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If we could observe, for each individual, <math>y_{1}(i)</math> and <math>y_{0}(i)</math> among a large representative sample of the population, we could estimate the ATE simply by taking the average value of <math>y_{1}(i)-y_{0}(i)</math> across the sample. However, we can not observe both <math>y_{1}(i)</math> and <math>y_{0}(i)</math> for each individual since an individual cannot be both treated and not treated. For example, in the drug example, we can only observe <math>y_{1}(i)</math> for individuals who have received the drug and <math>y_{0}(i)</math> for those who did not receive it. This is the main problem faced by scientists in the evaluation of treatment effects and has triggered a large body of estimation techniques.
If we could observe, for each individual, <math>y_{1}(i)</math> and <math>y_{0}(i)</math> among a large representative sample of the population, we could estimate the ATE simply by taking the average value of <math>y_{1}(i)-y_{0}(i)</math> across the sample. However, we can not observe both <math>y_{1}(i)</math> and <math>y_{0}(i)</math> for each individual since an individual cannot be both treated and not treated. For example, in the drug example, we can only observe <math>y_{1}(i)</math> for individuals who have received the drug and <math>y_{0}(i)</math> for those who did not receive it. This is the main problem faced by scientists in the evaluation of treatment effects and has triggered a large body of estimation techniques.
如果我们能观察到一个大型代表性样本中每个个体的< math> y _ {1}(i) </math> 和 < math > y _ {0}(i) </math> ，我们可以简单地通过取样本中 <math> y _ {1}(i)-y _ {0}(i) </math> 的平均值来估计平均治疗效果。然而，我们不能同时观察每个个体的<math> y _ {1}(i)-y _ {0}(i) </math>，因为每个个体不能同时被处理和不被处理。例如，在药物例子中，我们只能观察到个体接受过药物治疗的< math> y _ {1}(i) </math> 和个体未接受药物的 < math> y _ {0}(i) </math> 。这是研究学者在评估治疗效果时面临的主要问题，并因此引发了大量估计技术的研究。
如果我们能观察到一个大型代表性样本中每个个体的<math> y _ {1}(i) </math> 和 <math> y _ {0}(i) </math> ，我们可以简单地通过取样本中 <math> y _ {1}(i)-y _ {0}(i) </math> 的平均值来估计平均治疗效果。然而，我们不能同时观察每个个体的<math> y _ {1}(i)-y _ {0}(i) </math>，因为每个个体不能同时被处理和不被处理。例如，在药物例子中，我们只能观察到个体接受过药物治疗的<math> y _ {1}(i) </math> 和个体未接受药物的 <math> y _ {0}(i) </math> 。这是研究学者在评估治疗效果时面临的主要问题，并因此引发了大量估计技术的研究。
== Estimation ==
== Estimation ==