by excluding 1 and 2 from the LASSO penalty. One could then try to estimate and do inference about 1 and 2 by applying ordinary least squares with as the outcome, and 1 and 2 and any selected ββ ββ control variables as regressors. The problem with this approach can be seen by noting that, in LASSO, ββ ββ YY DD DD any variable that is highly correlated to the 1 or 2 treatment variables is likely to be dropped since including such a variable does not add much predictive power for the outcome given that the DD DD treatment variables 1 and 2 are already in the model. As a consequence, the exclusion of a variable YY that is highly correlated to the treatment variables 1 and 2 can lead to substantial omitted-variables DD DD bias in the coefficients 1 and 2 . Similarly, if one applied a variable selection method to only equations DD DD (2a) and (2b) for predicting 1 and 2 , one could miss variables that have moderate-sized coefficients in ββ ββ predicting 1 and 2 , but large direct effects on . Such an omission may again lead to non-negligible DD DD omitted-variable bias in the coefficients 1 and 2 .<sup>9</sup> DD DD YY In more intuitive terms, the DML method consists of estimating the coefficients ββ ββ 1 and 2 from the regression coefficients of the “residualized” on the “residualized” 1 and 2 variables, where the ββ ββ residuals are constructed from the difference between the actual value of (and the actual values of YY DD DD 1 and 2 ) and the fitted value from a regression of (and 1 and 2 ) on .<sup>10</sup> As in the conventional YY (or ad-hoc) approach summarized earlier, the maintained (and untested) assumption for causal DD DD YY DD DD ZZ inferences based on the DML regression equations (2), (2a) and (2b) is that 1 2] = 0 , and for .<sup>11</sup> The key difference between the ad-hoc approach and DML that allows for EE[εε|ZZ, DD , DD causal inferences is the assumption that ii 1 or 2 are conditionally exogenous (or as good as randomly EE[vv |ZZ] = 0 ii= 1,2 assigned conditional on the covariates ) is more defensible. The vector of the confounding variables DD DD is of much higher dimension than the vector which is composed of the few confounding variables ZZ ZZ chosen in ad-hoc manner. Thus, in the ad-hoc approach the estimates of 1 and 2 may subject to φφ omitted variable bias thus preventing one from making causal inferences. ββ ββ In contrast to the ad-hoc approach, the vector in the DML approach, as applied in this study, consists of 375 possible control variables for the analysis using the MICS from the Republic of Congo and 109 ZZ

> 9 The DML approach has a more relaxed sparsity requirement than double selection (Belloni et al. 2014b), that adds robustness since the sample is split and coefficients are obtained from one sample and used in another independent sample (see the Appendix for more details).

> 10 The sequence of steps involved in applying the DML method is presented in detail in the Appendix.

> 11 An alternative approach is to employ instrumental variables for the 1 and 2 treatment variables and apply a variant of the DML approach explored here. The difficulty common to most applications of the instrumental variable approach is to identify variables that are excluded from equation (2) but correlated with the treatment variables DD DD 1 and 2 and uncorrelated with the error term . DD DD εε

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