seminar-RDD-part02.knit

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# 高级计量暑期班</br></br>(Seminar of Advanced Econometrics)

### 胡华平

### 西北农林科技大学

### 经济管理学院数量经济教研室

### huhuaping01@hotmail.com

### 2022-06-28

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class: center, middle, duke-green,hide_logo
name:chapter-RDD02

# RDD PART 02：断点效应评估

### [1.RDD原理是什么？(How Does It Work?)](#principal)

### [2.RDD该如何实施？(How Is It Performed?)](#perform)

### [3.RDD怎样高级进阶？(How the Pros Do It?)](#pro)

---
layout: false
class: center, middle, duke-orange,hide_logo
name: principal

# 1.RDD原理是什么？</br></br>(How Does It Work?)

### [1.1 RDD直观解释](#rdd-explain)

### [1.2 RDD相关概念](#rdd-concept)

### [1.3 RDD因果关系分析](#rdd-causality)

### [1.4 RDD基本假设](#rdd-assumption)

### [1.5 RDD的基本过程](#rdd-process)

---
layout: true

<div class="my-footer"><span><a href="https://home.huhuaping.com/">https://home.huhuaping.com </a> &emsp;&emsp; <a href="#chapter-RDD02"> RDD Part02 断点效应评估</a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#principal"> 1.RDD原理是什么？ </a> </span></div>

---

### （引子）RDD典型分析：地理区隔与收入变化

.case[

**示例**：

- 圣地亚哥(San Diego)是美国南部一个大城市，占地面积超过300平方英里。 它也很富裕，截至2019年，家庭平均年收入超过85000美元，比全国平均水平高出约50%。

- 当您向南进入城市的其他区域时，一些南部地区的收入会少一些。例如用，当你往南到达的圣伊西德罗(San Ysidro)地区时（靠近墨西哥边境），家庭收入已经下降到50000-55000美元左右。你越往南走，期望家庭收入就越低。

- 但是，当我们越过边境进入墨西哥的蒂华纳(Tijuana, Mexico)时会发生什么？一旦越过边境进入墨西哥的蒂华纳(Tijuana)。 你会发现家庭收入，突然和急剧地下降到20000美元左右。

]

---

### （引子）RDD典型分析：地理区隔与收入变化

.fyi[

**思考**：

- 我们从圣地亚哥(San Diego)市中心开车到南部区域圣伊西德罗(San Ysidro)，只有16英里距离，收入下降了25%。但是，只要继续往南步行几英尺越过边境进入墨西哥境内的蒂华纳(Tijuana, Mexico)，家庭收入则发生急剧下降。

- 当然，对于圣地亚哥南部的家庭，地理位置可能有所不同，这可以解释收入的一些差异。但是在边界线附近两端，家庭收入会出现显著跳跃，这是地理位置因素所难以解释的。

]

---
name: rdd-explain

### 1.1 RDD直观解释：一句话理解

断点回归设计（.red[R]egression .red[D]iscontinued .red[D]esign, RDD）：

.notes[

> RDD是一种用于检验**因果关系**（causal relationship）假设的分析方法([Thistlethwaite and Campbell, 1960](#bib-thistlethwaite1960))

]

---

### 1.1 RDD直观解释：复杂一点的解释

RDD主要用于如下情形([Cattaneo and Titiunik, 2021](#bib-cattaneo2021))：

.notes[

- 被研究对象（units）上可以观测到一个**运行变量**（running variable）

- 基于某些规则（rule）研究者可以给出运行变量上的一个（或若干个）**断点值**（cutoff），并据此对所有被研究对象设定**分配水平**（assignment level）：包括**处置条件**（treatment condition）和**控制条件**（control condition）。

- 在断点值以上的被研究对象将被分配**处置条件**（treatment condition），并被定义为**处置组**（treated group）；在断点值以下的被研究对象将被分配**控制条件**（control condition），并被定义为**控制组**（controlled group）

- 在满足某些**假设条件**下，断点附近处置条件分配概率的断点式变化，可以揭示出**处置条件**对**结果变量**（目标变量）的因果关系。

]

---
name: rdd-concept

### 1.2 RDD相关概念：结果变量、运行变量、混淆变量

- **结果变量（output variable）**：研究的目标变量，一般记为
`$Y$`

> 例如，结果变量为观测到的病人是否猝死。
  
- **运行变量(Runing variable)**<sup>a</sup>：是一个可以观测得到的变量。一方面它将决定被研究对象（units）是否被处置（treated）；另一方面它本身也会影响到结果变量。一般记为
`$X$`

> 例如，医生测量病人的血压，如果收缩压高于135，医生会给病人开降压药，这里病人的血压就是运行变量。

- **混淆变量(Confound variable)**：是哪些不能被直接观测得到的变量，它们可能会同时影响到**运行变量**（进而干扰到马上要定义的**处置变量**）以及**结果变量**。一般记为
`$U$`

.footnote[

<sup>a</sup> 也被称为分派变量（assigning variable），或者强制变量（forcing variable）

]
 
---

### 1.2 RDD相关概念：断点和处置变量

- **断点（Cutoff）**：是运行变量中的一个具体取值，根据它的取值我们可以来决定对象是否需要处置。 这一取值一般记为
`$X=c_0$`

> 以血压为例，假定断点值设置为收缩压135。如果你的血压高于135，就应该吃药。 如果低于135，就无须吃药。

- **处置变量（Treatment variable）**： 根据运行变量和断点值的关系，定义得到的关于是否要分配处置水平的虚拟变量。一般定义为：

`$$\begin{align}
D=
  \begin{cases}
  0 \quad \text{if} \quad X< c_0 \\
  1 \quad \text{if} \quad X  \geq c_0 
  \end{cases}
\end{align}$$`

> 例如，给定运行变量
`$X$`为病人血压，断点值为
`$c_0=135$`，那么处置变量即为**是否用药**。具体地，所有血压值
`$\quad X  \geq c_0$`的病人都会进行**用药处置**，也即虚拟变量赋值
`$D=1 \quad (\text{if} \quad X  \geq c_0)$`；否则就**不用药**，虚拟变量赋值为0。

---

### 1.2 RDD相关概念：谱宽

- **谱宽**(Band width)：是断点值附近的一个**邻域**的区间范围的长度，一般记为
`$h$`，此时这个领域的区间范围定义为
`$b \equiv [c_0-h, c_0 +h], \quad \text{and} \quad h >0$`

.case[
**示例**：

- 研究者可以任意给定运行变量（血压）的一个**谱宽**为
`$h=10$`，则**断点值**附近的一个**邻域**的区间范围为
`$b \equiv [c_0-h, c_0 +h]=[135-10, 135 +10]=[125, 145]$`

]

???

我们有理由认为，在边境线两边的家庭几乎是相同的，除了边境线。 但是，距离更远的人(比如圣地亚哥San Diego市中心vs.墨西哥境内更远的人)可能会因为边界以外的原因而有所不同。 带宽就是您愿意考虑的可比较的边境线两边附近的空间范围。距离美墨边境线各10英尺? 各1000英尺? 各80英里?

---
name: rdd-causality

### 1.3 RDD因果关系分析：随机控制实验

.notes[

- **随机控制实验**（Randomized controlled experiments）：也称为随机对照实验，可以通过严格控制其他影响因素的变动，而准确分析特定一个影响因素对结果变量
`$Y$`的作用。绝大部分自然科学研究都基于这一实验设计理念。

- **准自然实验**（Quasi-experiment or Natural experiment）：对于社会科学家而言，严格的随机控制实验往往无法获得或极难实施。但是在特定条件下，也还是可以得到某种“近似”（as if）随机性的数据生成机制（DGP）。

- **局部随机性实验**（Local randomized
experiment）：在某些情形下，全局性（global）的随机对照实验难以满足或事实，但是却可以在局部范围内（local）进行近似随机的对照实验([Hausman and Rapson, 2018](https://doi.org/10.1146/annurev-resource-121517-033306))。

]

---

### （示例）局部随机控制实验

.case[
**分数与录取案例**：

- 高校根据高考成绩划定投档线和录取线，如果某省理工类一本录取最低控制分数线为450，该省内的一所重点高校N理工类最低录取分数线为520分。

- 那么该重点高校N最低录取线（520分）附近以下，例如516-519分之间未被录取的很多学生，与略高于最低录取线，例如520-522分之间被成功录取的很多学生，这两类学生群体理论上并无明显差异。

- 那么我们就可以基于这一局部观察，设计局部随机控制性实验分析。

]

---
exclude: true

### 1.3 RDD因果关系分析：因果路径图

.footnote[

- [DAG制图可参看](https://evalf20.classes.andrewheiss.com/example/dags/)
]

---

### 1.3 RDD因果关系分析：断点与局部随机

.pull-left[

<div class="figure" style="text-align: center">
<img src="../pic/chpt02-RD-DGG01.jpg" alt="a)数据生成机制DGP" width="780" height="200" />
<p class="caption">a)数据生成机制DGP</p>
</div>

]

.pull-right[

<div class="figure" style="text-align: center">
<img src="../pic/chpt02-RD-DGG02.jpg" alt="b)RDD因果关系解析" width="403" height="200" />
<p class="caption">b)RDD因果关系解析</p>
</div>

]

- 图a)展示的是常见的数据生成机制（DGP）。因为**混淆变量**
`$U$`的存在，使得难以有效分析出**处置变量**
`$D$`对**结果变量**
`$Y$`的作用关系（影响效应）。

- 图b)展示的是在RDD框架下，研究者能够很大程度上剥离**混淆变量**
`$U$`的干扰，并有效分析出**处置变量**
`$D$`对**结果变量**
`$Y$`的作用关系（影响效应）。

---

### 1.3 RDD因果关系分析：可观测事实与反事实

**可观测事实**（observed facts）：在给定研究对象某种分配条件下（例如处置条件或控制条件），可以分别得到处置组对象（treated group， **T**）和控制组对象（controlled group, **C**），就能分别观测到结果变量的表现，也即**可观测事实**。

> **可观测结果**（observed outcome）：此时，处置组和控制组的结果变量容易被观测得到，分别可记为
`$[Y_i^1\mid D=1]$`以及
`$[Y_i^0\mid D=0]$`

**反事实**（Counterfactual）：对于.red[处置组]的研究对象，如果不给它们分配处置条件，那么它们的结果变量会是如何呢？同理，对于.red[控制组]的研究对象，如果给它们分配处置条件，那么它们的结果变量又会是如何呢？显然，这些都是假想情形，实际并未发生的事实。

> **潜在结果**（Potential outcome）：此时，处置组和控制组的结果变量不能被直接观测得到，表现为**潜在结果**，我们分别可记为
`$[Y_i^0\mid D=1]$`以及
`$[Y_i^1\mid D=0]$`

---
exclude: true

## Code script: Hensen bruce fig 21.1

- 此代码仅用于生成本地图片，为提高渲染效率代码文件仅需运行一次。

- 如更新代码，请更改`eval = TRUE`

---

### （示例）图形演示：可观测事实与反事实

---
exclude: true

### 1.3 RDD因果关系分析：可观测事实与反事实(not use)

- 处置组对象的结果变量的期望：

`$$\begin{align}
E(Y_i^1\mid X_i=x) & \equiv 
E\left(Y_i^1\mid X_i\geq c_0\right) + E(Y_i^1 \mid X_i < c_0)   \\
& \equiv E\left(Y_i^1\mid D=1\right) + E(Y_i^1 \mid D=0) \\
& \equiv E\left(Y^1\mid c^+\right) + E(Y^1 \mid c^-)
\end{align}$$`

