We estimate separate Basic and UP models for caseloads, recipients, and child recipients. In addition, we estimate an average monthly benefit (AMB) equation for the combined programs. All equations are methodologically identical, differing only in the specification of their variables. In this section we first describe the "generic" structure of the equations.

Each equation estimated has the following general form:

Equation 3.1: D*ln*(Y_{st}) = a_{1 }Z1_{st} + ... +a _{J}ZJ_{st} + b_{1 }DX1_{st} + ... + _{b}_{K}_{D}XK_{st}

+ g_{2}Q2_{t} + ... + g_{4}Q4_{t} + d_{80}T80_{t} + ... + d_{94}T94_{t} + e_{st}

where:

- D
*ln*(Y_{st}) is the change in the natural logarithm of the caseload, recipients, child recipients, or average monthly benefits in State "s" and from quarter "t-1" to quarter "t;" - Z1
_{st}+ ... + ZJ_{st}are dummy (binary) explanatory variables and current year values of selected "flow" variables (see below); - a
_{1}... a_{J}are the coefficients of the dummy variables;? - DX1
_{st},..., DXK_{st}are quarter to quarter changes in continuous explanatory variables. These include both current and lagged values for selected variables; - b
_{1},...,b_{K}are the coefficients of the continuous explanatory variables; - Q2
_{t}... Q4_{t}are quarterly dummy variables (the first quarter is the base quarter). Each dummy variable equals .25 for the quarter indicated by its name, and zero for all other quarters; - g
_{2}... g_{4}are the coefficients of the quarterly dummies, interpreted as the difference between the annualized growth rate of Y in the reference quarter and the first quarter, other things constant; - T80
_{t}... T94_{t}_{}are year dummies, equal to .25 for every quarter during the reference calendar year, and 0 in all other quarters; - d
_{80}... d_{94}are the coefficients of the year dummies, interpreted as the annual rate of growth in the first quarter of the reference year after controlling for other explanatory variables;^{ }and - e
_{st}is the regression disturbance.

The model is specified in changes in order to eliminate state "fixed effects" -- factors that vary across states, but not over time. "Time," or "year" effects are captured by the time dummies. Hence, we are essentially relying on covariation in changes across states to estimate the model's parameters. Purely cross-sectional covariation is not used at all, and use of purely time-series covariation to estimate parameters other than the year and quarterly dummy coefficients is minimal.^{(1)}^{}

We specify the dependent variable as a change in the logarithm because we expect the effects of changes in the explanatory variables to be proportional to the size of the caseload, rather than independent of the caseload size. For instance, a change in the maximum monthly benefit and a change in the unemployment rate are both expected to have an impact on caseload size that is proportional to the size of the caseload -- the more cases, or potential cases, that may be affected, the larger is the effect.

Some continuous explanatory variables are also in logarithms; for such variables the corresponding coefficient is an elasticity -- the percent change in the dependent variable per percent change in the independent variable. Those not in logarithms are all rates of some sort (e.g., the average benefit reduction rate).^{(2)}

Continuous variables are specified in changes in most cases. Some "flow" variables -- representing potential case openings (e.g., immigrants) or closings -- are specified in levels. ^{(3)} The regression disturbance is assumed to follow a first-order autoregressive process, with different parameters in each state. Formally:

Equation 3.2: e_{st} = r_{s}e_{st}_{-1} + v_{st},

where r_{s} is the autocorrelation coefficient for state s (-1 < r_{s} < 1), and v_{st} is a random variable that is independent over time, but not across states. We assume that the variances and contemporaneous covariances of the v_{st} are constant over time. This model is sometimes referred to as the "Parks" model (Parks, 1967).^{(4)}

The Parks methodology has an important limitation -- the number of explanatory variables can be no larger than the number of time-series observations.^{(5)} This prevented us from using the methodology for estimating models within subperiods and testing for stability across subperiods.^{}^{(6)} We did, however, estimate an UP model for all states using just the sub-period following the federal mandate, implemented in the fourth quarter of 1991, using an alternative specification. We assumed in this case that contemporaneous cross-state covariances of the disturbances are zero and that the autocorrelation coefficient is the same for all states. We also weighted each observation by the state's population.^{(7)} To assess the influence of the alternative estimation methodology on the findings, we re-estimated our final caseload models for the full period via this method and compared the results to the Park estimates.