# _Estimating the Average Treatment Effect on the Treated_

We exploit the above CIA in several ways to estimate the effect of being in the sprayed region during years in which Colombia agreed not to spray the exclusion area. First, we start by running the regression: Here,<sup></sup> _t_<sup>identifies the effect of being in the sprayed region (relative to the exclusion region)</sup> during year _t_ as long as the conditional expectation of the outcome is linear in the covariates. We compute standard errors clustering at the cell level and robust to heteroskedasticity.

Column 1 in table 3 presents the regression estimates on cultivation separately by year and also pools together the years 2006 to 2010 and 2008 to 2010, when Colombia effectively stopped spraying the exclusion area. Our estimates show that cultivation fell in the sprayed area relative to the exclusion region in all years since 2006, and the effects are all significant at traditional levels. Pooling the years 2006 to 2010 together (or 2008 to 2010), we find that cells in the sprayed area had 0.24 less hectares of coca per square km (standard error = 0.015) as a consequence of the spraying. As before, we also present standard errors robust against spatial correlation among cells within 5 km of each other. These are presented in square brackets only below the main estimates in table 3. Though, in this case, they are considerably larger than the traditional ones, they do not change any of our conclusions.

Consistency of the previous estimates requires the conditional expectation of cultivation and aerial spraying to be linear in the covariates. To relax this assumption, we follow several strategies in which we control nonparametrically for the propensity score,  _i_  _P T_ [ _i_  1| _Zi_ ,{ _Yit_  , _Sit_  } _t_   2005 . We estimate the propensity score,<sup>�</sup> _i_<sup>, using a probit model (not</sup> reported to save space), making this approach semi-parametric. In column 2 we reweight the regression in equation 5 by the propensity score (see Hirano, Imbens, and Ridder 2003). In particular, we weight observations in the sprayed area by  _p_ / (1 _p_ ) , where _p_ is the fraction of cells in this area, and observations in the exclusion area by <sup>�</sup> _i_ / (1  � _i_ ) , with<sup>�</sup> _i_<sup>being the estimated propensity score of the grid. This method ensures</sup> that all covariates are balanced and set to the distribution of the sprayed region. Once reweighted, the regression estimate equals the average treatment effect of being in an area currently sprayed.<sup>10</sup>

Besides reweighting by the propensity score, we also control linearly for all covariates in the regression. This is known as a double-robust regression, as it provides consistent estimates as long as the propensity score is correctly specified or the linear covariates are an accurate model for the conditional expectation of spraying and cultivation (see Imbens and Wooldridge 2009). As can be seen from the results in column 2, the results change little relative to column 1, suggesting that the linear controls were already capturing most of the relevant heterogeneity in cultivation and spraying dynamics before the intervention.<sup>11</sup>

> 10 We can also estimate the average treatment effect, but this requires weighting by 1/<sup></sup> � _i_ the observations in the sprayed area. However, there are values with very low estimated propensity scores that make this exercise imprecise. In any case, our results are similar. Moreover, the average treatment effect on the treated seems like the relevant object to evaluate the actual effect of spraying on cultivation.

11 We report the usual regression standard errors clustering at the grid level. These errors ignore the fact that the propensity score is estimated in a previous stage. However, as suggested