- 控制组对象的结果变量的期望：

`$$\begin{align}
E(Y_i^0\mid X_i=x) & \equiv 
E\left(Y_i^0\mid X_i\geq c_0\right) + E(Y_i^0 \mid X_i < c_0)   \\
& \equiv E\left(Y_i^0\mid D=1\right) + E(Y_i^0 \mid D=0) \\
& \equiv E\left(Y^0\mid c^+\right) + E(Y^0 \mid c^-)
\end{align}$$`

- 处置变量对结果变量的因果效应：

`$$\begin{align}
\tau &= E(Y_i^1\mid X_i=x)  - E(Y_i^0\mid X_i=x)  \\
&= \left[ E\left(Y^1\mid c^+\right) + E(Y^1 \mid c^-) \right] -
\left[ E\left(Y^0\mid c^+\right) + E(Y^0 \mid c^-) \right]
\end{align}$$`

---

### 1.3 RDD因果关系分析：可观测事实与反事实（表达式）

- 处置条件下结果变量（可观测的和潜在的）的期望：

`$$\begin{align}
& \equiv \mathbb{E}\left(Y_i^1\mid X_i\geq c_0\right) + \mathbb{E}(Y_i^0 \mid X_i \geq c_0)   \\
& \equiv \mathbb{E}\left(Y_i^1\mid D=1\right) + \mathbb{E}(Y_i^0 \mid D=1) \\
& \equiv \mathbb{E}\left(Y^1\mid c^+\right) + \mathbb{E}(Y^0 \mid c^+)
\end{align}$$`

- 控制条件下结果变量（可观测的和潜在的）的期望：

`$$\begin{align}
& \equiv \mathbb{E}\left(Y_i^1\mid X_i < c_0\right) + \mathbb{E}(Y_i^0 \mid X_i < c_0)   \\
& \equiv \mathbb{E}\left(Y_i^1\mid D=0\right) + \mathbb{E}(Y_i^0 \mid D=0) \\
& \equiv \mathbb{E}\left(Y^1\mid c^- \right) + \mathbb{E}(Y^0 \mid c^-)
\end{align}$$`

- 处置变量对结果变量的因果效应：

`$$\begin{align}
\tau &= 
\left[ \mathbb{E}\left(Y^1\mid c^+\right) + \mathbb{E}(Y^0 \mid c^+) \right] -
\left[ \mathbb{E}\left(Y^1\mid c^- \right) + \mathbb{E}(Y^0 \mid c^-) \right]
\end{align}$$`

---

### 1.3 RDD因果关系分析：断点处置效应ATE（表达式）

- 处置变量对结果变量的因果效应：

`$$\begin{align}
\tau &= 
 \mathbb{E}\left(Y_i\mid X_i \geq c\right) -
\mathbb{E}\left(Y_i\mid X_i < c\right)  \\ &= 
 \mathbb{E}\left(Y^1_i\mid X_i \geq c\right) -
\mathbb{E}\left(Y^0_i\mid X_i < c\right) \\
&= 
 \mathbb{E}\left(Y^1_i\right) -
\mathbb{E}\left(Y^0_i\right)
\end{align}$$`

---

### 1.3 RDD因果关系分析：断点处置效应ATE（图示）

- 潜在结果变量的条件均值为常数的情形：

???

source: fig2.a from Cattaneo M D, Idrobo N, Titiunik R. A Practical Introduction to Regression Discontinuity Designs: Extensions[J]. , 2021: 106.

---

### 1.3 RDD因果关系分析：断点处置效应ATE（图示）

- 潜在结果变量的条件均值并**不是**常数的情形：

???

source: fig2.b from Cattaneo M D, Idrobo N, Titiunik R. A Practical Introduction to Regression Discontinuity Designs: Extensions[J]. , 2021: 106.

---

### 1.3 RDD因果关系分析：局部断点处置效应

- 断点处置效应具有局部性特征（the local nature of RD effect）。

???

source: fig3 from Cattaneo M D, Idrobo N, Titiunik R. A Practical Introduction to Regression Discontinuity Designs: Extensions[J]. , 2021: 106.

---
name: rdd-assumption

### 1.4 RDD基本假设：连续性假设

**假设1**：**结果变量**的期望值在**断点**处需要满足**连续性假设**（continuity assumption）：

- **结果变量**的期望值在**断点**处连续，也即
`$E[Y_i(1)|X_i = x]$`和
`$E[Y_i(0)|X_i = x]$`，可是作为
`$x$`的函数（
`$f(x)$`），且在
`$x=c_0$`出连续。（见下图）

- **断点**值
`$c_0$`本身需要满足**外生性**（exogeneity）条件。也即，**断点**值
`$c_0$`在触发**处置变量**D的时候，不会有**其他变量**在同时期来干预这种“触发行为”。

- 在上述条件下，运行变量
`$X$`对结果变量
`$Y$`将**不再**具有直接影响（
`$X \rightarrow Y$`），而是通过处置变量
`$D$`发生间接作用（
`$X \rightarrow D \rightarrow Y$`）。

- **连续性假设**（continuity assumption）应该是RDD**最关键**的一个假设条件，而且这符合**经验事实**。

> .blue[**大自然不会跳跃！**<sup>[a]</sup>  ---达尔文《物种起源》]

.footnote[

<sup>[a]</sup> 事物的发展变化总是**渐进式**的，而不会陡然改变。常言道“量变引发质变”。

]

---

### （示例）条件期望函数CEF的连续性假设

---

### 1.4 RDD基本假设：断点性假设

**假设2**：被研究对象被分配（assign）.red[处置条件]（treated condition）<sup>[1]</sup>的**条件概率**（Conditional Probability of Receiving Treatment）
`$P(D_i= 1 \mid X_i=c_0)$`在断点处是**不连续的**（也即间断的）。

常见的处置分配概率不连续**模式**包括：

- **骤变不连续**（Sharp discontinuity）：处置条件分配的概率在断点处被完全决定。

- **模糊不连续**（Fuzzy discontinuity）：处置条件分配的概率在断点处**不能**被完全决定。

.footnote[

<sup>[1]</sup> 回顾**分配水平**（assign level）具有两个水平：处置条件（treated condition）和控制条件（controlled condition）

]

---
exclude: true

### （示例）RDD的断点性假设

---
exclude: true

## Code chunk：模拟断点处置

---

### （示例）断点性假设：骤变（Sharp）不连续

- 处置条件的骤变（Sharp）不连续示例：小学入学年龄严格要求出生日期（
`$X$`）在
`$c_0=9月1日$`之前。

---

### （示例）断点性假设：模糊（Fuzzy）不连续

- 处置条件的模糊（Fuzzy）不连续：小学入学年龄要求出生日期（
`$X$`）在
`$X \in [8月1日,9月30日]$`期间，家长可以自己选择孩子是否上小学。

---
name: rdd-process

### 1.5 RDD的基本过程：概览

如果暂时忽略各种细节，一个最简化的RDD分析过程包括：

- 设定断点两边对结果变量的预测模型方法（predictive model）

- 选择局部谱宽（bandwidth）

- 估计并计算因果效应

---
exclude: true

## Code script: 模拟数据RDD过程

- 参考来源：[github](https://github.com/NickCH-K/causaldata)

---

### （示例）RDD的基本过程1/4：原始数据

---

### （示例）RDD的基本过程2/4：断点两边拟合

- 这里采用了**LL**方法拟合局部均值（local mean）

---

### （示例）RDD的基本过程3/4：选定一个谱宽Bandwidth

- 我们暂时不关心远离断点处的观测值（因为混淆变量会产生作用）

- 最优化的谱宽选择可以基于某些准则，例如BIC等

---

### （示例）RDD的基本过程4/4：断点处估计因果效应

- 谱宽范围内、断点两边的估计结果，表现出了“跳跃”效果（jumps）

---
layout: false
class: center, middle, duke-orange,hide_logo
name: perform

# 2.RDD该如何实施？</br></br>(How Is It Performed?)

### [2.1 平均和断点处置效应](#ate)

### [2.2 骤变RDD的估计](#sharp-est)

### [2.3 骤变RDD谱宽选择](#sharp-bw)

### [2.4 骤变RDD推断](#sharp-refer)

### [2.5 RDD协变量分析](#rdd-cov)

---
layout: true

<div class="my-footer"><span><a href="https://home.huhuaping.com/">https://home.huhuaping.com </a>  &emsp;&emsp; <a href="#chapter-RDD02"> RDD Part02 断点效应评估 </a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#perform"> 2.RDD该如何实施？ </a> </span></div>

---

### （引子）符号表达体系

- 结果变量
`$Y$`

- 运行变量
`$X$`，断点值
`$c_0$`

- 处置变量
`$D$`：

`$$\begin{align}
D=
  \begin{cases}
  0 \quad \text{if} \quad X< c_0 \\
  1 \quad \text{if} \quad X  \geq c_0 
  \end{cases}
\end{align}$$`

- 实验组对象**T**
`$(D=1)$`；控制组对象**C**
`$(D=0)$`

---
name: ate

### 2.1 平均和断点处置效应：定义

- 当个体
`$i$`被分配为“处置条件”时，其结果变量为
`$Y_1$`为；当个体
`$i$`被分配为“控制条件”时，其结果变量为
`$Y_0$`。

- 此时，个体
`$i$`的**处置效应**(treatment effect)记为
`$\theta=Y_1 -Y_0$`，因为其具有随机性，也被称为**随机处置效应**(random treatment effect)

- 给定一个可观测的协变量
`$X$`（运行变量），我们可以得到个体
`$i$`的**条件处置效应**(conditional treatment effect)，并记为：

`$$\theta |(X=x)=(Y_1 -Y_0)|(X=x)$$`

- 对于
`$X=x$`处的多个个体，我们可以得到它们的条件**平均处置效应**（average treatment effect, ATE），并记为：

`$$\theta(x) \equiv \mathbb{E}(\theta \mid X=x)$$`

---

### （示例）个体和平均处置效应

---

### 2.1 平均和断点处置效应：条件期望函数CEF

给定结果变量的**条件期望函数**(conditional expect function, CEF)<sup>a</sup>如下：

`$$m(x) \equiv \mathbb{E}(Y|X=x)$$`

则可以分别得到**控制条件**和**处置条件**下的**条件期望函数**：

`$$\begin{align}
  \begin{cases}
    m_0(x) = \mathbb{E}(Y_0|X=x)\\
    m_1(x) = \mathbb{E}(Y_1|X=x)
  \end{cases}
\end{align}$$`

进而，我们可以把**条件平均处置效应**(conditional ATE)表达为：

`$$\begin{align}
\theta(x) &\equiv \mathbb{E}(\theta \mid X=x) \\
& = \mathbb{E}[(Y_1 -Y_0) \mid X=x] \\
& = \mathbb{E}[(Y_1\mid X=x ) -(Y_0\mid X=x)] \\
& = m_1(x)  -m_0(x) 
\end{align}$$`

.footnote[

<sup>a</sup> 这里先表达为**隐函数**形式，也即其具体函数表达式未知。
]

