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PSM-DID：双重差分倾向匹配得分---
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1 、简介
现代计量经济学和统计学的发展为我们的研究提供了可行的工具。倍差法来源于计量经济学的综列数据模型，是政策分析和工程评估中广为使用的一种计量经济方法。主要是应用于在混合截面数据集中，评价某一事件或政策的影响程度。该方法的基本思路是将调查样本分为两组，一组是政策或工程作用对象即“作用组”，一组是非政策或工程作用对象即“对照组”。根据作用组和对照组在政策或工程实施前后的相关信息，可以计算作用组在政策或工程实施前后某个指标（如收入）的变化量（收入增长量），同时计算对照组在政策或工程实施前后同一指标的变化量。然后计算上述两个变化量的差值（即所谓的“倍差值”）。这就是所谓的双重差分估计量（Difference in Differences，简记DD或DID），因为它是处理组差分与控制组差分之差。该法最早由Ashenfelter（1978）引入经济学，而国内最早的应用或为周黎安、陈烨（2005）。
常用的倍差法主要包括双重倍差法和三重倍差法。双重差分法（Difference-in-difference,DID）有几种其他的称谓：倍差法、差分再差分等。该方法的原理非常简单，它要求数据期至少有两期，所有的样本被分为两类：实验组和控制组，其中实验组在第一期是没有受到政策影响，此后政策开始实施，第二期就是政策实施后的结果，控制组由于一直没有受政策干预，因此其第一期和第二期都是没有政策干预的结果。双重差分方法的测算也非常简单，两次差分的效应就是政策效应。
双重差分法的假定，为了使用OLS一致地估计方程，需要作以下两个假定。
假定1：此模型设定正确。特别地，无论处理组还是控制组，其时间趋势项都是。此假定即“平行趋势假定”（parallel trend assumption）。DID最为重要和关键的前提条件：共同趋势（Common Trends）
双重差分法并不要求实验组和控制组是完全一致的，两组之间可以存在一定的差异，但是双重差分方法要求这种差异不随着时间产生变化，也就是说，处理组和对照组在政策实施之前必须具有相同的发展趋势。
假定2：暂时性冲击与政策虚拟变量不相关。这是保证双向固定效应为一致估计量（consist estimator）的重要条件。在此，可以允许个体固定效应与政策虚拟变量相关（可通过双重差分或组内变换消去，或通过LSDV法控制）。
DID允许根据个体特征进行选择，只要此特征不随时间而变；这是DID的最大优点，即可以部分地缓解因 “选择偏差”（selection bias）而导致的内生性（endogeneity）。
2、命令介绍
下载安装命令方法为：
ssc install diff, replace 下载安装方法（外部命令）
语法格式为：
diff outcome_var [if] [in] [weight] ,[ options]
模型必选项介绍：
其中“outcome_var”表示结果变量
“treat(varname) ”为必选项，用来指定处理变量
“period(varame)”用来指定实验期虚拟变量（1=实验期，0=非实验期）
可选项介绍：
cov(varlist)，协变量，加上kernel可以估计倾向得分
kernel， 执行双重差分倾向得分匹配
id(varname)，kernel选项要求使用
bw(#) ，核函数的带宽，默认是0.06
ktype(kernel)，核函数的类型
qdid(quantile)，执行分位数双重差分
pscore(varname) 提供倾向得分
logit，进行倾向得分计算，默认probit回归
ddd(varname)，三重差分
SE/Robust
cluster(varname) 计算聚类标准误。
robust 稳健标准误
3、最低工资法能否会降低对低技能工人的需求？
案例数据介绍:cardkrueger1994
背景介绍:在这种情况下,作者研究提高最低工资的影响在新泽西州——治疗组在快餐行业的就业水平。他们将接受治疗的这一组餐厅员工数量的变化与相邻州宾夕法尼亚州(对照组)的员工数量的变化进行了比较。他们在1992年2月收集了基线，并在11月收集了后续数据。
1992年4月，新泽西州通过最低工资法案，将最低工资从4.25美元提高到5.05美元，而相邻的宾夕法尼亚州的最低工资却保持不变。因此，Card and Kruger考虑了一个自然实验，即将新泽西州作为实验组，而宾州作为控制组，收集了两州不同快餐店在实施新法前后前后雇佣人数的数据，并采用双重差分法进行估计。
该数据集共包含522家快餐，并涉及两个时期（1992年2月和1992年11月，以t表示，分别赋值为0和1）。treated用以区分实验组和控制组，其中1表示新泽西，0表示宾州。因变量为fte（full time employment），用以刻画快餐店的雇佣人数。数据集还包括其余4个控制变量，均为快餐店的品牌，包括bk（Burger King），kfc（Kentuky Fried Chiken ），roys（Roy Rogers），wendys（Wendy's）。
首先我们先定义t和treated的交互项，并用进行双重差分估计：
use "http://fmwww.bc.edu/repec/bocode/c/CardKrueger1994.dta"
生成实验组和法案实施时期的交互项
gen gd=t*treated // (定义交叉项gd)
手工进行DID估计，并使用稳健标准误
reg fte gd treated t, r
结果为：
gen gd=t*treated
. reg fte gd treated t, r
Linear regression Number of obs = 801
F(3, 797) = 1.43
Prob > F = 0.2330
R-squared = 0.0080
Root MSE = 9.003
------------------------------------------------------------------------------
| Robust
fte | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
gd | 2.913982 1.736818 1.68 0.094 -.4952963 6.323261
