双重差分

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Difference in differences (DID or DD) is a statistical technique used in econometrics and quantitative research in the social sciences that attempts to mimic an experimental research design using observational study data, by studying the differential effect of a treatment on a 'treatment group' versus a 'control group' in a natural experiment. It calculates the effect of a treatment (i.e., an explanatory variable or an independent variable) on an outcome (i.e., a response variable or dependent variable) by comparing the average change over time in the outcome variable for the treatment group to the average change over time for the control group. Although it is intended to mitigate the effects of extraneous factors and selection bias, depending on how the treatment group is chosen, this method may still be subject to certain biases (e.g., mean regression, reverse causality and omitted variable bias).

双重差分法（DID 或 DD）是一种用于计量经济学和社会科学定量研究的统计技术，它试图利用观察性研究数据来模拟实验研究设计，通过研究自然实验中的“治疗组”和“对照组”之间的差异性效果。它通过比较治疗组和对照组的结果变量在一段时间的平均变化，计算出治疗（即解释变量或自变量）对结果（即反应变量或因变量）的影响。虽然该方法旨在减轻外部因素和选择偏差的影响，但取决于治疗组的选择方式，该方法仍可能受到某些偏差的影响（例如，均值回归、反向因果关系和遗漏变量偏差）。

In contrast to a time-series estimate of the treatment effect on subjects (which analyzes differences over time) or a cross-section estimate of the treatment effect (which measures the difference between treatment and control groups), difference in differences uses panel data to measure the differences, between the treatment and control group, of the changes in the outcome variable that occur over time.

与受试者治疗效果的时间序列估计（分析随时间变化的差异）或治疗效果的横截面估计（衡量治疗组和对照组之间的差异）不同，双重差分法使用面板数据来衡量治疗组和对照组的结果变量随时间变化的差异。

General definition

【图1：Illustration of Difference in Differences＋双重差分法的说明】

Difference in differences requires data measured from a treatment group and a control group at two or more different time periods, specifically at least one time period before "treatment" and at least one time period after "treatment." In the example pictured, the outcome in the treatment group is represented by the line P and the outcome in the control group is represented by the line S. The outcome (dependent) variable in both groups is measured at time 1, before either group has received the treatment (i.e., the independent or explanatory variable), represented by the points P₁ and S₁. The treatment group then receives or experiences the treatment and both groups are again measured at time 2. Not all of the difference between the treatment and control groups at time 2 (that is, the difference between P₂ and S₂) can be explained as being an effect of the treatment, because the treatment group and control group did not start out at the same point at time 1. DID therefore calculates the "normal" difference in the outcome variable between the two groups (the difference that would still exist if neither group experienced the treatment), represented by the dotted line Q. (Notice that the slope from P₁ to Q is the same as the slope from S₁ to S₂.) The treatment effect is the difference between the observed outcome (P₂) and the "normal" outcome (the difference between P₂ and Q).

双重差分法要求从治疗组和对照组在两个或两个以上不同时间段测量数据，特别是“治疗”前以及“治疗”后的至少一个时间段。在图中的示例中，治疗组的结果用线P表示，对照组的结果用线S表示。两组的结果（因）变量都是在时间1，即任何一组接受治疗（即自变量或解释变量）前测量的，分别由点P1和S1表示。治疗组之后接受或经历治疗，并在时间2再次测量两组。并非所有治疗组和对照组在时间2的差异（即P2和S2的差异）都可以解释为是治疗的效果，因为治疗组和对照组在时间1的开始时间不同。因此，DID计算出两组的结果变量之间的“正常”差异（如果两组均未接受治疗，差异仍然存在），由虚线Q表示（注意：P1到Q的斜率与S1到S2的斜率相同）。治疗效果是观察结果（P2）和“正常”结果（P2和Q之间的差异）之间的差异。