---

### （示例）条件期望函数CEF与平均处置效应

---

### 2.1 平均和断点处置效应：CEF连续性假设

结果变量的条件期望函数在断点处的**连续性**（continuity）假设：

> 给定断点值为
`$x=c$`，假设结果变量的条件期望函数
`$m(x)$`在断点处
`$x=c$`连续。

> 这也意味着在*控制条件**和**处置条件**下的**条件期望函数**也在断点处是连续的。也即
`$m_0(x)$`和
`$m_1(x)$`在断点处
`$x=c$`连续。

.note[

**定义**：我们把条件函数的
**极限**（
`$z$`从右边向
`$x$`值取极限，和
`$z$`从左边向
`$x$`值取极限）定义如下

`$$\begin{align}
m(x+)&=\lim_{z \downarrow x} m(z)\\
m(x-)&=\lim_{z \uparrow x} m(z)
\end{align}$$`

]

---

### 2.1 平均和断点处置效应：定理

**断点处置效应**定理：给定处置分配规则为
`$D=1\{X \geq c\}$`，而且假定结果变量满足断点处的连续性假设，也即结果变量的条件期望函数
`$m(x)$`在断点处
`$x=c$`连续，那么**断点处置效应**为：

`$$\bar{\theta}=\lim_{z \downarrow c} m(z) - \lim_{z \uparrow c} m(z)=m(c+)-m(c-)$$`

---

### 2.1 平均和断点处置效应：证明

**证明**：首先，我们进一步定义结果变量：

`$$\begin{align}
Y \equiv Y_0 \cdot\mathbb{1}\{x<c\} + Y_1 \cdot\mathbb{1}\{x\geq c\}
\end{align}$$`

两边对
`$X=x$`取期望，且根据结果变量的条件期望函数的定义，则有：

`$$\begin{align}
\mathbb{E}(Y|X=x) &= \mathbb{E}(Y_0|X=x) \cdot\mathbb{1}\{x<c\} + \mathbb{E}(Y_1|X=x) \cdot\mathbb{1}\{x\geq c\} \\ 
\Rightarrow m(x) &= m_0(x)\cdot\mathbb{1}\{x<c\} + m_1(x) \cdot\mathbb{1}\{x\geq c\} 
\end{align}$$`

根据前面关于**条件处置效应**的定义及**连续性假设**，则有：

.pull-left[
`$$\begin{align}
\theta(x) &\equiv \mathbb{E}(\theta \mid X=x) \\
& = \mathbb{E}[(Y_1 -Y_0) \mid X=x] \\
& = \mathbb{E}[(Y_1\mid X=x ) -(Y_0\mid X=x)] \\
& = m_1(x)  -m_0(x) 
\end{align}$$`
]

.pull-rgith[
`$$\begin{align}
\theta(c) &=  m_1(c)  -m_0(c) \\
&= \lim_{x\downarrow c}{m(x)} - \lim_{x\uparrow c}{m(x)}  &&\leftarrow \text{(连续性假设)} \\
& = m(c+) - m(c-)
\end{align}$$`
]

---
name: sharp-est

### 2.2 骤变RDD的估计：边界估计问题

**断点回归设计**（RDD）属于典型的**边界估计**（boundary estimation）问题，这里我们将优先采用**局部线性回归**（local linear regression, LLR）方法进行估计。

> 这里，我们将使用到非参数的**核函数**（kernel  function）方法来除了回归的权重问题。

给定如下条件：

- 变量集

`$$\begin{align}
Z_{i}(x)=\left(
  \begin{array}{c}
  \mathbb{1} \\
  X_{i}-x
  \end{array}
\right)
\end{align}$$`

- 核函数（kernel function）
`$K(u)$`

- 谱宽（bandwidth）
`$h$`

---

### 2.2 骤变RDD的估计：局部线性回归估计（CEF）

此时，可以证明**局部线性**方法下的系数估计为（证明略）：

- 对于断点左侧
`$x < c$`，系数估计为<sup>a</sup>：

`$$\begin{align}
\boldsymbol{\widehat{\beta}_{0}}(x)=\left(\sum_{i=1}^{n} K\left(\frac{X_{i}-x}{h}\right) Z_{i}(x) Z_{i}(x)^{\prime}\cdot \mathbb{1}\left\{X_{i}<c\right\}\right)^{-1}\left(\sum_{i=1}^{n} K\left(\frac{X_{i}-x}{h}\right) Z_{i}(x) Y_{i}\cdot \mathbb{1}\left\{X_{i}<c\right\}\right)
\end{align}$$`

- 对于断点左侧
`$x \geq c$`，系数估计为<sup>b</sup>：

`$$\begin{align}
\boldsymbol{\widehat{\beta}_{1}}(x)=\left(\sum_{i=1}^{n} K\left(\frac{X_{i}-x}{h}\right) Z_{i}(x) Z_{i}(x)^{\prime} \cdot\mathbb{1}\left\{X_{i} \geq c\right\}\right)^{-1}\left(\sum_{i=1}^{n} K\left(\frac{X_{i}-x}{h}\right) Z_{i}(x) Y_{i} \cdot\mathbb{1}\left\{X_{i} \geq c\right\}\right)
\end{align}$$`

.footnote[

<sup>a</sup> <sup>b</sup> 需要注意的是，这里我们得到的都是系数向量（vector）。

]

---

### 2.2 骤变RDD的估计：局部线性回归估计（断点效应）

根据结果变量条件期望函数
`$m(x)$`的定义，我们可以使用上述系数估计
`$\boldsymbol{\widehat{\beta}_{0}}(x),\boldsymbol{\widehat{\beta}_{1}}(x)\}$`，进一步得到结果变量条件期望函数的估计结果<sup>a</sup>：

`$$\begin{align}
\widehat{m}(x)=\left[\boldsymbol{\widehat{\beta}_{0}}(x)\right]_{1} \cdot  \mathbb{1}\{x<c\}+\left[\boldsymbol{\widehat{\beta}_{1}}(x)\right]_{1} \cdot \mathbb{1}\{x \geq c\}
\end{align}$$`

因此，根据**断点处置效应定理**，可以得到在断点
`$x=c$`处对总体平均处置效应
`$\bar{\theta}$`的样本估计结果
`$\hat{\theta}$`：

`$$\begin{align}
\widehat{\theta}=\left[\boldsymbol{\widehat{\beta}_{1}}(c)\right]_{1}-\left[\boldsymbol{\widehat{\beta}_{0}}(c)\right]_{1}=\hat{m}(c+)-\widehat{m}(c-)
\end{align}$$`

.footnote[

<sup>a</sup> 条件期望函数CEF只需要用到系数向量（vector）的第一个元素，因此用了下标<sub>1</sub>表达。

]

---

### 2.2 骤变RDD估计：简单线性回归方法

- 容易证明**骤变RDD断点处置效应**也可以通过如下简单线性回归方法
等价地得到
`$\widehat{\theta}$`的对应估计值：

`$$\begin{align}
Y=\beta_{0}+\beta_{1} X+\beta_{3}(X-c) D+\theta D+e
\end{align}$$`

需要注意的是：

- 上述等价模型，只是等价前面的基于**非正规化矩形核函数**（unnormalized Rectangular）谱宽下的**局部线性LL**断点处置估计效应值。

- 简单地，上述等价模型需要进行样本数据集的重新定义。具体地，运行变量的范围需要调整到
`$X\in [c-h^{\ast}, c+h^{\ast}]$`，其中
`$h^{\ast}=\sqrt{3}h=\sqrt{3}\times 8$`

---
name: sharp-bw

### 2.3 骤变RDD谱宽选择：基本问题

.notes[

- 基于**边界估计**的**局部线性回归**方法本质上需要进行**非参数估计**，这尤其体现在核函数的**谱宽**（bandwidth）估计。

- 目前还没有达成一致意见的最优谱宽选择方法。因此在进行LLR估计之前，研究者不得不多尝试多种**数据导向**（data based）的谱宽选择工具。

- 谱宽估计是一项具有挑战性的工作，有些具体估计方法会异常复杂。

]

当然，这里可以建议使用两种谱宽选择方法：

- **多项式**(polynomial, PN)谱宽选择法([Fan, Gijbels, Hu, et al., 1996](#bib-fan1996))：这是一种经验方法。

- **交叉验证**（cross validation, CV）谱宽选择法

---

### 2.3 骤变RDD谱宽选择：多项式法

- 首先构造包含
`$q$`阶多项式和断点漂移项的模型：

`$$\begin{align}
m(x)=\beta_{0}+\beta_{1} x+\beta_{2} x^{2}+\cdots+\beta_{q} x^{q}+\beta_{q+1} D
\end{align}$$`

- 然后，通过估计得到的条件期望函数
`$\widehat{m}(x)$`计算二阶求导结果：

`$$\begin{align}
\widehat{m}^{\prime \prime}(x)=2 \widehat{\beta}_{2}+6 \widehat{\beta}_{3} x+12 \widehat{\beta}_{4} x^{2}+\cdots+q(q-1) \widehat{\beta}_{q} x^{q-2}
\end{align}$$`

- 再计算常量
`$\overline{B}$`，其中
`$[\xi_1, \xi_2]$`是运行变量
`$X$`内部的一个评价区间：

`$$\begin{align}
\widehat{B}=\frac{1}{n} \sum_{i=1}^{n}\left(\frac{1}{2} \widehat{m}^{\prime \prime}\left(X_{i}\right)\right)^{2} \mathbb{1}\left\{\xi_{1} \leq X_{i} \leq \xi_{2}\right\}
\end{align}$$`

- 最后，对于任意**正规化核**（normalized kernel），可以计算得到**谱宽**：

`$$\begin{align}
h_{\text{FG}}=0.58 \cdot \left(\frac{\widehat{\sigma}^{2}\left(\xi_{2}-\xi_{1}\right)}{\widehat{B}}\right)^{1 / 5} n^{-1 / 5}
\end{align}$$`

---

### 2.3 骤变RDD谱宽选择：多项式法

根据核函数的不同，多项式法(polynomial)计算公式略有不同：

- 对于**非规化矩形核**（un-normalized rectangular kernel）
`$K(u)=1/2, \text {for} |u|\leq 1$`：

`$$\begin{align}
h_{\text{pn}}=1\cdot \left(\frac{\widehat{\sigma}^{2}\left(\xi_{2}-\xi_{1}\right)}{\widehat{B}}\right)^{1 / 5} n^{-1 / 5}
\end{align}$$`

- 对于**非规化三角核**（un-normalized rectangular kernel）
`$K(u)=1-|u|, \text {for} |u|\leq 1$`：

`$$\begin{align}
h_{\text{pn}}=1.42\cdot \left(\frac{\widehat{\sigma}^{2}\left(\xi_{2}-\xi_{1}\right)}{\widehat{B}}\right)^{1 / 5} n^{-1 / 5}
\end{align}$$`

---

### 2.3 骤变RDD谱宽选择：交叉验证法

.notes[

- **交叉验证**（cross validation, CV）方法：主要形式是把训练集分成两部分，一部分用来训练模型，另一部分用来验证模型。

- **交叉验证**方法包括：**留出法**（holdout）、**留一法**（Leave-one-out,
LOO）、K折法（K-fold）、**自助法**（Bootstrap）等。

- 这里介绍的**交叉验证**谱宽选择法主要采用**留一法**（Leave-one-out,LOO）。
]

