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第57行： − 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，对于每个个体，在一个大的代表性样本中，对于每个个体，< 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 ATE。然而，我们不能同时观察每个个体的数学和数学，因为个体既不能治疗也不能不治疗。例如，在药物的例子中，我们只能观察到接受过药物的个体的 < math > y _ {1}(i) </math > 和未接受药物的 < math > y _ {0}(i) </math > 。这是科学家在评价治疗效果时面临的主要问题，并引发了大量的估计技术。
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In order to define formally the ATE, we define two potential outcomes : <math>y_{0}(i)</math> is the value of the outcome variable for individual <math>i</math> if they are not treated, <math>y_{1}(i)</math> is the value of the outcome variable for individual <math>i</math> if they are treated. For example, <math>y_{0}(i)</math> is the health status of the individual if they are not administered the drug under study and <math>y_{1}(i)</math> is the health status if they are administered the drug.
In order to define formally the ATE, we define two potential outcomes : <math>y_{0}(i)</math> is the value of the outcome variable for individual <math>i</math> if they are not treated, <math>y_{1}(i)</math> is the value of the outcome variable for individual <math>i</math> if they are treated. For example, <math>y_{0}(i)</math> is the health status of the individual if they are not administered the drug under study and <math>y_{1}(i)</math> is the health status if they are administered the drug.
为了正式定义 ATE，我们定义了两个潜在的结果: <math>y_{0}(i)</math > 是个体 <math> i </math> 没有被处理的结果变量的取值，<math> y _ {1}(i) </math> 是个体 <math> i </math> 被处理的结果变量的取值。例如，<math>y_{0}(i)</math > 是个体 <math> i </math> 没有注射研究药物的个体健康状态，<math>y_{1}(i)</math > 是个体 <math> i </math> 被注射药物的健康状态。
为了正式定义 ATE，我们定义了两个潜在的结果: <math>y_{0}(i)</math > 是个体 <math> i </math> 没有被处理的结果变量的取值，<math> y _ {1}(i) </math> 是个体 <math> i </math> 被处理的结果变量的取值。例如，<math>y_{0}(i)</math > 是个体 <math> i </math> 没有被注射研究药物的个体健康状态，<math>y_{1}(i)</math > 是个体 <math> i </math> 被注射药物的健康状态。
The treatment effect for individual <math>i</math> is given by <math>y_{1}(i)-y_{0}(i)=\beta(i)</math>. In the general case, there is no reason to expect this effect to be constant across individuals. The average treatment effect is given by
The treatment effect for individual <math>i</math> is given by <math>y_{1}(i)-y_{0}(i)=\beta(i)</math>. In the general case, there is no reason to expect this effect to be constant across individuals. The average treatment effect is given by
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。
== Estimation ==
== Estimation ==