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.