The age distribution of the population has changed substantially during the period under examination as the "baby boom" generation has aged. Since the proportion of households on AFDC varies by the age of household head, we developed variables to capture the effects of population age distribution changes on AFDC participation. These variables also capture the effects of population growth. For each program (Basic and UP), we developed three "expected participation" variables  one each for caseloads, total recipients, and child recipients. Construction of these variables is discussed in Section C.1, below.
A second, related, issue is that some explanatory variables in the model can be expected to change with changes in the age distribution (e.g., the unemployment rate), but such changes would not be expected to have an impact on AFDC participation  although they might be associated with changes in participation because AFDC participation is affected by changes in the age distribution. When feasible, we adjusted such variables to remove the effect of age distribution changes. We describe the construction of the adjusted variables in Section C.2.

1. "Expected" Participation

The value of the expected participation variable for a specific State and quarter is the level of participation we would expect if agespecific participation rates were the same as national monthly average agespecific participation rates in 1990. We chose 1990 as the base year because the national agespecific data needed to construct the variable is better in that year than in others, due to the Decennial Census.
For the Basic caseload, the agespecific participation rate is defined as the number households in the Basic program headed by women in the age group divided by the number of women in the age group. Agespecific participation rates for the UP caseload are defined analogously, but using the number of households in the UP caseload, the age of the adult male in the household, and male population data. For expected recipients and expected child recipients in each program, we used the same scheme to classify households into age groups. The agespecific participation rate for recipients is the number of recipients in households in the age category divided by the number of women (Basic) or men (UP) in the age group, and the agespecific participation rate for child recipients is defined in the same way, but only including children. National agespecific participation rates for 1990 were estimated using the 1990 Survey of Income and Program Participation (see Chapter Four).
We constructed annual expected participation variables for each state by computing a weighted sum of the 1990 national agespecific participation rates, with the weight for each age group equal to the State's population of the relevant sex in the age category in the current year:
Equation 3.3: A^{*}_{st} = S_{a} A_{a90} P_{ast}
where A^{*}_{st} is "expected" AFDC participation (i.e., expected caseload, recipients, or child recipients in one of the programs) in State s and year t, A_{a90} is the 1990 national AFDC participation rate in age group a, and P_{ast} is the size of the population of the relevant sex in State s and year t that is in age group a. The final step was to convert the annual series to quarterly series, using the methodology discussed in Section E, below.
The change in the logarithm of each expected participation variable is used as an explanatory variable in the relevant participation equation. The coefficient of this variable can be interpreted as the percent change in the caseload associated with a one percent increase in the size of the population, holding the age distribution of the population and other explanatory variables constant. Hence, we would expect them to be close to one. In the initial models we estimated the coefficients of these variables were very significant, but not significantly different from one. In the models reported here, we have constrained the coefficient of these variables to be one.


2. Ageadjusted Explanatory Variables

Each ageadjusted variable is the logarithm of the ratio of the unadjusted variable to its "expected" value. Expected values are computed analogously to the computation of expected participation: they are a weighted average of agespecific national values for 1990, with the weight for each age group equal to the share of the State's population in the age category in the current year. Mathematically, the index variable (X_{st }) is defined by:
Equation 3.4: X_{st }= ln(W_{st} / W^{*}_{st})
where W_{st} is the unadjusted variable, and W^{*}_{st} is the "expected" value of the variable, defined as:
Equation 3.5: W^{*}_{st} = S_{a} W_{a90} P_{ast}/P_{st}
where W_{a90} is the 1990 national value of the variable for age group a and P_{st }is the total population of State s in period t. Thus, for example, the unemployment rate index for State A in 1982 is the log of the actual rate divided by the rate we expect given the age distribution of State A's working age population in 1982 and national unemployment rates by age in 1990. Annual figures were converted to quarterly series as described in Section E, below.
The complexity of the construction of each ageadjusted variable may diminish the ability of policy makers and others to understand and use the findings. The interpretation of the results and their potential use are not as difficult as they may first appear, however, and the results may be substantially more useful if adjusted variables are used than if unadjusted ones are used.
To illustrate, consider the logarithm of the ageadjusted unemployment rate as it would appear in a typical model:
Equation 3.6: ln(A_{st}) = ... + b_{u }ln(u_{st}/u_{st}*) + ... = ... + b_{u }ln(u_{st})  b_{u }ln(u_{st}*) + ....
where u_{st} is the unemployment rate and u_{st}* is the "expected" rate, defined as in Equation 3.5. It is apparent from the second representation of the unemployment term above that the coefficient of the ageadjusted variable can be interpreted as the elasticity of the caseload with respect to unemployment holding the other explanatory variables constant, and provided that the unemployment rate change is not due to a change in expected unemployment  i.e., not due to a change in the age distribution of the population. This is no different than what the interpretation would be if we used the logarithm of the unadjusted unemployment rate as the explanatory variable, except in that case the interpretation would apply to changes due to changes in the age distribution of the population as well as any others. This also illustrates the reason for making an age adjustment: we would not expect a change in unemployment that is due to a change in the age distribution of the population to have the same impact on AFDC participation as a change that is due to the business cycle. In fact, we would expect it to have no effect other than the effect that is accounted for by the expected participation variables.
Continuing the illustration, contingency loans to States, intended to help them finance their AFDC payments during a recession, could be tied to the unemployment rate, with the maximum loan amount related to the gap between the unemployment rate and some "standard" unemployment rate. Some specific value for the unemployment rate would be the simplest choice for the standard rate, but the fact that unemployment rates vary because of changes in the age distribution of the population, not just because of the business cycle, means that the "standard" that would be appropriate for a given age distribution would be inappropriate for another one. The expected unemployment rate could be used as the standard instead, thereby recognizing the effect of a change in the age distribution of the population on the unemployment rate. Under the latter system, the maximum loan amount would be insulated from changes in the unemployment rate that are caused by changes in the age distribution of the population rather than by the business cycle.
