Each age-adjusted 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 age-specific 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 (Xst ) is defined by:
Equation 3.4: Xst = ln(Wst / W*st)
where Wst is the unadjusted variable, and W*st is the "expected" value of the variable, defined as:
Equation 3.5: W*st = Sa Wa90 Past/Pst
where Wa90 is the 1990 national value of the variable for age group a and Pst 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 age-adjusted 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 age-adjusted unemployment rate as it would appear in a typical model:
Equation 3.6: ln(Ast) = ... + bu ln(ust/ust*) + ... = ... + bu ln(ust) - bu ln(ust*) + ....
where ust is the unemployment rate and ust* 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 age-adjusted 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.