**留一法**（Leave-one-out,LOO）选择谱宽的基本步骤：

- 初步选定一个临近断点的区间
`$[\xi_1, \xi_2]$`（去中心化后centered X的范围）

- 任意选择初始谱宽

- 通过**留一法**计算模型预测残差及其**残差平方和**

- 以**最小化**残差平方和为目标，分析谱宽的变化趋势<sup>a</sup>，并最终确定谱宽bandwidth。

.footnote[

<sup>a</sup> 可以绘制CV标准（如均方误差AMSE）与谱宽关系的图示法进行观察。

]

---

### 2.3 骤变RDD谱宽选择：方法评析

- 谱宽估计的噪点（noise）会进入到RDD估计进程中去，因此谱宽选择显得非常重要。

- 无论是**多项式法**还是**交叉验证法**，确定最终谱宽时，都考虑到了**全局性**准确度。

.fyi[
> 这意味着它们都用到了更多的样本数据，因此谱宽估计会比较稳定。
]

- 另一种**局部性**的谱宽选择方法，主要考察**断点**附近(near-by)的准确度。

.fyi[

> 因为局部性存在多种可能，所以这类方法得到的谱宽会更加不稳定。具体参看([Imbens and Kalyanaraman, 2012](#bib-imbens2012c); [Arai and Ichimura, 2018](#bib-arai2018))。

]

- 通过**改变**谱宽值，来对RDD估计进行**稳健性检查**是很必要的。

.fyi[
> 更大的谱宽，一般会使得断点效应估计系数**方差减小**(reduce variance)，置信区间变窄，但同时也会**增加偏误**(increase bias)。

]

---

### 2.3 骤变RDD谱宽选择：方法评析

.fyi[

谱宽选择的**经验法则**：

- 实践操作中，我们往往需要同时结合**多项式法**和**交叉验证法**来确定一个谱宽
`$\tilde{h}$`。

- 在上述基础上，我们还需要适当调减谱宽值，例如
`$h = 25\%\cdot\tilde{h}$`，以减少估计偏误。

]

---
name: sharp-refer

### 2.4 骤变RDD推断：理论估计偏误和方差

基于**局部线性回归**LLR估计结果，对断点处置效应参数
`$\bar{\theta}$`的推断陈述（inferential statement），都会受到其中**非参数估计**偏差的影响。

可以证明，**局部线性回归**（LLR）的估计量
`$\hat{m}(x)$`在**标准正则条件**（standard regularity conditions）下将服从**渐近正态分布**。

- 此时，RDD估计量
`$\hat{\theta}$`的渐近偏误（bias）和渐近方差分别为：

`$$\begin{align}
\operatorname{bias}[\widehat{\theta}]
&=\frac{h^{2} \sigma_{K^{*}}^{2}}{2}\left(m^{\prime \prime}(c+)-m^{\prime \prime}(c-)\right)\\
\operatorname{var}[\widehat{\theta}]
&=\frac{R_{K}^{*}}{n h}\left(\frac{\sigma^{2}(c+)}{f(c+)}+\frac{\sigma^{2}(c-)}{f(c-)}\right)
\end{align}$$`

---

### 2.4 骤变RDD推断：样本方差

上述理论方差，我们可以通过两个**边界回归**（断点两边）的系数估计量的**渐近方差**求和计算得到。我们首先给定如下条件：

- 变量集：

`$$\begin{align}
Z_{i}=\left(
  \begin{array}{c}
  \mathbb{1} \\
  X_{i}-c
  \end{array}
\right)
\end{align}$$`

- **核函数**（kernel function）
`$K_i=k\left(\frac{X_i - c}{h}\right)$`

- **谱宽**（bandwidth）
`$h$`

- **留一法**<sup>a</sup>得到的模型预测**残差**（leave-one-out prediction error）
`$\tilde{e}_i$`

.footnote[

<sup>a</sup> **留一法**(Leave One Out, LOO) 是一种 常见的交叉验证方法，其中每个观察集都被视为验证集test，其余的
`$(n-1)$`观测值被视为训练集training。此处原理类似，每次都去掉一个数据进行估计，然后根据估计结果进行预测，然后得到预测误差。
]

---

### 2.4 骤变RDD推断：样本方差

此时，我们可以得到**局部线性回归**LLR估计系数
`$\hat{\theta}$`的**方差协方差矩阵**分别为：

`$$\begin{aligned}
&\widehat{\boldsymbol{V}}_{0}=\left(\sum_{i=1}^{n} K_{i} Z_{i} Z_{i}^{\prime} \cdot  \mathbb{1}\left\{X_{i}<c\right\}\right)^{-1}\left(\sum_{i=1}^{n} K_{i}^{2} Z_{i} Z_{i}^{\prime} \tilde{e}_{i}^{2} \cdot  \mathbb{1}\left\{X_{i}<c\right\}\right)\left(\sum_{i=1}^{n} K_{i} Z_{i} Z_{i}^{\prime} \cdot  \mathbb{1}\left\{X_{i}<c\right\}\right)^{-1} \\
&\widehat{\boldsymbol{V}}_{1}=\left(\sum_{i=1}^{n} K_{i} Z_{i} Z_{i}^{\prime} \cdot  \mathbb{1}\left\{X_{i} \geq c\right\}\right)^{-1}\left(\sum_{i=1}^{n} K_{i}^{2} Z_{i} Z_{i}^{\prime} \tilde{\boldsymbol{e}}_{i}^{2} \cdot  \mathbb{1}\left\{X_{i} \geq c\right\}\right)\left(\sum_{i=1}^{n} K_{i} Z_{i} Z_{i}^{\prime} \cdot \mathbb{1}\left\{X_{i} \geq c\right\}\right)^{-1}
\end{aligned}$$`

进一步地，估计系数
`$\hat{\theta}$`的渐进方差为上述两个矩阵第一个对角元素之和：

`$$\text{Var}{(\hat{\theta})}=\left[\widehat{\boldsymbol{V}}_{0}\right]_{11}+\left[\widehat{\boldsymbol{V}}_{1}\right]_{11}$$`

---

### 2.4 骤变RDD推断：置信区间和置信带

最后，我们可以分别对断点两侧计算**逐点置信区间**（Pointwise Conﬁdence Interval），并相应构建**置信带**。

`$$\begin{align}
\widehat{m}(x) \pm z_{1-\alpha/2}(n-1) \cdot \sqrt{\widehat{V}_{\widehat{m}(x)}}\\
\widehat{m}(x) \pm 1.96 \sqrt{\widehat{V}_{\widehat{m}(x)}}
\end{align}$$`

---
exclude: true

## Code script: Hansen fig21-1b

- 此代码仅用于生成本地图片，为提高渲染效率代码文件仅需运行一次。

- 如更新代码，请更改`eval = TRUE`

---
name: case-wage

### （死亡率案例）背景说明1/2

.case[

**援助项目与儿童死亡率**：

- 案例基于([Ludwig and Miller, 2007](#bib-ludwig2007))的研究，他们重点评估了美国联邦政府脱贫援助项目（Head Start）的**骤变RDD**政策效应。

- 该援助项目于1965年实施，为3-5岁贫困孩子及其家庭提供学前教育、健康和社会服务等方面的资金援助。对于该援助项目经费，联邦政府将决定通过公开竞标，分配给提交援助申请的中标县。

- 为了保障援助项目的针对性，联邦政府将重点考虑资助被认定的300个贫困县。其中**贫困县**是基于1960年美国统计测度得到的**贫困线水平**（poverty rate）予以划定。

- 最终，300个贫困县中，有80%的县获得了项目资助；而其他提交申请的县中（非贫困县），有43%的县也获得了项目资助。

]

---

### （死亡率案例）背景说明2/2

.case[

**援助项目与儿童死亡率**(续)：
- ([Ludwig and Miller, 2007](#bib-ludwig2007))重点关注**援助项目**对中长期**儿童死亡率**影响。其中儿童死亡率定义为：1973-1983年间、儿童年龄范围在8-18岁、儿童死亡原因为Head Start定义的相关原因（如结核病等）。因而而援助项目希望努力消减这些儿童死亡情形的发生。

- 我们关注的问题：**脱贫援助项目**（Head Start）对**儿童死亡率**（`Y=mortality rate`）的因果效应。我们将采用骤变RDD非参数回归估计，**运行变量**为县贫困率（`X=poverty rate`），**断点值**(cut-off)设定为
`$c=59.1984$`。将使用**子样本数据**的样本数为n=2783。

]

---

### （死亡率案例）样本数据集

.pull-left[

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class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>obs<\/th>\n      <th>X<\/th>\n      <th>Y<\/th>\n      <th>D<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"targets":2,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n  }"},{"targets":3,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n  }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render"],"jsHooks":[]}</script>

]

.pull-right[

- 样本数据的描述性统计如下：

```
       X            Y             D       
 Min.   :15   Min.   :  0   Min.   :0.00  
 1st Qu.:24   1st Qu.:  0   1st Qu.:0.00  
 Median :34   Median :  0   Median :0.00  
 Mean   :37   Mean   :  2   Mean   :0.11  
 3rd Qu.:47   3rd Qu.:  3   3rd Qu.:0.00  
 Max.   :82   Max.   :136   Max.   :1.00  
```

]

---

### （死亡率案例）样本数据集：分组描述性统计

<div id="htmlwidget-aa88adbc9fe279ae36e1" style="width:100%;height:auto;" class="datatables html-widget"></div>
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---

### （死亡率案例）样本数据散点图

---

### （死亡率案例）谱宽选择及CEF估计的规则策略

- 规则1：我们设定**先验谱宽**为
`$h=8$`，断点值设定为
`$c =59.1984\%$`。

- 规则2：分别设定断点两边**箱组中心点**序列值（center of bins）。我们将采用非对称箱组设置方法：

> 
- 控制组（断点左边）的评估范围为
`$[15, 59.2]$`，序列间隔为`0.2`。评估总箱组数为
`$g1=222$`，待评估序列值为
`$15.0, 15.2, 15.4, 15.6, 15.8, \cdots,58.6, 58.8, 59.0, 59.2$`。
- 处置组（断点右边）的评估范围为
`$[59.2, 82]$`，序列间隔为`0.2`。评估总箱组数为
`$g2=115$`，待评估序列值为
`$59.2, 59.4, 59.6, 59.8, 60.0, \cdots,81.4, 81.6, 81.8, 82.0$`。

- 规则3：基于**三角核函数**（triangle kenerl）采用局部线性估计法，分别对断点两侧进行条件期望函数CEF
`$m(x)$`进行估计，并得到估计值
`$\widehat{m}(x)$`（见下面附表和附图）。

---

### （死亡率案例）CEF m(x)估计：计算附表

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### （死亡率案例）CEF m(x)估计图示：断点左侧（控制组）

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### （死亡率案例）CEF m(x)估计图示：断点右侧（处置组）

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### （死亡率案例）CEF m(x)估计图示：断点两侧（对比）

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### （死亡率案例）CEF方差估计：计算方差、标准差

- 直接使用谱宽<sup>a</sup>
`$h=8$`进行局部线性LL估计，并利用**留一法**法计算得到**预测误差**
`$\tilde{\boldsymbol{e}}$`，并最终分别得断点两侧的协方差矩阵（见下式），从而进一步计算得到CEF估计值的方差和标准差（见后面附表）。

.footnote[

<sup>a</sup> 这里我们没有再次评估条件方差估计中的最优谱宽，而是简单直接地使用了CEF估计时的谱宽。但是我们还是要注意，二者的最优谱宽可以完全不相同！