= 正式定义 =

Consider the model

考虑以下模型

[math]\displaystyle{ y_{it} ~=~ \gamma_{s(i)} + \lambda_t + \delta I(\dots) + \varepsilon_{it} }[/math]

y_{it} ~=~ \gamma_{s(i)} + \lambda_t + \delta I(\dots) + \varepsilon_{it}

where [math]\displaystyle{ y_{it} }[/math] is the dependent variable for individual [math]\displaystyle{ i }[/math] and time [math]\displaystyle{ t }[/math], [math]\displaystyle{ s(i) }[/math] is the group to which [math]\displaystyle{ i }[/math] belongs (i.e. the treatment or the control group), and [math]\displaystyle{ I(\dots) }[/math] is short-hand for the dummy variable equal to 1 when the event described in [math]\displaystyle{ (\dots) }[/math] is true, and 0 otherwise. In the plot of time versus [math]\displaystyle{ Y }[/math] by group, [math]\displaystyle{ \gamma_s }[/math] is the vertical intercept for the graph for [math]\displaystyle{ s }[/math], and [math]\displaystyle{ \lambda_t }[/math] is the time trend shared by both groups according to the parallel trend assumption (see Assumptions below). [math]\displaystyle{ \delta }[/math] is the treatment effect, and [math]\displaystyle{ \varepsilon_{it} }[/math] is the residual term.

其中， y _ { it }是个体 i 和时间 t的因变量，s (i)是i所属的组(即治疗组或对照组) 。 I(\dots) 则是哑变量的简称，当 (\dots) 中所描述的事件为真时等于1，否则等于0。在时间与 Y 的分组图中，gamma _ s 是 s 组的图形的垂直截距，而 lambda _ t 是根据平行趋势假设，两组共享的时间趋势(见下文假设)。Δ 是治疗效果，varepsilon _ { it }是残差项。

Consider the average of the dependent variable and dummy indicators by group and time:

考虑按组和时间划分的因变量和虚拟指标的平均值:

[math]\displaystyle{ \begin{align} n_s & = \text{ number of individuals in group } s \\ \overline{y}_{st} & = \frac{1}{n_s} \sum_{i=1}^n y_{it} \ I(s(i) ~=~ s), \\ \overline{\gamma}_s & = \frac{1}{n_s} \sum_{i=1}^n \gamma_{s(i)} \ I(s(i) ~=~ s) ~=~ \gamma_s, \\ \overline{\lambda}_{st} & = \frac{1}{n_s} \sum_{i=1}^n \lambda_t \ I(s(i) ~=~ s) ~=~ \lambda_t, \\ D_{st} & = \frac{1}{n_s} \sum_{i=1}^n I(s(i) ~=~\text{ treatment, } t \text{ in after period}) \ I(s(i) ~=~ s) ~=~ I(s ~=~\text{ treatment, } t \text{ in after period}) , \\ \overline{\varepsilon}_{st} & = \frac{1}{n_s} \sum_{i=1}^n \varepsilon_{it} \ I(s(i) ~=~ s), \end{align} }[/math]

\begin{align} n_s & = \text{ number of individuals in group } s \\ \overline{y}_{st} & = \frac{1}{n_s} \sum_{i=1}^n y_{it} \ I(s(i) ~=~ s), \\ \overline{\gamma}_s & = \frac{1}{n_s} \sum_{i=1}^n \gamma_{s(i)} \ I(s(i) ~=~ s) ~=~ \gamma_s, \\ \overline{\lambda}_{st} & = \frac{1}{n_s} \sum_{i=1}^n \lambda_t \ I(s(i) ~=~ s) ~=~ \lambda_t, \\ D_{st} & = \frac{1}{n_s} \sum_{i=1}^n I(s(i) ~=~\text{ treatment, } t \text{ in after period}) \ I(s(i) ~=~ s) ~=~ I(s ~=~\text{ treatment, } t \text{ in after period}) , \\ \overline{\varepsilon}_{st} & = \frac{1}{n_s} \sum_{i=1}^n \varepsilon_{it} \ I(s(i) ~=~ s), \end{align}

开始{ align } n _ s & = text { number of individuals in group } s overline { y }{ st } & = frac {1}{ n _ s } sum { i = 1} ^ n y _ it } i (s (i) ~ = ~ s) ，overline { gamma _ s & = frac {1}{ n _ s } sum { i = 1 ^ n _ γ { s (i)} i (s (i) ~ = ~ s) = ~ gamma _ s,1} ^ n lambda _ t i (s (i) ~ = ~ s) ~ = ~ lambda _ t,1} ^ n i (s (i) ~ = ~ text { in after period }) i (s (i) ~ = ~ i (s ~ = ~ text { treatment，}) ,在这里输入译文在这里输入译文在这里输入译文在这里输入译文在这里输入译文

and suppose for simplicity that [math]\displaystyle{ s=1,2 }[/math] and [math]\displaystyle{ t=1,2 }[/math]. Note that [math]\displaystyle{ D_{st} }[/math] is not random; it just encodes how the groups and the periods are labeled. Then

