Selection into Identification in Fixed Effects Models, with Application to Head Start

Source: Panel Study of Income Dynamics 1968–2011 waves.

Notes: Panel A of this table presents comparisons of the characteristics of individuals in switching families and nonswitching families. Columns 1, 2, and 3, respectively, show the mean characteristics of individuals in families that are switchers, individuals in families that are not switchers, and individuals that attended Head Start (HS) in nonswitcher families. Column 3 presents the p-value for the test that Columns 1 and 2 are equal. Column 4 shows the estimates from a regression of each row heading on an indicator for being in a switcher family, with the corresponding p-value shown in Column 5, with standard errors clustered on id1968. All controls from the main specification are included, except the variable shown in the row heading. All estimates are weighted to be representative of 1995 population; see text for details. Panel B shows the mean and standard deviation (in parentheses) of the inverse of the post-regression propensity score weights when the target is Head Start participants. This gives a measure of how aligned the characteristics of switchers are with the characteristics of Head Start participants, the population of interest. An average value of one implies perfect alignment, while a higher value implies that the characteristics of switchers are overrepresented relative to the characteristics of Head Start participants. Pr(switch) and Pr(Head Start) are estimated from a multinomial logit model of these outcomes on family size and other covariates described in the text.

Source: Panel Study of Income Dynamics 1968–2011 waves.

Notes: This table shows the change in the composition of the PSID sample moving from all individuals and estimating a model without controls (“All (no FFE)”), to individuals that have at least one other sibling in the sample and estimating a model without controls (“Siblings Sample (no FFE)”), to individuals in families that have variation in Head Start attendance and estimating a model with family fixed effects (“Switchers sample (FFE)”). Panel A shows the share of individuals in each sample that come from a family with one child (zero siblings), two children, etc. Panel B shows the variance in Head Start for each family size and sample. For the switcher sample, this is calculated net of family fixed effects. Panel C shows the “regression weight” given to each family size in a given sample, denoted as ω_z and defined formally in Section III. The shares and regression weights do not sum to one for the “all sample” because this sample also includes an additional category of individuals who have an unknown number of siblings (due to a missing mother ID). These individuals account for 8.5 percent of the “all sample.”

Sources: Panel Study of Income Dynamics 1968–2011 waves and Children of the National Longitudinal Study of Youth.

Notes: Panel A of this table shows the coefficients from regressions of outcomes on a series of indicators for whether an individual attended Head Start interacted with an indicator for the number of children in one’s family. The data source and specification varies across columns. Columns 1 and 2 use our main PSID sample, and the outcome is attainment of some college. Columns 3–5 use the CNLSY79 sample, and the outcomes are indicators for graduating from high school, being idle, and having a learning disability, respectively. Column 1 includes controls, but not mother fixed effects, and standard errors are clustered at the family ID level. Columns 2–5 include mother fixed effects, and standard errors are clustered by mother ID. The number of Head Start switchers is equal to the number of individuals in families that have variation in Head Start. “Effective Obs. (CX Individs.)” is the equivalent number of cross-sectional units that provide the same amount of variation as switchers. “Effective Obs. (Individs. 2-Person Families)” is the equivalent number of individuals in two-person switching families that provide the same amount of variation as switchers. Both of these are calculated using Equation 3, where the denominator is the variance of Head Start, residualized by the family mean of the covariates in the analysis, or 0.125, respectively. Panel B shows the weighted average of the coefficients when using regression weights, ω_z (defined in Section III), determined by the overall distribution of families (“All”), the distribution of 2+ child families (“Siblings”), and the distribution of 2+ child families that have variation in Head Start attendance (“Switchers”).

↵* p < 0.10,

↵** p < 0.05,

↵*** p < 0.01.

Notes: This table shows the results from 3,000 Monte Carlo simulations. Each panel of the table shows results from a different DGP and/or different covariates used in the p-score, and each row within panel is for a different target population. The true DGP is linear and is discussed in Section IV.D. Panel A shows results where Head Start has a constant treatment effect (TE) for all individuals. Panel B shows results where Head Start (HS) has no effect on individuals from small families (three or fewer children) and a large effect for families with many children (four or more children). Panels C and D show results where treatment effects are linear in X_ig. Column 1, “True Beta,” presents the true average increase in the probability of completing some college for participants in Head Start in the sample, which is a function of the DGP and sample composition. Columns 2 and 3 present the bias of various estimation strategies, defined as the difference between the estimated effects of Head Start and the true beta. The estimated effects come from a LPM, propensity-score weighted LPM, respectively. Column 4 presents the ratio of the mean squared error (MSE) of the reweighting estimators relative to LPM. Reweighted estimates are obtained using in-regression weighting, with weights adjusting for the representativeness of switchers (using the variable(s) indicated in each of the panel headings as predictors in the multinomial logit step) and the conditional variance of Head Start within families. All betas are multiplied by 1,000.

↵* p < 0.01.