]

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### （死亡率案例）CEF方差估计：计算方差估计值（附表）

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---

### （死亡率案例）CEF的置信区间和置信带

- 进一步计算**局部线性估计**下的**逐点置信区间**（Pointwise Conﬁdence Interval）（见后面附表），并得到置信带（见后面附图）。

`$$\begin{align}
\widehat{m}(x) \pm z_{1-\alpha/2}(n-1) \cdot \sqrt{\widehat{V}_{\widehat{m}(x)}}\\
\widehat{m}(x) \pm 1.96 \sqrt{\widehat{V}_{\widehat{m}(x)}}
\end{align}$$`

---

### （死亡率案例）CEF的置信区间和置信带（附表）

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---

### （死亡率案例）CEF的置信区间和置信带：断点左侧（控制组）

---

### （死亡率案例）CEF的置信区间和置信带：断点右侧（处置组）

---

### （死亡率案例）CEF的置信区间和置信带：断点两侧侧（对比）

---

### （死亡率案例）RDD断点处置效应：计算结果

- 根据**断点处置效应定理**，可以得到在断点
`$x=c=59.1984$`处对总体平均处置效应
`$\bar{\theta}$`的样本估计结果
`$\hat{\theta}$`：

`$$\begin{align}
\widehat{\theta}
&=\left[\boldsymbol{\widehat{\beta}_{1}}(c)\right]_{1}-\left[\boldsymbol{\widehat{\beta}_{0}}(c)\right]_{1}\\
&=\hat{m}(c+)-\widehat{m}(c-)\\
&=3.3096
-1.8035
=-1.5060
\end{align}$$`

- 断点处置效应估计值为
`$\hat{\theta}=-1.5060$`。

.small[
> 
- 断点左边的条件期望(CEF)的估计值
`$\widehat{m}(c-)=3.31$`；
- 断点右边的条件期望(CEF)的估计值
`$\widehat{m}(c+)=1.8$`；

]

- **结论**：援助项目的实施，减低了儿童死亡率，使得10万个孩子中约1.51个小孩免于遭受死亡。相比不实施项目援助，儿童死亡率由3.3096，下降到1.8035，降幅接近50%。

---

### （死亡率案例）RDD断点处置效应：估计误差及显著性检验

- 进一步地，估计系数
`$\hat{\theta}$`的**渐进方差**为两个方差协方差矩阵第一个对角元素之和：

`$$\begin{align}
\text{Var}{(\hat{\theta})}
&=\left[\widehat{\boldsymbol{V}}_{0}\right]_{11}+\left[\widehat{\boldsymbol{V}}_{1}\right]_{11}\\
&= 0.3673 + 0.1417
= 0.5090\\
se({(\hat{\theta})}) &= \sqrt{\text{Var}{(\hat{\theta})}}
= \sqrt{0.5090}
=0.7134
\end{align}$$`

.small[
> 
- 断点左边的条件期望(CEF)的估计值
`$\widehat{m}(c-)=3.3096$`；
- 断点右边的条件期望(CEF)的估计值
`$\widehat{m}(c+)=1.8035$`；

]

- **结论**：援助项目的实施，减低了儿童死亡率，使得10万个孩子中约-1.5060个小孩免于遭受死亡。相比不实施项目援助，儿童死亡率由3.3096，下降到1.8035，降幅接近50%。

---
exclude: true

### Code chunk：RDD equivalent simple OLS regression

---

### （死亡率案例）等价线性回归：调整运行变量范围

- 如前所述，**骤变RDD断点处置效应**也可以通过如下简单线性回归方法等价地得到
`$\widehat{\theta}$`的对应估计值：

`$$\begin{align}
Y=\beta_{0}+\beta_{1} X+\beta_{3}(X-c) D+\theta D+e
\end{align}$$`

- 简单地，上述等价模型需要进行样本数据集的重新定义。具体地，**运行变量**
`$X$`的范围需要调整到
`$X\in [c-h^{\ast}, c+h^{\ast}]$`，其中
`$h^{\ast}=\sqrt{3}h=\sqrt{3}\times 8=13.86$`

---

### （死亡率案例）等价线性回归：调整后的数据集

.pull-left[

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]

.pull-right[

- 样本数据的描述性统计如下：

```
       X            Y            D             XcD      
 Min.   :45   Min.   : 0   Min.   :0.00   Min.   : 0.0  
 1st Qu.:50   1st Qu.: 0   1st Qu.:0.00   1st Qu.: 0.0  
 Median :55   Median : 0   Median :0.00   Median : 0.0  
 Mean   :56   Mean   : 3   Mean   :0.34   Mean   : 1.8  
 3rd Qu.:62   3rd Qu.: 4   3rd Qu.:1.00   3rd Qu.: 2.4  
 Max.   :73   Max.   :65   Max.   :1.00   Max.   :13.8  
```

]

---

### （死亡率案例）等价线性回归：分组描述性统计

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---

### （死亡率案例）等价线性回归：OLS估计结果

`$$\begin{equation}
\begin{alignedat}{999}
&\widehat{Y}=&&-1.0987&&+0.0758X_i&&+0.0331XcD_i&&-1.5454D_i\\ 
&(s)&&(2.9382)&&(0.0564)&&(0.1060)&&(0.7375)\\ 
&(t)&&(-0.37)&&(+1.34)&&(+0.31)&&(-2.10)\\ 
&(over)&&n=757&&\hat{\sigma}=5.1830 && &&\\ 
&(fit)&&R^2=0.0059&&\bar{R}^2=0.0019 && &&\\ 
&(Ftest)&&F^*=1.48&&p=0.2191 && &&
\end{alignedat}
\end{equation}$$`

- 用上述等价回归法估计得到的断点处置效应估计值为
`$\widehat{\theta}=-1.5454$`，样本t统计量为
`$t^{\ast}=-2.10$`，对应的概率值为
`$p=0.0180$`，表明是统计显著的。

---
name: rdd-cov

### 2.5 协变量RDD：基本问题

- 回顾**断点处置效应**定理：

> 给定处置分配规则为
`$D=1\{X \geq c\}$`，而且假定结果变量满足断点处的连续性假设，也即结果变量的条件期望函数
`$m(x)$`在断点处
`$x=c$`连续，那么**断点处置效应**为：

`$$\bar{\theta}=m(c+)-m(c-)$$`

.fyi[

- 根据前面的讨论，就效应**估计**和**推断**而言，RDD分析中完全没有必要引入其他协变量（
`$Z$`）进入模型。

- 当然，为了提高模型预测**准确度**，我们可以引入一些额外的、有价值的协变量。

]

---

### 2.5 协变量RDD：符号表达

- 给定变量集为：
`$(Y,X,Z)$`，其中
`$Z$`为含有
`$k$`个元素的协变量向量（covariates vector）

- 同前，
`$Y_0$`和
`$Y_1$`分别为**控制条件**和**处置条件**下的**结果变量**（观测的或反事实的）

- 并进一步假定条件期望函数CEF是如下的**线性形式**，且断点两边的方程中协变量系数是相同的
`$\beta^{\prime}$`：

`$$\begin{align}
&\mathbb{E}\left[Y_{0} \mid X=x, Z=z\right]=m_{0}(x)+\beta^{\prime} z \\
&\mathbb{E}\left[Y_{1} \mid X=x, Z=z\right]=m_{1}(x)+\beta^{\prime} z
\end{align}$$`

- 那么，结果变量
`$Y$`的条件期望函数CEF将可以表达为：

`$$\begin{align}
m(x, z)=m_{0}(x) \cdot \mathbb{1}\{x< c\}+m_{1}(x) \cdot \mathbb{1}\{x \geq c\}+\beta^{\prime} z 
\end{align}$$`

- 此时，可以证明**断点处置效应**结果为：

`$$\overline{\theta} = m(c+,z) - m(c-,z)$$`

---

### 2.5 协变量RDD：估计方法

RDD协变量估计方法有很多种，这里重点讨论([Robinson, 1988](#bib-robinson1988))提出了一种**半参数效率**估计方法，主要步骤如下：

- **步骤1**：直接采用前面的RDD局部线性回归方法（RDD LLR），用
`$Y_i$`对
`$X_i$`进行回归，并得到第1阶段的**结果变量**的拟合值
`$\widehat{m}_i = \widehat{m}_i(X_i)$`

- **步骤2**：依次做
`$Z_{i1}$`对
`$X_i$`、
`$Z_{i2}$`对
`$X_i$`、
`$\ldots$`的局部线性回归Z（LL），并分别得到**协变量**的拟合值
`$\widehat{g}_{1i},\widehat{g}_{2i},\ldots,\widehat{g}_{ki}$`

- **步骤3**：做
`$Y_i -m_{i}$`对
`$Z_{i1}-\widehat{g}_{1i},Z_{i2}-\widehat{g}_{2i},\ldots,Z_{ik}-\widehat{g}_{ki}$`的回归，并得到**估计系数**
`$\hat{\beta}$`及其**标准误**

- **步骤4**：构造残差
`$\hat{e}_i=Y_i - Z^{\prime}_i\hat{\beta}$`

- **步骤5**：再次采用RDD局部线性回归方法（LLR），进行
`$\hat{e}_i$`对
`$X_i$`的回归，并计算得到非参数估计量
`$\widehat{m}(x)$`，断点效应估计值
`$\hat{\theta}$`及其标准误。

---
exclude: true

## Code script: Hansen fig21-2a  covirates

- 此代码仅用于生成本地图片，为提高渲染效率代码文件仅需运行一次。

- 如更新代码，请更改`eval = TRUE`

---

### （死亡率案例）背景说明

.case[

**案例说明**：

我们继续使用前面([Ludwig and Miller, 2007](#bib-ludwig2007))的研究案例，来评估美国联邦政府脱贫援助项目（Head Start）对儿童死亡率的**骤变RDD**政策效应。现在我们考虑使用两个协变量（covariates）：

- 县级**黑人人口占比**（`black pop percentage`）
`$Z_a$`

- 县级**城镇人口占比**（`urban pop percentage`）
`$Z_a$`

]

- 上述两个协变量，本质上可以视作为**收入**变量（`income`）的**代理变量**（proxy）。

- 下面我们将使用([Robinson, 1988](#bib-robinson1988))的**半参数效率**估计方法来评估项目援助的断点处置效应（RDD ATE）。

---

### （死亡率案例）样本数据集

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class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>obs<\/th>\n      <th>X<\/th>\n      <th>Y<\/th>\n      <th>Za<\/th>\n      <th>Zb<\/th>\n      <th>D<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"targets":2,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n  }"},{"targets":3,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n  }"},{"targets":4,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 1, 3, \",\", \".\", null);\n  }"},{"targets":5,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 1, 3, \",\", \".\", null);\n  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]

.pull-right[

- 样本数据的描述性统计如下：

```
       X            Y             Za           Zb            D       
 Min.   :15   Min.   :  0   Min.   : 0   Min.   :  0   Min.   :0.00  
 1st Qu.:24   1st Qu.:  0   1st Qu.: 0   1st Qu.:  0   1st Qu.:0.00  
 Median :34   Median :  0   Median : 2   Median : 28   Median :0.00  
 Mean   :37   Mean   :  2   Mean   :11   Mean   : 29   Mean   :0.11  
 3rd Qu.:47   3rd Qu.:  3   3rd Qu.:15   3rd Qu.: 48   3rd Qu.:0.00  
 Max.   :82   Max.   :136   Max.   :83   Max.   :100   Max.   :1.00  
```

