In this section we describe previous efforts to model AFDC caseloads that used pooled state-level data for multiple states across states and over time. These models can be viewed as individual state time-series models that have been linked by using the same explanatory variables in all states and, with some exceptions, constraining the coefficients of each variable to be the same for all states. An important aspect of these models concerns the specification of the regression disturbance.

The general specification for this class of models is:

Equation 2.1: Y_{ts} = a+ b'X_{ts} + e_{ts}

where:

Y_{ts} is the dependent variable for year "t" in state "s" (a measure of program participation);

X_{ts}** **is a vector of explanatory variables;

a is the intercept;

b is a vector of coefficients for the explanatory variables (assumed constant across states and over time); and

e_{ts} is the regression disturbance.

There are various subclasses of pooled models, which are defined through the specification of the disturbance. For our purposes the most important subclass is "fixed effects" models. These models assume that the disturbance, _{ts}, is the sum of three terms: a "state fixed effect" that is different for each state but doesn't vary over time; a "time fixed effect" that is different each year but doesn't vary across states; and a random effect. The subclass can be specified as:

Equation 2.2: Y_{ts} = b'X_{ts} + a_{s} + t_{t} + u_{ts}

where:

a_{s} is the state fixed effect;

t_{t} is the time fixed effect for time period t; and

u_{ts} is the random disturbance.

We find it useful to think of the state fixed effect as the intercept term for the state; i.e., under this specification the regression intercept varies across states. This term will "explain," in a statistical sense, all of the cross-state variation in the average of the caseload variable overtime. Another way to state this is that it captures all factors that account for cross-state variation in the dependent variable that do not change over time.

Another way to interpret the fixed state effects specification is to recognize that it is equivalent to modeling the relationship between *changes* in the caseload variable to *changes* in the explanatory variables, with no state fixed effect (assuming appropriate specification of the other parts of the disturbance). The change model can be obtained by first differencing Equation 2.2:

Equation 2.3: DY_{ts} = b'DX_{ts} + Dt_{t} + Du_{ts}

where the prefix D indicates the one period change in the variable. Because fixed state effects don't change over time, they drop out of the change specification. From this specification it is evident that we are ignoring the cross-section relationship between the levels of the variables in estimating the models' coefficients. Note that any X variables that vary only across states, and not over time, will drop out of the model, too.

The fixed time effects capture factors that change over time, but have an equal influence on the caseload variable in all states. One important example is the implementation of a new federal AFDC policy in all states at one time (example, OBRA81). To the extent that such a change has an equal impact on the participation variable in all states, its impact will be captured in the fixed time effects. The implementation of a new policy may be captured in fixed time effects over a number of periods because the impact may not be fully realized in the first period of implementation, or because it may be partly realized in anticipation of implementation. Unit impacts that are not uniform in all states will not be captured; unless captured by explanatory variables, they become part of the regression disturbance.

Fixed time effects are usually implemented by including a time dummy for each period, omitting the dummy for an arbitrarily chosen base period. These dummies "use up" all of the information that national time-series models would use to estimate the model, as is evident from the fact that a national time-series model with a dummy variable for each period could not be estimated. Assuming state fixed effects are also included, the coefficient estimates for the explanatory variables are based on the relationship between *deviations* of *changes* in the dependent variable from the national average change and *deviations* of *changes* in the explanatory variables from changes in their respective national averages. Thus, fixed effect results can be quite different than those obtained from national time series alone.

The other commonly used subclass of pooled models is known as "random effects" models. As in fixed effect models, the disturbance is usually assumed to have three components -- one that varies across states, one that varies across time periods, and a third that varies across both. The critical difference between random effects and fixed effects models is that the state and time components of the error term are assumed to be uncorrelated with the X variables in the former, but not in the latter. The uncorrelated assumption is built into estimators for random effects models. If the assumption is correct, the estimator will be more efficient than fixed effects estimators, but if it is incorrect the estimator may be biased, perhaps substantially so.

Fixed effects models are more commonly used for studying program participation at the state level than are random effects models. Fixed state effects are important because there are many time-invariant characteristics of states (e.g., geography) that could have an impact on participation and might well be correlated with explanatory variables. Fixed time effects may or may not be important, depending on whether significant national factors changed over the period under investigation, and whether those changes are associated with changes in the explanatory variables.

The two subclasses of pooled models may be mixed. The AFDC participation studies discussed below provide examples of models with fixed state effects, but not fixed time effects.