坛友们好，最近阅读了一篇Journal of Urban economics杂志上分析我国2011年限购政策效应的DID文章（Tsur Somerville et al., 2020），对文内的模型设置有诸多不解之初，还望赐教。
1. 基准模型设置中是否忽视了多期因素？
文中选取四个样本城市，各城市出台限购政策的时间不一（2010年4月、2011年2月、2011年1月、2011年3月），本文的数据维度是月份。这里的After没有标明下标i是否可取？对估计结果会产生什么影响？
\[ y_{i,t}=\alpha +\beta _{1}\ast Treat_{i}\ast After_{t}+ \beta _{2}\ast After_{t}+\beta _{3}\ast X_{i,t}+\mu_{i}+\delta_{t}+\epsilon_{i,t}\]
2. 模型变式1中，作者通过加入Before放宽了平行趋势假定吗？
在文中引入Before的含义是什么，如何使得政策干预前处理组和对照组的平均效应有所差异？这不是动摇了DID的基本假设吗？原文表述为：
allows for non-parallel trends in the data in the period prior to the treatment.
We do this by allowing the pre-treatment mean effect for the projects in districts where there will be restrictions to vary from that of non-treatment districts. Formally，加入Teat_i和Before_{t,'}的交乘项，Before_{t,'} has the value of one for the second three months of the six month pre-restriction period annd zero otherwise.
\[y_{i,t}=\alpha +\beta _{1}\ast Treat_{i}\ast Before_{t}^{’}+\beta _{2}\ast Treat_{i}\ast After_{t}+\beta _{3}\ast After_{t}+\beta _{4}\ast X_{i,t}+\mu_{i}+\delta_{t}+\epsilon_{i,t}\]
3. 模型变式2中，加入Trend的含义是什么？
原文表述为：
imposes a functional form to address trend differences in the dependent variable before and after the introduction in restrictions across the two groups of districts.
We allow for different trends between the restricted and non-restricted areas, through Trend and Treat*Trend...
\[y_{i,t}=\alpha +\beta _{1}\ast Trend + \beta _{2}\ast Treat_{i}\ast Trend +\beta _{3}\ast Treat_{i}\ast Trend\ast After_{t}+ \beta _{4}\ast After_{t}+\beta _{5}\ast X_{i,t}+\mu _{i}+\lambda _{month}+\epsilon _{i,t}\]