]

---

### （死亡率案例）样本数据集：分组描述性统计

<div id="htmlwidget-15da17347e5dc6885cf0" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-15da17347e5dc6885cf0">{"x":{"filter":"none","vertical":false,"caption":"<caption>处置组和控制组描述性统计(g=2783)<\/caption>","data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17"],["zb_sd","zb_min","zb_mean","zb_max","za_sd","za_min","za_mean","za_max","y_sd","y_min","y_mean","y_max","x_sd","x_min","x_mean","x_max","n"],[26.9891060580212,0,31.0054720908391,100,12.8137145723433,0,7.87018883069652,75.5,5.85285505830407,0,2.23391840468932,136.054428100586,12.053219794862,15.2085123062134,33.2935062410746,59.1890487670898,2489],[18.0871433734761,0,13.3904762157828,93.6999969482422,26.3735391723135,0,33.9108843657689,83.4000015258789,4.51299091351648,0,2.42157824469261,29.8953666687012,5.25628600018557,59.198413848877,65.8677754304847,81.5702743530273,294]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>stats<\/th>\n      <th>D0<\/th>\n      <th>D1<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"targets":2,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 2, 3, \",\", \".\", null);\n  }"},{"targets":3,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 2, 3, \",\", \".\", null);\n  }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0}],"pageLength":10,"scrollX":true,"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render"],"jsHooks":[]}</script>

---

### （死亡率案例）协变量RDD：规则策略

在进行协变量RDD分析之前，我们设定如下的规则策略：

- 规则1：我们设定**先验谱宽**为
`$h=8$`，断点值设定为
`$c =59.1984\%$`。

- 规则2：分别设定断点两边**箱组中心点**序列值（center of bins）。我们将采用非对称箱组设置方法：

- 规则3：如果使用局部线性估计法（LL），则采用**三角核函数**（triangle kenerl）。

- 规则4：我们将使用([Robinson, 1988](#bib-robinson1988))的**半参数效率**估计方法来评估断点处置效应（RDD ATE）。

---

### （死亡率案例）协变量RDD：第1阶段LLR估计残差

- **步骤1**：直接采用前面的局部线性回归方法（LLR），用
`$Y_i$`对
`$X_i$`进行LL回归，得到第1阶段的**结果变量**的拟合值
`$\widehat{m}_i = \widehat{m}_i(X_i)$`，并进一步构造**留一法**残差<sup>a</sup> `$Y_i - \widehat{m}_i$`

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.footnote[
<sup>a</sup> 这个阶段的残差序列用`e`命名。
]

---

### （死亡率案例）协变量RDD：第2阶段LLR估计残差

- **步骤2**：同上，依次做
`$Z_a$`对
`$X_i$`、
`$Z_b$`对
`$X_i$`的**局部线性回归**（LLR），并分别得到**协变量**的拟合值
`$\widehat{g}_{1i},\widehat{g}_{2i}$`，及其对应残差<sup>a</sup>
`$(Z_a-\widehat{g}_{1i}),(Z_b-\widehat{g}_{2i})$`

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.footnote[
<sup>a</sup> 这个阶段的两个残差序列分别用`Ra`和`Rb`命名。
]

---

### （死亡率案例）协变量RDD：第3阶段OLS估计(模型)

- **步骤3**：利用前面两个阶段的残差，做
`$Y_i -m_{i}$`对
`$Z_{i1}-\widehat{g}_{1i},Z_{i2}-\widehat{g}_{2i},\ldots,Z_{ik}-\widehat{g}_{ki}$`的**无截距**的普通最小二乘回归（OLS），并得到**估计系数**
`$\hat{\beta}$`及其**标准误**

`$$\begin{align}
(Y_i -m_{i}) &= \hat{\beta}_1(Z_{ia}-\widehat{g}_{1i})+\hat{\beta}_2(Z_{ib}-\widehat{g}_{2i})\\
e&=\hat{\beta}_{1}R_a + \hat{\beta}_{2}R_a 
\end{align}$$`

---

### （死亡率案例）协变量RDD：第3阶段OLS估计（结果）

- 上述模型，未矫正标准误下OLS估计的结果如下<sup>a</sup> ：

`$$\begin{equation}
\begin{alignedat}{999}
&\widehat{e}=&&+0.0265Ra_i&&-0.0094Rb_i\\ 
&(s)&&(0.0083)&&(0.0045)\\ 
&(t)&&(+3.19)&&(-2.08)\\ 
&(p)&&(0.0014)&&(0.0377)\\ 
&(over)&&n=2783&&\hat{\sigma}=5.7091
\end{alignedat}
\end{equation}$$`

- 上述模型，进行稳健标准误矫正OLS估计的结果如下<sup>b</sup> ：

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.footnote[
<sup>a b</sup> 两种OLS估计程序下，回归系数都相同，只是系数对应的标准误不一样。这里我们仅需要用到回归系数，因此不影响后续步骤。
]

---

### （死亡率案例）协变量RDD：构造残差

- **步骤4**：利用前面的OLS估计系数，我们就可以构造得到残差
`$\hat{e}_i=Y_i - Z^{\prime}_i\hat{\beta}$`

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LL估计得到的残差(n=2783)<\/caption>","data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99","100","101","102","103","104","105","106","107","108","109","110","111","112","113","114","115","116","117","118","119","120","121","122","123","124","125","126","127","128","129","130","131","132","133","134","135","136","137","138","139","140","141","142","143","144","145","146","147","148","149","150","151","152","153","154","155","156","157","158","159","160","161","162","163","164","165","166","167","168","169","170","171","172","173","174","175","176","177","178","179","180","181","182","183","184","185","186","187","188","189","190","191","192","193","194","195","196","197","198","199","200","201","202","203","204","205","206","207","208","209","210","211","212","213","214","215","216","217","218","219","220","221","222","223","224","225","226","227","228","229","230","231","232","233","234","235","236","237","238","239","240","241","242","243","244","245","246","247","248","249","250","251","252","253","254","255","256","257","258","259","260","261","262","263","264","265","266","267","268","269","270","271","272","273","274","275","276","277","278","279","280","281","282","283","284","285","286","287","288","289","290","291","292","293","294","295","296","297","298","299","300","301","302","303","304","305","306","307","308","309","310","311","312","313","314","315","316","317","318","319","320","321","322","323","324","325","326","327","328","329","330","331","332","333","334","335","336","337","338","339","340","341","342","343","344","345","346","347","348","349","350","351","352","353","354","355","356","357","358","359","360","361","362","363","364","365","366","367","368","369","370","371","372","373","374","375","376","377","378","379","380","381","382","383","384","385","386","387","388","389","390","391","392","393","394","395","396","397","398","399","400","401","402","403","404","405","406","407","408","409","410","411","412","413","414","415","416","417","418","419","420","421","422","423","424","425","426","427","428","429","430","431","432","433","434","435","436","437","438","439","440","441","442","443","444","445","446","447","448","449","450","451","452","453","454","455","456","457","458","459","460","461","462","463","464","465","466","467","468","469","470","471","472","473","474","475","476","477","478","479","480","481","482","483","484","485","486","487","488","489","490","491","492","493","494","495","496","497","498","499","500","501","502","503","504","505","506","507","508","509","510","511","512","513","514","515","516","517","518","519","520","521","522","523","524","525","526","527","528","529","530","531","532","533","534","535","536","537","538","539","540","541","542","543","544","545","546","547","548","549","550","551","552","553","554","555","556","557","558","559","560","561","562","563","564","565","566","567","568","569","570","571","572","573","574","575","576","577","578","579","580","581","582","583","584","585","586","587","588","589","590","591","592","593","594","595","596","597","598","599","600","601","602","603","604","605","606","607","608","609","610","611","612","613","614","615","616","617","618","619","620","621","622","623","624","625","626","627","628","629","630","631","632","633","634","635","636","637","638","639","640","641","642","643","644","645","646","647","648","649","650","651","652","653","654","655","656","657","658","659","660","661","662","663","664","665","666","667","668","669","670","671","672","673","674","675","676","677","678","679","680","681","682","683","684","685","686","687","688","689","690","691","692","693","694","695","696","697","698","699","700","701","702","703","704","705","706","707","708","709","710","711","712","713","714","715","716","717","718","719","720","721","722","723","724","725","726","727","728","729","730","731","732","733","734","735","736","737","738","739","740","741","742","743","744","745","746","747","748","749","750","751","752","753","754","755","756","757","758","759","760","761","762","763","764","765","766","767","768","769","770","771","772","773","774","775","776","777","778","779","780","781","782","783","784","785","786","787","788","789","790","791","792","793","794","795","796","797","798","799","800","801","802","803","804","805","806","807","808","809","810","811","812","813","814","815","816","817","818","819","820","821","822","823","824","825","826","827","828","829","830","831","832","833","834","835","836","837","838","839","840","841","842","843","844","845"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.footnote[
<sup>a</sup> 这个步骤构造出来的残差序列`RZ`。
]

---

### （死亡率案例）协变量RDD：LLR估计CEF（附表）

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### （死亡率案例）协变量RDD：断点左侧CEF（控制组）

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### （死亡率案例）协变量RDD：断点右侧CEF（处置组）

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### （死亡率案例）协变量RDD：断点两侧（对比）

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### （死亡率案例）协变量RDD：标准差和置信区间（附表）

- 同前，进一步计算得到CEF估计值的方差和标准差以及95%置信区间

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### （死亡率案例）协变量RDD：断点左侧置信带（控制组）

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### （死亡率案例）协变量RDD：断点右侧置信带（处置组）

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### （死亡率案例）协变量RDD：断点两侧

---

### （死亡率案例）协变量RDD：断点处置效应

- 根据**断点处置效应定理**，可以得到在断点
`$x=c=59.1984$`处对总体平均处置效应
`$\bar{\theta}$`的样本估计结果
`$\hat{\theta}$`：

- 断点处置效应估计值为
`$\hat{\theta}=-1.5617$`。

.small[
> 
- 断点左边的条件期望(CEF)的估计值
`$\widehat{m}(c-)=2.8209$`；
- 断点右边的条件期望(CEF)的估计值
`$\widehat{m}(c+)=1.2592$`；

]

- **结论**：援助项目的实施，减低了儿童死亡率，使得10万个孩子中约-1.5617个小孩免于遭受死亡。相比不实施项目援助，儿童死亡率由2.8209，下降到1.2592，降幅接近50%。

---

### （死亡率案例）协变量RDD：估计误差及显著性检验

- 进一步地，估计系数
`$\hat{\theta}$`的**渐进方差**为两个方差协方差矩阵第一个对角元素之和：

- 因此RDD断点处置效应估计值
`$\hat{\theta}$`的**标准误**为
`$se({\hat{\theta}}) =0.7122$`；最后我们可以计算得到**RDD断点处置效应**对应的t统计量：
`$t^{\ast}=\frac{\hat{\theta}}{se(\hat{\theta})}=-2.19$`，其概率值为
`$p=0.0283$`.