为简单起见，假设 s = 1,2， t = 1,2。请注意， d _ { st }不是随机的，它只是编码了组和时期的标记方式。那么

[math]\displaystyle{ \begin{align} & (\overline{y}_{11} - \overline{y}_{12}) - (\overline{y}_{21} - \overline{y}_{22}) \\[6pt] = {} & \big[ (\gamma_1 + \lambda_1 + \delta D_{11} + \overline{\varepsilon}_{11}) - (\gamma_1 + \lambda_2 + \delta D_{12} + \overline{\varepsilon}_{12}) \big] \\ & \qquad {} - \big[ (\gamma_2 + \lambda_1 + \delta D_{21} + \overline{\varepsilon}_{21}) - (\gamma_2 + \lambda_2 + \delta D_{22} + \overline{\varepsilon}_{22}) \big] \\[6pt] = {} & \delta (D_{11} - D_{12}) + \delta(D_{22} - D_{21}) + \overline{\varepsilon}_{11} - \overline{\varepsilon}_{12} + \overline{\varepsilon}_{22} - \overline{\varepsilon}_{21}. \end{align} }[/math]

\begin{align} & (\overline{y}_{11} - \overline{y}_{12}) - (\overline{y}_{21} - \overline{y}_{22}) \\[6pt] = {} & \big[ (\gamma_1 + \lambda_1 + \delta D_{11} + \overline{\varepsilon}_{11}) - (\gamma_1 + \lambda_2 + \delta D_{12} + \overline{\varepsilon}_{12}) \big] \\ & \qquad {} - \big[ (\gamma_2 + \lambda_1 + \delta D_{21} + \overline{\varepsilon}_{21}) - (\gamma_2 + \lambda_2 + \delta D_{22} + \overline{\varepsilon}_{22}) \big] \\[6pt] = {} & \delta (D_{11} - D_{12}) + \delta(D_{22} - D_{21}) + \overline{\varepsilon}_{11} - \overline{\varepsilon}_{12} + \overline{\varepsilon}_{22} - \overline{\varepsilon}_{21}. \end{align}

开始{ align } & (overline { y } _ {11}-overline { y } _ {12})-(overline { y } _ {21}-overline { y } _ {22})[6 pt ] = {} & big[(gamma _ 1 + lambda _ 1 + delta d _ {11} + overline { varepsilon } _ {11})-(gamma _ 1 + lambda _ 2 + delta d _ {12} + overline { varepsilon } _ {12}) big ]大[(gamma _ 2 + lambda _ 1 + delta d _ {21} + overline { varepsilon }{21})-(gamma _ 2 + lambda _ 2 + delta d _ {22} + overline { varepsilon }{22})[6 pt ] = {} & delta (d _ {11}-d _ {12}) + delta(d _ {22}-d _ {21})+ overline { varepsilon } _ {11}-overline { varepsilon } _{12} + overline { varepsilon } _ {22}-overline { varepsilon } _ {21}.结束{ align }

The strict exogeneity assumption then implies that

严格的'''<font color="#ff8000"> 外生性假设Strict exogeneity assumption</font>'''则意味着

[math]\displaystyle{ \operatorname{E} \left [ (\overline{y}_{11} - \overline{y}_{12}) - (\overline{y}_{21} - \overline{y}_{22}) \right ] ~=~ \delta (D_{11} - D_{12}) + \delta(D_{22} - D_{21}). }[/math]

\operatorname{E} \left [ (\overline{y}_{11} - \overline{y}_{12}) - (\overline{y}_{21} - \overline{y}_{22}) \right ] ~=~ \delta (D_{11} - D_{12}) + \delta(D_{22} - D_{21}).

