In the context of treatment effect estimation, this paper proposes a new methodology to recover the counterfactual distribution when there is a single (or a few) treated unit and possibly a high-dimensional number of potential controls observed in a panel structure. The methodology accommodates, albeit does not require, the number of units to be larger than the number of time periods (high-dimensional setup). As opposed to model only the conditional mean, we propose to model the entire conditional quantile function (CQF) in the absence of intervention and estimate it using the pre-intervention period using a penalized regression. We derive non-asymptotic bounds for the estimated CQF valid uniformly over the quantiles, allowing the practitioner to re-construct the entire contractual distribution. Moreover, we bound the probability coverage of this estimated CQF which can be used to construct valid confidence intervals for the (possibly random) treatment effect for every post-intervention period or simultaneously. We also propose a new hypothesis test for the sharp null of no-effect based on the $\mathcal{L}^p$ norm of deviation of the estimated CQF to the population one. Interestingly, the null distribution is quasi-pivotal in the sense that it only depends on the estimated CQF, $\mathcal{L}^p$ norm, and the number of post-intervention periods, but not on the size of the post-intervention period. For that reason, critical values can then be easily simulated. We illustrate the methodology is by revisiting the empirical study in Acemoglu et al (2016).

翻译：在治疗效果估计方面,本文件建议采用新方法,在出现单一(或少数)处理过的单位时,恢复反事实分布,并有可能在小组结构中观察到高维数量的潜在控制。该方法虽然不要求,但考虑到单位数量大于时段数(高维设置)。与只模拟有条件平均值相比,我们提议在没有干预的情况下模拟整个有条件量化函数(CQF),并使用受罚回归法来估计整个有条件量化函数(CQF)。我们为CQF估算值的简单值统一适用于量数,我们得出了不易接受的CQF界限,使执业者能够重新构建整个合同分布。此外,我们将这一估计的CQF的概率范围限制在为每个干预期或同时设定有效的(可能随机的)治疗效果间隔。我们还提议以 $\mathcal{L ⁇ p$标准为基础,对“CQF”估计的精确值偏离值标准值标准值进行新的假设值测试。 估计CFreval=L值的精确度的数值在后值分析中只能根据CF估计值的数值进行。