This note revisits the role of time-invariant observed covariates in the Synthetic Control (SC) method. We first derive conditions under which the original result of Abadie et al (2010) regarding the bias of the SC estimator remains valid when we relax the assumption of a perfect match on observed covariates and assume only a perfect match on pre-treatment outcomes. We then show that, even when the conditions for the first result are valid, a perfect match on pre-treatment outcomes does not generally imply an approximate match for all covariates. This will only be true for those that are both relevant and whose effects (over time) are not collinear with the effects of other observed and unobserved covariates. Taken together, our results show that a perfect match on covariates should not be required for the SC method, as long as there is a perfect match on a long set of pre-treatment outcomes.

中文翻译：

---

协变量在综合控制方法中的作用

本说明重新审视了时不变协变量在合成控制（SC）方法中的作用。我们首先得出条件，当我们放宽对观察到的协变量的完全匹配假设并仅对治疗结果进行完全匹配时，Abadie等人（2010）关于SC估计量偏差的原始结果仍然有效。然后，我们表明，即使第一个结果的条件有效，治疗前结果的完美匹配通常也并不意味着所有协变量都近似匹配。这仅适用于既相关又影响（随时间变化）与其他观察到的和未观察到的协变量的影响共线的那些。综上所述，我们的结果表明，SC方法不需要协变量的完美匹配，