操作数名{ e }左[(上行{ y } _ {11}-上行{ y } _ {12})-(上行{ y } _ {21}-上行{ y } _ {22})右] ~ = ~ delta (d _ {11}-d _ {12}) + delta (d _ {22}-d _ {21})。

Without loss of generality, assume that [math]\displaystyle{ s = 2 }[/math] is the treatment group, and [math]\displaystyle{ t = 2 }[/math] is the after period, then [math]\displaystyle{ D_{22}=1 }[/math] and [math]\displaystyle{ D_{11}=D_{12}=D_{21}=0 }[/math], giving the DID estimator

Without loss of generality, assume that s = 2 is the treatment group, and t = 2 is the after period, then D_{22}=1 and D_{11}=D_{12}=D_{21}=0, giving the DID estimator

在不失一般性的前提下，假设 s = 2是治疗组，t = 2是后一期，则 d {22} = 1，d {11} = d {12} = d {21} = 0，得到 DID 估计量

[math]\displaystyle{ \hat{\delta} ~=~ (\overline{y}_{11} - \overline{y}_{12}) - (\overline{y}_{21} - \overline{y}_{22}), }[/math]

\hat{\delta} ~=~ (\overline{y}_{11} - \overline{y}_{12}) - (\overline{y}_{21} - \overline{y}_{22}),

hat { delta } ~ = ~ (overline { y } _ {11}-overline { y } _ {12})-(overline { y } _ {21}-overline { y } _ {22}) ,

which can be interpreted as the treatment effect of the treatment indicated by [math]\displaystyle{ D_{st} }[/math]. Below it is shown how this estimator can be read as a coefficient in an ordinary least squares regression. The model described in this section is over-parametrized; to remedy that, one of the coefficients for the dummy variables can be set to 0, for example, we may set [math]\displaystyle{ \gamma_1 = 0 }[/math].

这可以解释为 D _ { st }所示的治疗效果。下面展示了如何将这个估计值解读为普通最小二乘回归中的系数。本节描述的模型是过度参数化的; 为了弥补这一点，可以将哑变量的一个系数设置为0，例如，我们可以设置 gamma _ 1 = 0。

= 假设 =

Illustration of the parallel trend assumption

right|thumb|320px| Illustration of the parallel trend assumption

【图2：Illustration of the parallel trend assumption＋平行趋势假设的说明】

All the assumptions of the OLS model apply equally to DID. In addition, DID requires a parallel trend assumption. The parallel trend assumption says that [math]\displaystyle{ \lambda_2 - \lambda_1 }[/math] are the same in both [math]\displaystyle{ s=1 }[/math] and [math]\displaystyle{ s=2 }[/math]. Given that the formal definition above accurately represents reality, this assumption automatically holds. However, a model with [math]\displaystyle{ \lambda_{st} ~:~ \lambda_{22} - \lambda_{21} \neq \lambda_{12} - \lambda_{11} }[/math] may well be more realistic. In order to increase the likelihood of the parallel trend assumption holding, a difference-in-difference approach is often combined with matching. This involves 'Matching' known 'treatment' units with simulated counterfactual 'control' units: characteristically equivalent units which did not receive treatment. By defining the Outcome Variable as a temporal difference (change in observed outcome between pre- and posttreatment periods), and Matching multiple units in a large sample on the basis of similar pre-treatment histories, the resulting ATE (i.e. the ATT: Average Treatment Effect for the Treated) provides a robust difference-in-difference estimate of treatment effects. This serves two statistical purposes: firstly, conditional on pre-treatment covariates, the parallel trends assumption is likely to hold; and secondly, this approach reduces dependence on associated ignorability assumptions necessary for valid inference.

OLS模型的所有假设同样适用于DID。此外，使用DID方法还需要满足平行趋势假设。平行趋势假设认为在 s = 1和 s = 2中 lambda _ 2-lambda _ 1的值都是相同的。鉴于上面的正式定义准确地代表了现实，这个假设自动成立。然而，符合 lambda { st } ~ : ~ lambda {22}-lambda {21} neq lambda {12}-lambda {11}的模型可能更加符合现实。为了增加平行趋势假设成立的可能性，双重差分法往往与匹配法相结合。这就涉及将已知的“治疗”单元与模拟的反事实“控制”单元进行“匹配”: 即得到未接受治疗的特征等效单元。通过将结果变量定义为时间差异(治疗前后的观察结果的变化) ，并根据类似的治疗前历史对大样本中的多个单元进行匹配，所得出的ATE结果（即ATT：受治疗者的平均治疗效果）提供了一个稳健的治疗效果差异估计。这样做有两个统计学目的：首先，以治疗前协变量为条件，平行趋势假设很可能成立; 其次，这种方法减少了对有效推论所必需的相关可忽略性假设的依赖。

As illustrated to the right, the treatment effect is the difference between the observed value of y and what the value of y would have been with parallel trends, had there been no treatment. The Achilles' heel of DID is when something other than the treatment changes in one group but not the other at the same time as the treatment, implying a violation of the parallel trend assumption.

