Doubly robust estimation of multivariate fractional outcome means with multivalued treatments

Abstract.

This article suggests a doubly robust method of estimating potential outcome means for multivariate fractional outcomes when the treatment of interest is unconfounded and can take more than two values. The method involves maximizing a propensity score weighted multinomial quasi-log-likelihood function with a multinomial logit conditional mean. We show that this estimator, which we call weighted multivariate fractional logit (wmflogit), consistently estimates the potential outcome means if either the propensity score model or the conditional mean model is misspecified. Our simulations demonstrate this double robustness property for the case of shares generated using a Dirichlet distribution. Finally, we advocate for the use of wmflogit by applying it to estimate time-use shares of women participating in the Mexican conditional cash transfer program, Progresa, using Stata’s fmlogit command developed by Buis.

Keywords:

• Double robustness
• fractional outcomes
• multinomial
• unconfoundedness

JEL codes::

• C21
• C31

Acknowledgments

We would like to thank Dalia Ghanem and Karen del Mar Ortiz Becerra for providing us the Progresa dataset.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. A version, known as AIPW, was proposed and studied in a string of papers by Robins et al. (1994), Robins et al. (2000), and Scharfstein et al. (1999). More recently, improved versions of the AIPW estimator have been proposed in both the missing data and causal inference literatures.

2. Including pre-treatment outcomes in the covariate vector makes unconfoundedness even more plausible as an identifying assumption. As in the case of Hirano and Imbens (2001), unconfoundedness is also easy to justify when we have a rich set of controls in the data that make it approximately true.

3. This QLL can be obtained by applying the general result on weighting given in Lemma 3.2 of Sloczyński and Wooldridge (2018) to the multinomial QLL function.

4. As mentioned in Imbens (2000), one may estimate the propensity scores using discrete response models if there is no natural ordering among the treatment levels or ordered response models if there is a natural ordering to the alternatives.

5. See section A for a derivation of the first-order conditions for the wmflogit estimator.

6. Similar equations for the ATE and the ATT can also be found in Hirano et al. (2003) for the binary case.

7. These sample sizes ensure that we have enough variation in the different share categories across the three treatment levels.

8. Two school related activities, namely, min_esc and min_tar are not used in construction of any of the three activities.

9. Progresa used marginalization index to geographically target highly marginalized localities in the seven states of Mexico. These localities were then randomized to either be in the intervention or control group.