treated | -2.883534 1.403338 -2.05 0.040 -5.638209 -.1288592
t | -2.40651 1.594091 -1.51 0.132 -5.535623 .7226031
_cons | 19.94872 1.317281 15.14 0.000 17.36297 22.53447
------------------------------------------------------------------------------
.
上述结果显示，政策效应（did）在10%的显著性水平上显著，且系数为正（2.914），表明最低工资法案政策实施后，快餐店的雇佣人数不会减少，反而会在一定程度上增多。不过，这个结论未考虑其他控制变量的影响。
接着我们引入快餐品牌的虚拟变量作为控制变量，再次回归
reg fte gd treated t bk kfc roys,r
Linear regression Number of obs = 801
F(6, 794) = 57.30
Prob > F = 0.0000
R-squared = 0.1878
Root MSE = 8.1617
------------------------------------------------------------------------------
| Robust
fte | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
gd | 2.93502 1.543422 1.90 0.058 -.0946504 5.96469
treated | -2.323906 1.253701 -1.85 0.064 -4.784867 .1370549
t | -2.402678 1.410265 -1.70 0.089 -5.170966 .3656108
bk | .9168795 .9382545 0.98 0.329 -.9248729 2.758632
kfc | -9.204856 .8991089 -10.24 0.000 -10.96977 -7.439945
roys | -.8970458 1.041071 -0.86 0.389 -2.940623 1.146532
_cons | 21.16069 1.307146 16.19 0.000 18.59482 23.72656
------------------------------------------------------------------------------
使用diff命令进行操作，结果为：
*-2、双重差分
diff fte, t(treated) p(t) robust
****结果为：
*-----------------------------------result.begin--------------------------------
diff fte, t(treated) p(t) robust
DIFFERENCE-IN-DIFFERENCES ESTIMATION RESULTS
Number of observations in the DIFF-IN-DIFF: 801
Before After
Control: 78 77 155
Treated: 326 320 646
404 397
--------------------------------------------------------
Outcome var. | fte | S. Err. | |t| | P>|t|
----------------+---------+---------+---------+---------
Before | | | |
Control | 19.949 | | |
Treated | 17.065 | | |
Diff (T-C) | -2.884 | 1.403 | -2.05 | 0.040**
After | | | |
Control | 17.542 | | |
Treated | 17.573 | | |
Diff (T-C) | 0.030 | 1.023 | 0.03 | 0.976
| | | |
Diff-in-Diff | 2.914 | 1.737 | 1.68 | 0.094*
--------------------------------------------------------
R-square: 0.01
* Means and Standard Errors are estimated by linear regression
**Robust Std. Errors
**Inference: *** p<0.01; ** p<0.05; * p<0.1
*-----------------------------------result.over--------------------------------
4.2、DID with covariates带协变量的估计
diff fte, t(treated) p(t) cov(bk kfc roys)
diff fte, t(treated) p(t) cov(bk kfc roys) report
diff fte, t(treated) p(t) cov(bk kfc roys) report bs
结果为：
. diff fte, t(treated) p(t) cov(bk kfc roys)
DIFFERENCE-IN-DIFFERENCES WITH COVARIATES
DIFFERENCE-IN-DIFFERENCES ESTIMATION RESULTS
Number of observations in the DIFF-IN-DIFF: 801
Before After
Control: 78 77 155
Treated: 326 320 646
404 397
--------------------------------------------------------
Outcome var. | fte | S. Err. | |t| | P>|t|
----------------+---------+---------+---------+---------
Before | | | |
Control | 21.161 | | |
Treated | 18.837 | | |
Diff (T-C) | -2.324 | 1.031 | -2.25 | 0.024**