- 综上，RDD结果表明援助项目降低了儿童死亡率，使得10万个孩子中约1.51个小孩免于遭受死亡。并且t统计量检验表明，援助项目在降低了儿童死亡率上具有统计显著性（置信度超过95%）。

---
exclude: true

### Code chunk： compare result

- 比较Hansen Head Start案例基准RDD和协变量RDd

- 需要事先依次运行`fig21-1b.R`和`fig21-2a.R`

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### （死亡率案例）总结：系数和标准误比较

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- 结论1：与基准RDD相比，两个协变量的引入没有明显改变断点处置效应估计值大小。

- 结论2：但是是否引入协变量，对CEF估计值
`$\widehat{m}(x)$`的影响较大。可以看到基准RDD更陡峭，而协变量RDD更平缓。（见后面附图对比）

- 结论3：考虑到两个协变量可以视作收入的**代理变量**，可以看到黑人人口比重负向影响儿童死亡率，而城镇人口比重正向影响儿童死亡率。

---

### （死亡率案例）总结：CEF估计值图形比较1/2

<div class="figure" style="text-align: center">
<img src="seminar-RDD-part02_files/figure-html/unnamed-chunk-61-1.svg" alt="基准RDD：局部线性回归及断点效应估计" height="450" />
<p class="caption">基准RDD：局部线性回归及断点效应估计</p>
</div>

---

### （死亡率案例）总结：CEF估计值图形比较2/2

<div class="figure" style="text-align: center">
<img src="seminar-RDD-part02_files/figure-html/unnamed-chunk-62-1.svg" alt="协变量RDD：局部线性回归及断点效应估计" height="450" />
<p class="caption">协变量RDD：局部线性回归及断点效应估计</p>
</div>

---
layout: false
class: center, middle, duke-orange,hide_logo
name: pro

# 3.RDD怎样高级进阶？</br></br>(How the Pros Do It?)

### [3.1 模糊RDD](#fuzzy-rdd)

### [3.2 拐点RDD (RKD)](#rkd)

### [3.3 多断点RDD](#multi-cutoff)

### [3.4 安慰剂检验](#placebo-test)

---
layout: true

<div class="my-footer"><span><a href="https://home.huhuaping.com/">https://home.huhuaping.com </a> &emsp;&emsp; <a href="#chapter-RDD02"> RDD Part02 断点效应评估 </a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#pro"> 3.RDD怎样高级进阶？ </a> </span></div>

---
exclude: true

## Code script: Hansen fig21-3

---
name: fuzzy-rdd

### 3.1 模糊RDD分析

**模糊RDD**（fuzzy regression discontinuity design,FRDD）：是指处置条件的条件分配概率在断点处是不连续的（跳跃的），但又不是从0直接跳跃到1的一种RDD分析情形。

> **骤变RDD**中，在断点两边，处置条件的条件分配概率在断点处是跳跃的，而且是直接从0跳跃到1。

- 我们定义处置条件的**条件分配概率**为：

`$$p(x) = \mathbb{P}[D=1|X=x]$$`

- 那么在断点
`$x=c$`左右两边的极限**条件分配概率**则分别定义为：
`$p(c-),p(c+)$`。

- 因此，对于**模糊RDD**而言，则意味着：
`$p(c-)\neq p(c+)$`

---

### （示例）模糊RDD分析：处置水平的分配概率

---

### （示例）模糊RDD分析：反事实与条件期望函数

---

### 3.1 模糊RDD分析：断点ATE定理（表达）

.fyi[

**定理**：模糊RDD下的断点处置效应ATE。

- 假定
`$m_0(x)$`和
`$m_1(x)$`在点
`$x=c$`处连续，
`$p(x)$`在点
`$x=c$`处不连续，且
在断点附近处置变量
`$D$`与真实参数
`$\theta|X$`相互独立，则断点处置效应ATE为：

`$$\begin{align}
\bar{\theta}=\frac{m(c+)-m(c-)}{p(c+)-p(c-)}
\end{align}$$`

]

---

### 3.1 模糊RDD分析：断点ATE定理（证明）

此时，我们考虑如下的模型：

`$$\begin{align}
\begin{aligned}
Y &=Y_{0} \mathbb{1}\{D=0\}+Y_{1} \mathbb{1}\{D=1\} \\
&=Y_{0}+\theta \mathbb{1}\{D=1\}
\end{aligned}
\end{align}$$`

- 对两边在
`$x=c$`附近同时取期望：

`$$\begin{align}
\begin{aligned}
m(x) =m_{0}(x)+\mathbb{E}[\theta \mathbb{1}\{D=1\} \mid X=x] 
=m_{0}(x)+\theta(x) p(x)
\end{aligned}
\end{align}$$`

- 在
`$x=c$`处取极限，则有：

`$$\begin{align}
m(c+)=m_{0}(c)+\bar{\theta} p(c+); \quad \quad 
m(c-)=m_{0}(c)+\bar{\theta} p(c-) \\
\end{align}$$`

- 最后可以证明：

`$$\begin{align}
m(c+)- m(c-) = \bar{\theta} p(c+) -\bar{\theta} p(c-) 
\Rightarrow \quad  \quad
\bar{\theta} = \frac{m(c+)- m(c-)}{ p(c+) -  p(c-) }
\end{align}$$`

---

### 3.1 模糊RDD分析：断点ATE的估计（精简表达法）

- 对于分母部分，我们可以使用前面介绍的局部线性回归（LLR）对总体参数
`$m(c+)- m(c-)$`进行估计，得到断点
`$x=c$`附近的两边的估计值：

`$$\begin{align}
\widehat{m}(c+)- \widehat{m}(c-) 
& \equiv [\widehat{\beta}_1(c)]_1 - [\widehat{\beta}_0(c)]_1
\end{align}$$`

- 同理，分子部分我们也同样使用局部线性回归（LLR）对总体参数
`$p(c+)- p(c-)$`进行估计，得到断点
`$x=c$`附近的估计值：

`$$\begin{align}
\widehat{p}(c+)- \widehat{p}(c-) 
& \equiv [\widehat{\alpha}_1(c)]_1 - [\widehat{\alpha}_0(c)]_1
\end{align}$$`

- 最终，我们可以得到断点ATE的估计值为：

`$$\begin{align}
\widehat{\theta} &= \frac{\widehat{m}(c+)- \widehat{m}(c-)}{ \widehat{p}(c+) -  \widehat{p}(c-) }
\end{align}$$`

---

### 3.1 模糊RDD分析：断点ATE的估计（精简表达法）

对于模糊RDD断点ATE的估计：

`$$\begin{align}
\widehat{\theta} &= \frac{\widehat{m}(c+)- \widehat{m}(c-)}{ \widehat{p}(c+) -  \widehat{p}(c-) }
\end{align}$$`

- 上式实际上是两类断点估计的比率值。而且当
`$\widehat{p}(c+) -  \widehat{p}(c-) =1$`时，以上估计式即为**骤变RDD**的估计情形！

- 上述估计值的计算总共会需要进行4次**局部线性回归**，是不是需要都使用相同的**谱宽**（bandwidth），或者断点两侧是否要采用不同数量的**箱组**（bins），可以进行多次尝试！

---

### 3.1 模糊RDD分析：断点ATE的估计（IV表达法）

事实上，上述模糊RDD断点ATE的估计可以使用**工具变量法**（IV）等价得到。

- 简单地，可以把
`$D$`视作为
`$X$`的工具变量，然后把
`$Y$`对它们二者进行**局部加权工具变量估计**（Locally weighted IV estimation），从而得到断点ATE估计值
`$\widehat{\theta}$`。

- 断点处置效应能否被识别，有赖于在断点附近处概率
`$p(x)$`的跳跃性程度。如果跳跃不大，那么就会带来**弱工具变量**问题（Weak Instruments Problem）。

- ATE估计的标准误，其计算过程类似于IV回归法。我们先把估计量
`${\widehat{m}(c+)- \widehat{m}(c-)}$`的标准误定义为
`$s(\widehat{\theta})$`，那么就可以使用下式计算得到ATE估计
`$\widehat{\theta}$`的标准误：

`$$\begin{align}
s(ate)=\frac{s(\widehat{\theta})}{|\hat{p}(c+)-\widehat{p}(c-)|}
\end{align}$$`

---
exclude: true

## Code script: Huntington22-fig-20-16

---
name: rkd

### 3.2 拐点回归RKD分析：引子

.fyi[
**回顾与思考**：

- 断点回归设计（RDD）探讨的是**结果变量**
`$Y$`的条件期望值（均值）
`$\mathbb{E}(Y|X=x)\equiv m(x)$`在**断点**附近是否存在跳跃性（jump）的不连续。

- 那么，我们能不能分析除此之外，其他对象的跳跃性或不对称性呢？

> 例如结果变量的**标准差**是否跳跃？中位数是否跳跃？或者**箱组内**（bins）局部回归的判定系数
`$R^2$`是否跳跃？

]

---

### 3.2 拐点回归RKD分析：问题描述

**拐点回归设计**（Regression Kink Design, RKD）：是探讨**结果变量**
`$Y$`对**运行变量**
`$X$`的**斜率**（slope）是否存在显著改变（change）的一种处置效应回归分析设计框架。

- 处置条件只是改变了**斜率**，但是并没有引起结果变量的跳跃，也即结果变量在**拐点处**（kink point）还是连续的！

- 在有些情形下，研究者还可以关注**处置水平**D对**运行变量**X的变化率的**拐点效应**。

---

### （示例）拐点回归RKD分析：基于模拟数据

<div class="figure" style="text-align: center">
<img src="seminar-RDD-part02_files/figure-html/unnamed-chunk-67-1.svg" alt="结果变量对运行变量的变化率（斜率）具有拐点效应" height="400" />
<p class="caption">结果变量对运行变量的变化率（斜率）具有拐点效应</p>
</div>

---

### 3.2 拐点回归RKD分析：应用案例1

- 讨**结果变量**
`$Y$`对**运行变量**
`$X$`的变化率（**斜率**）具有**拐点效应**：

.case[
**案例1**：政府择机扶持科技公司。

- 政府在特定时间点开始，决定大量关注并投资科技公司。此时公司雇员人数为**结果变量**Y，时间观测为**运行变量**X，政府是否决定大量投资则为**处置变量**D。

- 这种情况下，公司雇员总人数Y可能并不会在**拐点**
`$x=c$`处立刻跳跃（不连续），但是我们预期在**拐点后**的公司的雇员增长率（Y对X的斜率）会比之前会有一个明显变化！

]

---

### 3.2 拐点回归RKD分析：应用案例2

- **结果变量**
`$Y$`对**运行变量**
`$X$`的变化率（**斜率**）具有**拐点效应**：

.case[

**案例2**：失业保险政策([Card, Lee, Pei, et al., 2015](#bib-card2015))。

- 案例背景为澳大利亚。公民的失业保险补贴水平大概为其正常工作收入的55%，并且有一个补贴最高上限值。因此，失业保险政策设计下，公民的正常工作收入会正向地影响失业保险补贴水平。工作收入越高，补贴会越多，直到达到一个补贴上限值。

- 此时，我们定义：公民的正常工作收入为**运行变量**X，公民是否能获得失业补贴为**处置变量**D。一个公民如果失业，把他从失业那一刻算起，直到他找到一份新工作，期间他所愿意的**等待时长**定义为**结果变量**Y，

]

---

### 3.2 拐点回归RKD分析：应用案例2

- **结果变量**
`$Y$`对**运行变量**
`$X$`的变化率（**斜率**）具有**拐点效应**：

.case[

**案例2(续)**：失业保险政策([Card, Lee, Pei, et al., 2015](#bib-card2015))。

- 这种情况下，政策如果给予更高的补贴水平，那么我们可以预期公民的就业等待时长Y可能会更长！因此，我们也可以预期，在达到最高补贴水平（拐点c）之前，公民正常的工作收入X越高，那么他的就业等待时长Y也会更长！