如右图所示，治疗效果是观察到的 y 值与未治疗的情况下y 值的平行趋势之间的差异。DID的致命缺点是当一组中治疗以外的某些因素发生了变化，而另一组在治疗的同时没有变化，这意味着违反了平行趋势假设。

To guarantee the accuracy of the DID estimate, the composition of individuals of the two groups is assumed to remain unchanged over time. When using a DID model, various issues that may compromise the results, such as autocorrelation and Ashenfelter dips, must be considered and dealt with.

为了保证DID估计的准确性，假定两组个体的组成在一段时间内保持不变。在使用 DID 模型时，必须考虑和处理可能影响结果的各种问题，如自相关和 Ashenfelter 倾斜。

= 实现 =

The DID method can be implemented according to the table below, where the lower right cell is the DID estimator.

DID 方法可以根据下表实现，其中右下角的单元格是 DID 估计器。

[math]\displaystyle{ y_{st} }[/math]	[math]\displaystyle{ s=2 }[/math]	[math]\displaystyle{ s=1 }[/math]	Difference
[math]\displaystyle{ t=2 }[/math]	[math]\displaystyle{ y_{22} }[/math]	[math]\displaystyle{ y_{12} }[/math]	[math]\displaystyle{ y_{12}-y_{22} }[/math]
[math]\displaystyle{ t=1 }[/math]	[math]\displaystyle{ y_{21} }[/math]	[math]\displaystyle{ y_{11} }[/math]	[math]\displaystyle{ y_{11}-y_{21} }[/math]
Change	[math]\displaystyle{ y_{21}-y_{22} }[/math]	[math]\displaystyle{ y_{11}-y_{12} }[/math]	[math]\displaystyle{ (y_{11}-y_{21})-(y_{12}-y_{22}) }[/math]

y_{st}	s=2	s=1	Difference
t=2	y_{22}	y_{12}	y_{12}-y_{22}
t=1	y_{21}	y_{11}	y_{11}-y_{21}
Change	y_{21}-y_{22}	y_{11}-y_{12}	(y_{11}-y_{21})-(y_{12}-y_{22})

开始! ！2! ！1! ！Difference
t=2	y_{22}	y_{12}	y_{12}-y_{22}
t=1	y_{21}	y_{11}	y_{11}-y_{21}
Change	y_{21}-y_{22}	y_{11}-y_{12}	(y_{11}-y_{21})-(y_{12}-y_{22})

Running a regression analysis gives the same result. Consider the OLS model

运行回归分析也会得到相同的结果。考虑 OLS 模型

[math]\displaystyle{ y ~=~ \beta_0 + \beta_1 T + \beta_2 S + \beta_3 (T \cdot S) + \varepsilon }[/math]

y ~=~ \beta_0 + \beta_1 T + \beta_2 S + \beta_3 (T \cdot S) + \varepsilon

Y ~ = ~ beta _ 0 + beta _ 1 t + beta _ 2 s + beta _ 3(t cdot s) + varepsilon

where [math]\displaystyle{ T }[/math] is a dummy variable for the period, equal to [math]\displaystyle{ 1 }[/math] when [math]\displaystyle{ t=2 }[/math], and [math]\displaystyle{ S }[/math] is a dummy variable for group membership, equal to [math]\displaystyle{ 1 }[/math] when [math]\displaystyle{ s=2 }[/math]. The composite variable [math]\displaystyle{ (T \cdot S) }[/math] is a dummy variable indicating when [math]\displaystyle{ S=T=1 }[/math]. Although it is not shown rigorously here, this is a proper parametrization of the model formal definition, furthermore, it turns out that the group and period averages in that section relate to the model parameter estimates as follows

其中 T 是表示时期的哑变量，当 s = 2时 T 等于1；S 是表示群体成员的哑变量，当 s = 2时 S 等于1。综合变量(T \cdot S)是一个哑变量，表示当 S = T = 1时的情况。虽然这里没有严格说明，但这是模型形式定义的适当参数化。此外，该部分中的组和周期平均值与模型参数估计有关，如下所示