After | | | |
Control | 18.758 | | |
Treated | 19.369 | | |
Diff (T-C) | 0.611 | 1.037 | 0.59 | 0.556
| | | |
Diff-in-Diff | 2.935 | 1.460 | 2.01 | 0.045**
--------------------------------------------------------
R-square: 0.19
* Means and Standard Errors are estimated by linear regression
**Inference: *** p<0.01; ** p<0.05; * p<0.1
.
4.3、双重差分倾向匹配得分Kernel Propensity Score Diff-in-Diff
diff fte, t(treated) p(t) cov(bk kfc roys) kernel rcs
diff fte, t(treated) p(t) cov(bk kfc roys) kernel rcs support
diff fte, t(treated) p(t) cov(bk kfc roys) kernel rcs support addcov(wendys)
diff fte, t(treated) p(t) kernel rcs ktype(gaussian) pscore(_ps)
diff fte, t(treated) p(t) cov(bk kfc roys) kernel rcs support addcov(wendys) bs reps(50)
结果为：
. diff fte, t(treated) p(t) cov(bk kfc roys) kernel rcs
KERNEL PROPENSITY SCORE MATCHING DIFFERENCE-IN-DIFFERENCES
Repeated Cross Section - rcs option
Matching iterations: control group at base line...
..............................................................................................
> ............................................................................................
> ............................................................................................
> ................................................
Matching iterations: control group at follow up...
..............................................................................................
> ............................................................................................
> ............................................................................................
> ..........................................
Matching iterations: treated group at baseline...
..............................................................................................
> ............................................................................................
> ............................................................................................
> ................................................
DIFFERENCE-IN-DIFFERENCES ESTIMATION RESULTS
Number of observations in the DIFF-IN-DIFF: 801
Before After
Control: 78 77 155
Treated: 326 320 646
404 397
--------------------------------------------------------
Outcome var. | fte | S. Err. | |t| | P>|t|
----------------+---------+---------+---------+---------
Before | | | |
Control | 20.040 | | |
Treated | 17.405 | | |
Diff (T-C) | -2.636 | 0.939 | -2.81 | 0.005***
After | | | |
Control | 17.341 | | |
Treated | 17.573 | | |
Diff (T-C) | 0.232 | 0.948 | 0.24 | 0.807
| | | |
Diff-in-Diff | 2.867 | 1.334 | 2.15 | 0.032**
--------------------------------------------------------
R-square: 0.01
* Means and Standard Errors are estimated by linear regression
**Inference: *** p<0.01; ** p<0.05; * p<0.1
.
