# IRLE. Evaluating Public Programs with Close Substitutes: The Case of Head Start. IRLE WORKING PAPER # December 2014

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1 IRLE IRLE WORKING PAPER # December 2014 Evaluating Public Programs with Close Substitutes: The Case of Head Start Patrick Kline and Christopher Walters Cite as: Patrick Kline and Christopher Walters. (2014). Evaluating Public Programs with Close Substitutes: The Case of Head Start. IRLE Working Paper No irle.berkeley.edu/workingpapers

11 where MT E h =E [Y i (h) Y i (c) U i (h, Z i, f) = U i (c), U i (c) > U i (n)] S c (f) + E [Y i (h) Y i (n) U i (h, Z i, f) = U i (n), U i (n) > U i (c)] (1 S c (f)), and S c (f) = P (U i (c) > U i (n) U i (h, Z i, f) = max {U i (c), U i (n)}) gives the share of children on the margin of participating in Head Start who prefer the competing program to preschool nonparticipation. Following the terminology in Heckman et al. (2008), the marginal treatment effect MT E h is the average effect of Head Start on test scores among households indifferent between Head Start and their next best alternative. This is essentially a marginal version of the result in (4), where integration is now over a set of children who may differ from current program participants in their mean impacts. The planner must balance the test score effects of improvements to the program feature against the costs. An (interior) optimum ensues when: g ( Ȳ ) MT E h = N h φ h (f) /N h (f) + φ h (f) φ c S c (f). (6) Condition (6) states that the dollar value of any gain in average test scores associated with improving the feature f is to be balanced against the direct provision cost per marginal enrollee N h φ h (f) /N h (f) of improving the program feature, plus the marginal increase in net social cost associated with expanding Head Start and contracting competing programs. 4 As in our analysis of optimal program scale, equation (6) shows that it is not necessary to separately identify the submtes that compose MT E h to determine the socially optimal value of f. It is sufficient to identify the average causal effect of Head Start for children on the margin of participation along with the average net social cost of an additional seat in this population. Identification using the HSIS We have shown that assessing optimal program scale requires knowledge of LAT E h and S c, while determining optimal program features requires knowledge of MT E h and S c. In the language of Heckman and Vytlacil (2001), LAT E h and MT E h are the policy-relevant treatment effects (PRTEs) associated with program expansion and structural reform respectively. Our empirical analysis uses the HSIS to identify these parameters. The HSIS data are a nationally-representative random sample of Head Start applicants, and HSIS offers are distributed randomly (US DHHS, 2010). As a result, compliance patterns and treatment effects in the HSIS should mimic patterns generated by random rationing of seats among program applicants. The HSIS is therefore ideal for estimating values of LAT E h and S c in the population of Head Start applicants. A complication arises if Head Start program expansion induces new households to apply to the 4 See Appendix B for derivations of equations (5) and (6). 10

20 estimates of LAT E h come from an instrumental variables regression of the form: Y i = Ḡ g=1 ( ) 1 {G i = g} βg 0 + βg h 1 {D i = h} + X iβ x + ɛ i, where G i is a categorical variable indexing groups defined by the intersection of whether the household s income is above the median, mother s education, age cohort, a composite measure of preschool quality, and a measure of transportation services at the Head Start site. X i is a vector of the same baseline covariates used in Table 5, included to increase precision. The interactions of group with Head Start attendance are instrumented by interactions of group and offer. Each parameter βg h captures the LAT E h for group g, which we term LAT E g h. Group-specific compliance share estimates Sc g come from corresponding IV regressions using 1 {D i = c} as the dependent variable. Groups with fewer than 100 observations are combined to form a single composite group. Figure 1 reveals that the relationship between LAT E g h and Sg c is sharply downward sloping. This indicates that the effects of Head Start participation are smaller among subpopulations that draw a large share of compliers from other preschools. Can this heterogeneity across groups be explained entirely by variation in the complier share S g c? To answer this question, we consider the null hypothesis that Head Start and competing preschools are equally effective and that the test score effects of Head Start relative to home care are constant across groups. We can write this hypothesis: LAT E g h = τ (1 Sg c ). (7) Note that this model implies that when Sc g equals one, the LAT E g h falls to zero. This restriction also implies that all of the variation across groups in LAT E g h is attributable to heterogeneity in the counterfactual enrollment choice of Head Start compliers. To test this hypothesis we conduct } optimal minimum distance estimation of (7) treating τ and the {S Ḡ compliance shares c 1, Sc 2,..., SḠc { } as unknown parameters and using the 2Ḡ estimates LAT E 1 h,..., LAT E Ḡ h, Ŝ1 c,..., ŜḠc as moments to match. 15 in Figure 1. Restricted estimates of the compliance shares and the best fitting line are portrayed The restrictions in model (7) are not rejected (p-value = 0.22), suggesting that a large portion of the variation across groups in LAT E g h is attributable to variation in the care environment from which compliers are drawn. The minimum distance estimate of τ is 0.315, which implies that if all Head Start compliers were drawn from home care the LAT E h would be roughly 30 percent greater than the first-year LAT E h estimate of This finding strongly suggests that the LAT E h can be altered by modifying substitution patterns in the market for preschool services. 15 This minimum distance approach bears a close connection to limited information maximum likelihood (LIML) estimation of a constant coefficient model where test scores are modeled as a function of a dummy for any preschool and the Ḡ group interactions with the offer dummy are used as excluded instruments (see Goldberger and Olkin, 1971). Our minimum distance estimator differs from LIML only because we use a cluster-robust covariance matrix to weight the reduced form moments. Note that by treating the compliance shares {S cg}ḡ g=1 as unknowns, we account for the effects of estimation error in those moments. An alternate approach would be to use the errors-in-variables procedure of Deaton (1985), which Devereux (2007) shows amounts to a jackknifed instrumental variables estimator. 19

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