[math]\displaystyle{ \begin{align} \hat{\beta}_0 & = \widehat{E}(y \mid T=0,~ S=0) \\[8pt] \hat{\beta}_1 & = \widehat{E}(y \mid T=1,~ S=0) - \widehat{E}(y \mid T=0,~ S=0) \\[8pt] \hat{\beta}_2 & = \widehat{E}(y \mid T=0,~ S=1) - \widehat{E}(y \mid T=0,~ S=0) \\[8pt] \hat{\beta}_3 & = \big[\widehat{E}(y \mid T=1,~ S=1) - \widehat{E}(y \mid T=0,~ S=1)\big] \\ & \qquad {} - \big[\widehat{E}(y \mid T=1,~ S=0) - \widehat{E}(y \mid T=0,~ S=0)\big], \end{align} }[/math]

\begin{align} \hat{\beta}_0 & = \widehat{E}(y \mid T=0,~ S=0) \\[8pt] \hat{\beta}_1 & = \widehat{E}(y \mid T=1,~ S=0) - \widehat{E}(y \mid T=0,~ S=0) \\[8pt] \hat{\beta}_2 & = \widehat{E}(y \mid T=0,~ S=1) - \widehat{E}(y \mid T=0,~ S=0) \\[8pt] \hat{\beta}_3 & = \big[\widehat{E}(y \mid T=1,~ S=1) - \widehat{E}(y \mid T=0,~ S=1)\big] \\ & \qquad {} - \big[\widehat{E}(y \mid T=1,~ S=0) - \widehat{E}(y \mid T=0,~ S=0)\big], \end{align}

\begin{align} \hat{\beta}_0 & = \widehat{E}(y \mid T=0,~ S=0) \\[8pt] \hat{\beta}_1 & = \widehat{E}(y \mid T=1,~ S=0) - \widehat{E}(y \mid T=0,~ S=0) \\[8pt] \hat{\beta}_2 & = \widehat{E}(y \mid T=0,~ S=1) - \widehat{E}(y \mid T=0,~ S=0) \\[8pt] \hat{\beta}_3 & = \big[\widehat{E}(y \mid T=1,~ S=1) - \widehat{E}(y \mid T=0,~ S=1)\big] \\ & \qquad {} - \big[\widehat{E}(y \mid T=1,~ S=0) - \widehat{E}(y \mid T=0,~ S=0)\big], \end{align}

where [math]\displaystyle{ \widehat{E}(\dots \mid \dots ) }[/math] stands for conditional averages computed on the sample, for example, [math]\displaystyle{ T=1 }[/math] is the indicator for the after period, [math]\displaystyle{ S=0 }[/math] is an indicator for the control group. Note that [math]\displaystyle{ \hat{\beta}_1 }[/math] is an estimate of the counterfactual rather than the impact of the control group. The control group is often used as a proxy for the counterfactual (see, Synthetic control method for a deeper understanding of this point). Thereby, [math]\displaystyle{ \hat{\beta}_1 }[/math] can be interpreted as the impact of both the control group and the intervention's (treatment's) counterfactual. Similarly, [math]\displaystyle{ \hat{\beta}_2 }[/math], due to the parallel trend assumption, is also the same differential between the treatment and control group in [math]\displaystyle{ T=1 }[/math]. The above descriptions should not be construed to imply the (average) effect of only the control group, for [math]\displaystyle{ \hat{\beta}_1 }[/math], or only the difference of the treatment and control groups in the pre-period, for [math]\displaystyle{ \hat{\beta}_2 }[/math]. As in Card and Krueger, below, a first (time) difference of the outcome variable [math]\displaystyle{ (\Delta Y_i = Y_{i,1} - Y_{i,0}) }[/math] eliminates the need for time-trend (i.e., [math]\displaystyle{ \hat{\beta}_1 }[/math]) to form an unbiased estimate of [math]\displaystyle{ \hat{\beta}_3 }[/math], implying that [math]\displaystyle{ \hat{\beta}_1 }[/math] is not actually conditional on the treatment or control group. Consistently, a difference among the treatment and control groups would eliminate the need for treatment differentials (i.e., [math]\displaystyle{ \hat{\beta}_2 }[/math]) to form an unbiased estimate of [math]\displaystyle{ \hat{\beta}_3 }[/math]. This nuance is important to understand when the user believes (weak) violations of parallel pre-trend exist or in the case of violations of the appropriate counterfactual approximation assumptions given the existence of non-common shocks or confounding events. To see the relation between this notation and the previous section, consider as above only one observation per time period for each group, then