4.4、 Quantile Diff-in-Diff 分位数双重差分法
diff fte, t(treated) p(t) qdid(0.25)
diff fte, t(treated) p(t) qdid(0.50)
diff fte, t(treated) p(t) qdid(0.75)
diff fte, t(treated) p(t) qdid(0.50) cov(bk kfc roys)
diff fte, t(treated) p(t) qdid(0.50) cov(bk kfc roys) kernel id(id) diff fte, t(treated) p(t) qdid(0.50) cov(bk kfc roys) kernel rcs
结果为
diff fte, t(treated) p(t) qdid(0.25)
DIFFERENCE-IN-DIFFERENCES ESTIMATION RESULTS
Number of observations in the DIFF-IN-DIFF: 801
Before After
Control: 78 77 155
Treated: 326 320 646
404 397
--------------------------------------------------------
Outcome var. | fte | S. Err. | |t| | P>|t|
----------------+---------+---------+---------+---------
Before | | | |
Control | 12.500 | | |
Treated | 11.000 | | |
Diff (T-C) | -1.500 | 1.584 | -0.95 | 0.344
After | | | |
Control | 11.500 | | |
Treated | 11.500 | | |
Diff (T-C) | -0.000 | 1.658 | 0.00 | 1.000
| | | |
Diff-in-Diff | 1.500 | 2.293 | 0.65 | 0.513
--------------------------------------------------------
R-square: 0.00
* Values are estimated at the .25 quantile
**Inference: *** p<0.01; ** p<0.05; * p<0.1
.
4.5、Balancing test of covariates.包含协变量的控制组与实验组之间差异检验
diff fte, t(treated) p(t) cov(bk kfc roys wendys) test
diff fte, t(treated) p(t) cov(bk kfc roys wendys) test id(id) kernel
diff fte, t(treated) p(t) cov(bk kfc roys wendys) test kernel rcs
diff fte, t(treated) p(t) cov(bk kfc roys wendys) test
TWO-SAMPLE T TEST
Number of observations (baseline): 404
Before After
Control: 78 - 78
Treated: 326 - 326
404 -
t-test at period = 0:
----------------------------------------------------------------------------------------------
Variable(s) | Mean Control | Mean Treated | Diff. | |t| | Pr(|T|>|t|)
---------------------+------------------+--------------+------------+---------+---------------
fte | 19.949 | 17.065 | -2.884 | 2.44 | 0.0150**
bk | 0.443 | 0.411 | -0.032 | 0.52 | 0.6035
kfc | 0.152 | 0.205 | 0.054 | 1.08 | 0.2818
roys | 0.215 | 0.248 | 0.033 | 0.61 | 0.5448
wendys | 0.190 | 0.136 | -0.054 | 1.22 | 0.2241
----------------------------------------------------------------------------------------------
*** p<0.01; ** p<0.05; * p<0.1
.
4.6. Triple differences (consider bk is a second treatment category).
三重差分法
diff fte, t(treated) p(t) ddd(bk)
diff fte, t(treated) p(t) ddd(bk)
TRIPLE DIFFERENCE (DDD) ESTIMATION RESULTS
Notation of DDD:
Control (A) treated = 0 and bk = 1
Control (B) treated = 0 and bk = 0
Treated (A) treated = 1 and bk = 1
Treated (B) treated = 1 and bk = 0
Number of observations in the DDD: 801
Before After
Control (A):34 35 69
Control (B):44 42 86
Treated (A):133 132 265
Treated (B):193 188 381
404 397
--------------------------------------------------------
Outcome var. | fte | S. Err. | |t| | P>|t|
----------------+---------+---------+---------+---------
Before | | | |
Control (A) | 25.654 | | |
Control (B) | 15.540 | | |
Treated (A) | 18.547 | | |
Treated (B) | 16.044 | | |
Diff (T-C) | -7.612 | 2.206 | 3.45 | 0.001***
After | | | |
Control (A) | 22.193 | | |
Control (B) | 13.667 | | |
Treated (A) | 19.913 | | |
Treated (B) | 15.930 | | |
Diff (T-C) | -4.543 | 2.214 | 2.05 | 0.040**
| | | |
DDD | 3.069 | 3.125 | 0.98 | 0.326
--------------------------------------------------------
R-square: 0.09
* Means and Standard Errors are estimated by linear regression
**Inference: *** p<0.01; ** p<0.05; * p<0.1
.