{0} + a_{1} j + a_{2} j^{2} 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.