其中 widehat { e }(\dots \mid \dots)代表在样本上计算的条件平均值，例如，T = 1是后时期的指标，S = 0是对照组的指标。请注意，hat { beta }1是对反事实的估计，而不是对照组的影响。对照经常被用作反事实的替代(见合成控制方法，以便更深入地理解这一点)。因此，hat { beta } _ 1可以被解释为对照组和干预(治疗)的反事实的影响。同样，由于平行趋势假设，T= 1时，治疗组和对照组之间也存在相同的差异，即\hat{\beta}_2。上述描述不应该被解释为仅是对照组对\hat{\beta}_1的(平均)效应，或者仅仅是治疗组和对照组在前期的差异，hat { beta } _ 2。正如 Card 和 Krueger 所说，结果变量的一阶差分(Delta y _ i = y _ { i，1}-y _ { i，0})消除了对时间趋势（即 hat { beta } _ 1）形成无偏估计的需要（hat { beta } _ 3），这意味着 hat { beta } _ 1实际上并不取决于治疗组或对照组。一致地，治疗组和对照组之间的差异将消除治疗差异(即，hat { beta } _ 2)的需要，进而形成对 hat { beta } _ 3的无偏估计。这种细微差别对于了解用户何时认为（微弱）违反平行预趋势或在存在非共同冲击或混杂事件的情况下违反适当的反事实近似假设是非常重要的。为了看清该符号与前面章节之间的关系，如上所述，考虑小组每个时间段只有一个观察值，那么

[math]\displaystyle{ \begin{align} \widehat{E}(y \mid T=1,~ S=0) & = \widehat{E}(y \mid \text{ after period, control}) \\ [3pt] \\ & = \frac{ \widehat{E}(y \ I(\text{ after period, control}) )}{ \widehat{P}(\text{ after period, control})} \\ [3pt] \\ & = \frac{ \sum_{i=1}^n y_{i,\text{after}} I(i \text{ in control}) } { n_{\text{control}} } = \overline{y}_{\text{control, after}} \\ [3pt] \\ & = \overline{y}_{\text{12}} \end{align} }[/math]

and so on for other values of [math]\displaystyle{ T }[/math] and [math]\displaystyle{ S }[/math], which is equivalent to

\begin{align}

\widehat{E}(y \mid T=1,~ S=0) & = \widehat{E}(y \mid \text{ after period, control}) \\

[3pt] \\ & = \frac{ \widehat{E}(y \ I(\text{ after period, control}) )}{ \widehat{P}(\text{ after period, control})} \\ [3pt] \\ & = \frac{ \sum_{i=1}^n y_{i,\text{after}} I(i \text{ in control}) } { n_{\text{control}} } = \overline{y}_{\text{control, after}} \\ [3pt] \\ & = \overline{y}_{\text{12}} \end{align}

and so on for other values of T and S, which is equivalent to

开始{ align } widehat { e }(y mid t = 1，~ s = 0) & = widehat { e }(y mid text { after period，control })[3 pt ] & = frac { widehat { e }(text { after period，control }(text { after period，control })}(y i (text { after period，control }))}{ wideh { widehat { p }(text }(after period，control }))[3 pt ]和 = frac { sum { i = 1} ^ nY _ { i，text { after } i (i text { in control })}{ n _ { text { control }} = overline { y }{ text { control，after }[3 pt ] & = overline { y }{ text {12}} end { align }等于

[math]\displaystyle{ \hat{\beta}_3 ~=~ (y_{11} - y_{21}) - (y_{12} - y_{22}). }[/math]

\hat{\beta}_3 ~=~ (y_{11} - y_{21}) - (y_{12} - y_{22}).

3 ~ = ~ (y _ {11}-y _ {21})-(y _ {12}-y _ {22}).

But this is the expression for the treatment effect that was given in the formal definition and in the above table.

