Abstract
A new semiparametric estimator for estimating conditional expectation functions from incomplete data is proposed, which integrates parametric regression with nonparametric matching estimators. Besides its applicability to missing data situations due to non-response or attrition, the estimator can also be used for analyzing treatment effect heterogeneity and statistical treatment rules, where data on potential outcomes is missing by definition. By combining moments from a parametric specification with nonparametric estimates of mean outcomes in the non-responding population within a GMM framework, the estimator seeks to balance a good fit in the responding population with low bias in the non-responding population. The estimator is applied to analyzing treatment effect heterogeneity among Swedish rehabilitation programmes.
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Notes
The estimation of average treatment effects has been intensively analyzed, in particular for active labour market programmes and rehabilitation programmes. See for example, Aakvik (2003), Abbring and van den Berg (2004) and the Special Issue on ‘Long term unemployment and social assistance’, Empirical Economics (1/2), 1998). The focus of this paper is on the heterogeneity in treatment effects, which could be exploited to improve the average effectiveness of policies through a better participant allocation.
E.g. in the case of panel attrition, X may refer to information collected in the baseline period.
The support restriction is incorporated by considering only observations with \(\widehat{p}_{i} > 0\), because S x ={x:f X∣D=1(x)>0}={x:p(x)>0}
More precisely, let \(\widehat{m}_{{vl}} {\left( \rho \right)}\) for ρ>0 be an estimator of the expectation E[Y v ∣p(X)=ρ, Λ l (X)=1], i.e. the expectation of the v-th variable of the outcome vector Y conditional on the propensity score in the l-th subpopulation. Let \(\widehat{m}_{l} {\left( \cdot \right)} = {\left( {\widehat{m}_{{1l}} {\left( \cdot \right)}, \ldots ,\widehat{m}_{{vl}} {\left( \cdot \right)}, \ldots \widehat{m}_{{Vl}} {\left( \cdot \right)}} \right)}^{\prime } \) be the element-wise-defined estimator of the outcome vector Y in the population l, i.e. of E[Y∣p(X)=ρ, Λ l (X)=1]. Stacking these estimators for the L subpopulations and multiplying element-wise with the population indicator function gives \( \widehat{m}_{{VL}} {\left( {\widehat{p}{\left( {X_{i} } \right)}} \right)} = {\left( {\widehat{m}^{\prime }_{1} {\left( {p{\left( {X_{i} } \right)}} \right)} \cdot \Lambda _{1} {\left( {X_{i} } \right)}, \ldots \widehat{m}^{\prime }_{l} {\left( {\widehat{p}{\left( {X_{i} } \right)}} \right)} \cdot \Lambda _{l} {\left( {X_{i} } \right)}, \ldots ,\widehat{m}^{\prime }_{L} {\left( {\widehat{p}{\left( {X_{i} } \right)}} \right)} \cdot \Lambda _{L} {\left( {X_{i} } \right)}} \right)}^{\prime } \).
When a standard propensity score matching routine is used, care should be exercised to ensure that the lower VL moments in (10) are summed over the same observations as in \(\widehat{\mu }\) and are scaled in the same way. For example, if the propensity score matching routine estimates the mean counterfactual outcome \({{\sum {\widehat{m}_{VL} \left( {\widehat p_i } \right)\left( {1 - D_i } \right)1\left( {\widehat p_i > 0} \right)} } \over {\sum {\left( {1 - D_i } \right)1} \left( {\widehat p_i > 0} \right)}}\) instead of \({{\sum {\widehat m_{VL} \left( {\widehat p_i } \right)\left( {1 - D_i } \right)1\left( {\widehat p_i > 0} \right)} } \over n}\), then also the VL must be scaled accordingly.
This includes one-to-one or pair matching.
Using Epanechnikov instead of Gaussian kernel, and vice versa, led to largely similar results.
The X data are scaled in the estimator to mean zero and variance one.
The expected outcomes vary considerably among these subpopulations. Whereas with DGP 1, the expected outcome is 13.1 for the respondents and 5.3 for the non-respondents, the outcome difference between respondents and non-respondents can be as large as 8.2 (for subpopulations ten and eleven) and as small as 0.8 (for subpopulation fourteen). Similar heterogeneity occurs for DGP 2 and 3. For instance, in DGP 2 the expected outcome for the respondents is usually larger than for the non-respondents, but this relationship is reversed in subpopulation five. In DGP 2, the expected outcomes for respondents and non-respondents are 2.2 and 1.5, respectively, and in DGP 3 these figures are 9.6 and 4.3.
Unless the past participants have been assigned randomly to the programmes.
Regularly employed individuals receive for the first two weeks sickness benefits from the employer and afterwards from the insurance office. Unemployed and self-employed individuals receive benefits directly from the insurance office. Sickness benefits amount to 80% of previous earnings, adjusted for the degree of lost working capacity and cut at an upper ceiling, and can be received for an unlimited period.
Medical and social rehabilitation are not coordinated by the insurance office.
The insurance offices themselves do not conduct rehabilitative activities.
A number of cases received more than one type of rehabilitation. Since neither it is known whether these measures where given in parallel or sequentially, nor the time sequence of these measures, these cases were assigned to the supposedly first or principal of the rehabilitative measures received. In most cases this has been medical rehabilitation, which is likely to be the first measure. The second priority is given to workplace rehabilitation, since workplace rehabilitation is usually full-time while educational training may operate alongside. For further details on the data see Frölich et al. (2004).
The reason for the latter is that the assessment refers to vocational rehabilitation.
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Acknowledgment
The author is also affiliated with the Institute for the Study of Labor (IZA), Bonn. I am grateful for discussions and comments to Bo Honoré, Francois Laisney, Michael Lechner, Ruth Miquel, Oivind Nilsen, Jeff Smith, the editor and three anonymous referees. This research was supported by the Swiss National Science Foundation (project NSF 4043-058311) and the Grundlagenforschungsfonds HSG (project G02110112).
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Appendices
Appendix A: Monte Carlo results
Appendix B: Swedish rehabilitation programmes
This Appendix contains additional tables on the estimation of optimal programme choices. Further results on alternative specifications are available in the supplementary Appendix.
Table B.1 gives the observed treatment outcomes and the nonparametrically estimated counterfactual outcomes for the 11 populations used in the GMM estimator in Section 4. The entry 48.3 in the top left, for example, indicates that among all the participants in No rehabilitation, a employment rate of 48.3% was observed. The potential No rehabilitation outcome for those who did not participate in No rehabilitation is estimated to be 43.6%. The respective figures for the 46–55 years old are 49.8 and 41.5%. The mean counterfactual outcomes are estimated separately for each population by ridge matching with the bandwidth value chosen by least-squares cross-validation from the grid {0.02,0.04,..,1}.
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Frölich, M. Semiparametric estimation of conditional mean functions with missing data. Empirical Economics 31, 333–367 (2006). https://doi.org/10.1007/s00181-005-0019-4
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DOI: https://doi.org/10.1007/s00181-005-0019-4