- 显然，在拐点之后（最高补贴之后），**等待时长**Y对正常**工作收入**X的比率应该会变得比拐点之前更加平缓（斜率更小）！——也即出现了Y对X的斜率具有**拐点效应**！同时，我们还可以预期到就业**等待时长**Y并不会在**拐点**
`$x=c$`处立刻跳跃（不连续）！

]

---

### 3.2 拐点回归RKD分析：应用案例3

- **处置变量**D对**运行变量**
`$X$`的变化率（**斜率**）具有**拐点效应**：

.case[

**案例3**：妇女育儿支持政策([Bana, Bedard, and Rossin-Slater, 2020](#bib-bana2020))。

- 案例背景为美国加利福尼亚州。政府制定了一项**妇女育儿家庭支持**政策（paid family leave）。对于符合条件的家庭，加州政府根据家庭正常工作收入，将补贴其家庭收入的55%直至一个最高最高上限值。

- 因此，**妇女育儿家庭支持**政策设计下，家庭的正常工作收入会正向地影响补贴水平。工作收入越高，补贴会越多，直到达到一个补贴上限值。

]

---

### 3.2 拐点回归RKD分析：应用案例3

- **处置变量**D对**运行变量**
`$X$`的变化率（**斜率**）具有**拐点效应**：

.case[

**案例3（续）**：妇女育儿支持政策([Bana, Bedard, and Rossin-Slater, 2020](#bib-bana2020))。

- 此时，我们定义：家庭的正常工作收入为**运行变量**X，家庭获得政策补贴水平为**处置变量**D。妇女获得的**育儿假时长**为结果变量Y。

- 显然，在拐点之前（最高补贴之前），家庭的**政策补贴水平**D越高，也意味着家庭正常**工作收入**X越高；而在拐点之后（最高补贴之后），家庭的**政策补贴水平**D会保持不变——也即意味着D对X的斜率为0！

- 因此，**处置变量**D对**运行变量**
`$X$`的变化率（**斜率**）具有**拐点效应**：

]

---
exclude: true

## Code script: Huntington22-fig-20-17 and 20-18

---

### （育儿支持案例）处置变量对运行变量的拐点效应

<div class="figure" style="text-align: center">
<img src="seminar-RDD-part02_files/figure-html/unnamed-chunk-69-1.svg" alt="处置变量对运行变量的变化率（斜率）具有拐点效应" height="400" />
<p class="caption">处置变量对运行变量的变化率（斜率）具有拐点效应</p>
</div>

---

### （育儿支持案例）结果变量对运行变量的拐点效应1/2

- 对于不打算再要孩子的家庭

<div class="figure" style="text-align: center">
<img src="seminar-RDD-part02_files/figure-html/unnamed-chunk-70-1.svg" alt="结果变量对运行变量的变化率（斜率）没有拐点效应" height="400" />
<p class="caption">结果变量对运行变量的变化率（斜率）没有拐点效应</p>
</div>

---

### （育儿支持案例）结果变量对运行变量的拐点效应2/2

- 对于打算再要孩子的家庭，不仅仅只是**拐点**效应，而是**断点**效应

<div class="figure" style="text-align: center">
<img src="seminar-RDD-part02_files/figure-html/unnamed-chunk-71-1.svg" alt="结果变量对运行变量的变化率（斜率）具有拐点效应" height="400" />
<p class="caption">结果变量对运行变量的变化率（斜率）具有拐点效应</p>
</div>

---

### 3.2 拐点回归RKD分析：拐点效应ATE估计

总体而言，关于拐点效应ATE的估计方法，与之前的RDD估计过程基本类似：

- 确定核函数

- 选择谱宽

- 局部线性回归LLR或局部多项式回归LPR

- 安慰剂效应检验

.fyi[

**编程提示**：

- 对于`R`或`stata`用户而言，可以使用分析包`rdrobust`

- 拐点回归RKD估计时，仅需要设定参数`deriv=1`即可

]

---
name: multi-cutoff

### 3.3 多断点RDD：引子

.fyi[
**回顾与思考**：

- 截止目前为止，我们已经接触了骤变断点（SRDD）、模糊断点（FRDD）、斜率改变拐点（RKD）

- 那么，我们能不能考虑政策存在多个断点（或拐点）的情形呢？

]

---

###（示例）多断点RDD应用案例

.case[

**应用案例**：

- 在**育儿支持案例**中，能拿到**最高**补贴支持的**家庭季度收入**（x=c），也是随着年度变化而进行调整的。例如在2004年这个收入水平划定为25000美元，而到了2005年则被划定在20000美元。

- 一国猪肉储备投放政策中，会考虑根据猪粮比价（X）变动，分别设定红色、橙色、蓝色和绿色预警窗口（多个断点），来决定如何干预市场（如生产收储或市场投放，以及数量多少等）。

]

---

###（示例）多断点RDD应用案例

.case[

**应用案例**（续）：

- 高考招生政策中，会考虑根据不同招生类型（如普通招录生、师范特招生、体育特招生等），设定不同的高考成绩录取线（多个断点）。而且普通高考招录的录取线，对于不同的省份也是不同的（例如，某省的某个高校在全国各省的招录录取线就会各不相同——一般本省录取线会更低）。

- 在多党派多家的政党选举中，某个政党能否竞选胜出执政，有的年度可能需要50.1%的投票率，但是有的年份可能只需要42.7%就能胜选。

]

---

### 3.3 多断点RDD：问题描述

- 很多情况下，**断点**（或拐点）本身就是政策指定者最为关注的议题

- 一些情形下，**多个断点**的政策设计具有很强的现实意义或价值。

**多断点分析**（Cutoffs cut off Analysis）：在运行变量X上，存在多个**断点**，断点值的划定，往往基于不同群体、不同地区，或不同时间段上的运行变量取值。

---

### 3.3 多断点RDD：断点ATE的估计

.fyi[
**回顾与启发**：

- 在经典的RDD估计中，**断点平均处置效应**（ATE）是把断点附近的处置效应做了简单平均（正如其名！）。

- 但是，在**多断点**的RDD情形下，事情变得复杂（不同的断点针对不同的群体），因此不能再直接、粗暴地进行**简单平均**——我们必须考虑到不同的群体区块！

]

.notes[

**编程提示**：

- 对于`R`或`stata`用户而言，可以使用分析包`rdmulti`

- 多断点RDD估计时，可以使用函数`rdmulti::rdmc()`进行分析

]

---
exclude: true

## Code script: Huntington22-fig-20-14

---
name: placebo-test

### 3.4 安慰剂检验：原理

**安慰剂检验**（Placebo Tests）：RDD分析的前提假设是，处置变量的作用是“干净的”、没有后门的（no back doors）。如果不使用结果变量Y，而是使用**协变量**作为“结果变量”进行正常的RDD估计，如果也表现出与之前同样显著的断点处置效应ATE，那么我们就要质疑我们的RDD设计框架了。

- 选择合理的协变量，将其视作为“结果变量”

- 进行常规的RDD分析流程

- 比较结果并得出检验结论。

> 理论上，上述操作不应该得到——**“显著存在断点处置效应”**——的结论！

---

### （政府转移支付案例）背景

.case[

**政府转移支付案例**：

- ([Manacorda, Miguel, and Vigorito, 2011](#bib-manacorda2011))分析了乌拉圭的一个大型扶贫项目，该项目削减了很大一部分贫困人口。论文关注的话题是：获得政府转移支付资金是否会让人们更有可能支持新成立的政府？

- 研究人员对一群接近收入临界值的人进行了调查，看看他们之后对政府的支持程度。收入低于临界值的人比收入高于临界值的人支持政府要更多吗？

]

---

### （政府转移支付案例）年龄协变量下的安慰剂检验

-  以协变量年龄作为结果变量进行安慰剂检验

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### （政府转移支付案例）教育协变量下的安慰剂检验

-  以协变量教育年数作为结果变量进行安慰剂检验

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# 本章参考文献

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<div class="my-footer"><span>huhuaping@  &emsp;&emsp; <a href="#chapter"> 第00章 课程说明 </a>
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<a href="#reference-ch02"> 本章参考文献 </a> </span></div>

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### 参考文献（References）：1/3

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<a name=bib-arai2018></a>[Arai, Y. and H.
Ichimura](#cite-arai2018) (2018). "Simultaneous Selection of
Optimal Bandwidths for the Sharp Regression Discontinuity
Estimator". In: _Quantitative Economics_ 9.1, pp. 441-482.

<a name=bib-bana2020></a>[Bana, S. H., K. Bedard, and M.
Rossin-Slater](#cite-bana2020) (2020). "The Impacts of Paid
Family Leave Benefits: Regression Kink Evidence from
California Administrative Data". In: _Journal of Policy
Analysis and Management_ 39.4, pp. 888-929.

<a name=bib-card2015></a>[Card, D., D. S. Lee, Z. Pei, et
al.](#cite-card2015) (2015). "Inference on Causal Effects in a
Generalized Regression Kink Design". In: _Econometrica_ 83.6,
pp. 2453-2483.

<a name=bib-cattaneo2021></a>[Cattaneo, M. D. and R.
Titiunik](#cite-cattaneo2021) (2021). "Regression
Discontinuity Designs" , p. 48.

<a name=bib-fan1996></a>[Fan, J., I. Gijbels, T. Hu, et
al.](#cite-fan1996) (1996). "A Study of Variable Bandwidth
Selection for Local Polynomial Regression". In: _Statistica
Sinica_, pp. 113-127.

]

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### 参考文献（References）：2/3

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<a name=bib-hausman2018></a>[Hausman, C. and D. S.
Rapson](#cite-hausman2018) (2018). "Regression Discontinuity
in Time: Considerations for Empirical Applications". In:
_Annual Review of Resource Economics_ 10.1, pp. 533-552. DOI:
[10.1146/annurev-resource-121517-033306](https://doi.org/10.1146%2Fannurev-resource-121517-033306).

<a name=bib-imbens2012c></a>[Imbens, G. and K.
Kalyanaraman](#cite-imbens2012c) (2012). "Optimal Bandwidth
Choice for the Regression Discontinuity Estimator". In: _The
Review of economic studies_ 79.3, pp. 933-959.

<a name=bib-ludwig2007></a>[Ludwig, J. and D. L.
Miller](#cite-ludwig2007) (2007). "Does Head Start Improve
Children's Life Chances? Evidence from a Regression
Discontinuity Design". In: _The Quarterly journal of
economics_ 122.1, pp. 159-208.

<a name=bib-manacorda2011></a>[Manacorda, M., E. Miguel, and
A. Vigorito](#cite-manacorda2011) (2011). "Government
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Journal: Applied Economics_ 3.3, pp. 1-28.

<a name=bib-robinson1988></a>[Robinson, P.
M.](#cite-robinson1988) (1988). "Root-N-consistent
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]

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### 参考文献（References）：3/3

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<a name=bib-thistlethwaite1960></a>[Thistlethwaite, D. L. and
D. T. Campbell](#cite-thistlethwaite1960) (1960).
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Post Facto Experiment." In: _Journal of Educational
psychology_ 51.6, p. 309.

]

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# 本章结束