但这是正式定义和上表中给出的治疗效果的表达式。

= Card and Krueger (1994) example =

Consider one of the most famous DID studies, the Card and Krueger article on minimum wage in New Jersey, published in 1994. Card and Krueger compared employment in the fast food sector in New Jersey and in Pennsylvania, in February 1992 and in November 1992, after New Jersey's minimum wage rose from $4.25 to $5.05 in April 1992. Observing a change in employment in New Jersey only, before and after the treatment, would fail to control for omitted variables such as weather and macroeconomic conditions of the region. By including Pennsylvania as a control in a difference-in-differences model, any bias caused by variables common to New Jersey and Pennsylvania is implicitly controlled for, even when these variables are unobserved. Assuming that New Jersey and Pennsylvania have parallel trends over time, Pennsylvania's change in employment can be interpreted as the change New Jersey would have experienced, had they not increased the minimum wage, and vice versa. The evidence suggested that the increased minimum wage did not induce a decrease in employment in New Jersey, contrary to what some economic theory would suggest. The table below shows Card & Krueger's estimates of the treatment effect on employment, measured as FTEs (or full-time equivalents). Card and Krueger estimate that the $0.80 minimum wage increase in New Jersey led to a 2.75 FTE increase in employment.

Card and Krueger estimate that the $0.80 minimum wage increase in New Jersey led to a 2.75 FTE increase in employment.

关于DID，最著名的研究之一便是 Card 和 Krueger 在1994年发表的关于新泽西州最低工资的文章。Card 和 Krueger 比较了1992年2月和1992年11月新泽西州最低工资从4.25美元上升到5.05美元之后，新泽西州和宾夕法尼亚州快餐部门的就业情况。仅在治疗前后观察新泽西州的就业情况变化，将无法控制一些被忽略变量，例如该地区的天气和宏观经济状况。通过将宾夕法尼亚州作为双重差分模型的对照，任何由新泽西州和宾夕法尼亚州的共同变量所引起的偏差都会被隐含的控制，即使这些变量是不可被观测到的。假设新泽西州和宾夕法尼亚州随着时间的推移有平行的趋势，那么宾夕法尼亚州的就业变化就可以解释为新泽西州在没有提高最低工资的情况下，会产生的变化，反之亦然。证据表明，提高最低工资并没有导致新泽西州就业率的下降，这与一些经济理论的说法恰恰相反。下表显示了 Card 和 Krueger 对就业治疗效果的估计，以 FTEs (或全职人力工时)衡量。Card 和 Krueger 估计，新泽西州0.80美元的最低工资增长导致了2.75个全职雇员的就业增加。

	New Jersey	Pennsylvania	Difference
February	20.44	23.33	−2.89
November	21.03	21.17	−0.14
Change	0.59	−2.16	2.75

	新泽西州	宾夕法尼亚州	差异
February	20.44	23.33	−2.89
November	21.03	21.17	−0.14
Change	0.59	−2.16	2.75

See also

• Design of experiments
• Average treatment effect
• Synthetic control method
• 实验设计
• 平均处理效应
• 合成控制方法

Further reading

• Angrist, J. D.; Pischke, J. S. (2008). Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press. pp. 227–243. ISBN 978-0-691-12034-8. https://books.google.com/books?id=ztXL21Xd8v8C&pg=PA227. 
• Cameron, Arthur C.; Trivedi, Pravin K. (2005). Microeconometrics: Methods and Applications. Cambridge university press. pp. 768–772. doi:10.1017/CBO9780511811241. ISBN 9780521848053. https://api.semanticscholar.org/CorpusID:120313863. 
• Imbens, Guido W.; Wooldridge, Jeffrey M. (2009). "Recent Developments in the Econometrics of Program Evaluation". Journal of Economic Literature. 47 (1): 5–86. doi:10.1257/jel.47.1.5.
• Bakija, Jon; Heim, Bradley (August 2008). "How Does Charitable Giving Respond to Incentives and Income? Dynamic Panel Estimates Accounting for Predictable Changes in Taxation". NBER Working Paper No. 14237. doi:10.3386/w14237.
• Conley, T.; Taber, C. (July 2005). "Inference with 'Difference in Differences' with a Small Number of Policy Changes". NBER Technical Working Paper No. 312. doi:10.3386/t0312.

= 进一步阅读 =

External links

• Difference in Difference Estimation, Healthcare Economist website

• Difference in Difference Estimation, Healthcare Economist website

= = 外部链接 =

• 差异估计差异，医疗经济学家网站

Category:Econometric modeling Category:Regression analysis Category:Design of experiments Category:Observational study Category:Causal inference Category:Subtraction

类别: 计量经济学模型 类别: 回归分析 类别: 实验设计 类别: 观察性研究 类别: 因果推断 类别: 减法

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