Standard errors when a regressor is randomly assigned
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回归变量随机分配时的标准误
- 研究问题
- 当关注的回归变量随机分配时，OLS 方差公式如何随误差依赖、处理效应异质性和组层面分配而变化？
- 方法与思路
- 作者分析随机回归变量与其他变量及误差项的独立关系，推导 OLS 的渐近方差，并扩展到组层面随机分配。
- 材料陈述的贡献
- 摘要称特定独立性条件下传统同方差公式仍有效；存在异质性时需调整，而跨单位处理效应独立可支持异方差稳健公式。
- 阅读关注点
- 应核对各方差简化所需的独立性条件，区分误差相关与处理效应相关，并检查个体及组层面随机化的适用范围。
- 阅读边界
- 所给为 arXiv 预印本，不能据此确认同行评审或期刊接收；摘要未给出完整渐近条件，不能推广为随机实验一律无需调整。
原文摘要
We examine asymptotic properties of the OLS estimator when the values of the regressor of interest are assigned randomly and independently of other regressors. We find that the OLS variance formula in this case is often simplified, sometimes substantially. In particular, when the regressor of interest is independent not only of other regressors but also of the error term, the textbook homoskedastic variance formula is valid even if the error term and auxiliary regressors exhibit a general dependence structure. In the context of randomized controlled trials, this conclusion holds in completely randomized experiments with constant treatment effects. When the error term is heteroscedastic with respect to the regressor of interest, the variance formula has to be adjusted not only for heteroscedasticity but also for correlation structure of the error term. However, even in the latter case, some simplifications are possible as only a part of the correlation structure of the error term should be taken into account. In the context of randomized control trials, this implies that the textbook homoscedastic variance formula is typically not valid if treatment effects are heterogenous but heteroscedasticity-robust variance formulas are valid if treatment effects are independent across units, even if the error term exhibits a general dependence structure. In addition, we extend the results to the case when the regressor of interest is assigned randomly at a group level, such as in randomized control trials with treatment assignment determined at a group (e.g., school/village) level.