10-dummy-model-slide.knit

# 计量经济学(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

### 2023-02-15

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---
class: center, middle, duke-orange,hide_logo
name: chapter10

# 第10章：虚拟变量回归模型

[10.0 相关知识回顾](#review)

[10.1 虚拟变量的设置规则](#rule)

[10.2 方差分析(ANOVA) 模型](#dummy-mod)

[10.3 只含有一个定性变量的ANOVA模型](#one)

[10.4 同时含有一个定性和一个定量变量的ANOVA 模型](#one-one)

]

[10.5 同时含有多个定性和定量变量的ANOVA 模型](#multiple)

[10.6 印度工人工资案例](#example)

[10.7 时间序列季节虚拟变量模型](#series)

[10.8 分段线性回归模型](#piecewise)

]

---
layout: false
class: center, middle, duke-softblue,hide_logo
name: review

# 10.0 相关知识回顾

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<div class="my-footer"><span>huhuaping@   <a href="#chapter10">第10章 虚拟变量回归模型</a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
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<a href="#review">10.0 相关知识回顾</a> </span></div>

---

## 变量类型

**定量变量**（Quantitative variable）一般也称为连续变量，是由测量或计数、统计所得到的量，可以通过数值表达，并具有直接的数值含义。

**定性变量**（qualitative variables）：又被称为指标变量（indicator variables）、分类变量（categorical variables），主要用于区分事物性质差异，往往用语义类别表达，没有直接的数值含义。

- 性别（男；女）

- 肤色（黄色；白色；黑色；其他）

- 种族、宗教、国籍、地区、政治动乱和党派等。

> **提问**：定性变量怎样表达出来？如何数量化？

---

## 变量尺度

**变量尺度**（Variable scale）：刻画的是变量的数值含义或数值关系。它将意味着在数值含义和关系上，变量是有层次级别的差异性。根据变量层级不同，具体可以分为由低到高的4个层级：

- **名义尺度（nominal scale）变量**：这类变量只用于属性分类，不具备任何数值含义或数值关系，也即不能加、减、乘、除，也不能比较大小。

- **序数尺度（order scale）变量**：这类变量具备很少的数值含义或数值关系，它可以比较大小，但不能进行加、减、乘、除。

- **区间尺度（interval scale）变量**：这类变量具备一定的数值含义或数值关系，它可以比较大小，也可以进行加、减，但不能进行乘、除。

- **比率尺度（ratio scale）变量**：这类变量具备最多的数值含义或数值关系，它可以比较大小，也可以进行加、减、乘、除。

---

### 区域经理年薪案例（数据）

<div id="htmlwidget-7dc9320d0de652a03d49" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-7dc9320d0de652a03d49">{"x":{"filter":"none","vertical":false,"caption":"<caption>变量类型和变量尺度<\/caption>","data":[["Zhang","Tom","Lee","Jeff","Kevin"],[25,35,27,42,38],[140,195,184,256,207],[2,3,1,5,4],["yellow","white","yellow","white","black"]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>staff<\/th>\n      <th>salary<\/th>\n      <th>sale<\/th>\n      <th>score<\/th>\n      <th>race<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

**区域经理年薪案例**中，公司有五名区域经理，分别负责不同的国际市场。

>变量
`$salary$`表示区域经理的年薪（万元）；变量
`$sale$`表示负责市场的销售额；变量
`$score$`表示客户对区域经理的评价（1表示很不满意，2表示不满意，3表示一般，4表示很满意，5表示非常满意）；变量
`$race$`表示区域市场主要消费群体的肤色（yellow表示黄色消费群体、white表示白色消费群体，black表示黑色消费群体）。

---

### 区域经理年薪案例（变量）

<div id="htmlwidget-80a75c85b99890c3b7cf" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-80a75c85b99890c3b7cf">{"x":{"filter":"none","vertical":false,"caption":"<caption>变量类型和变量尺度<\/caption>","data":[["Zhang","Tom","Lee","Jeff","Kevin"],[25,35,27,42,38],[140,195,184,256,207],[2,3,1,5,4],["yellow","white","yellow","white","black"]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>staff<\/th>\n      <th>salary<\/th>\n      <th>sale<\/th>\n      <th>score<\/th>\n      <th>race<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

根据以上定义，**区域经理年薪案例**中，可以认为年薪
`$salary$`、销售额
`$sale$`，以及客户评价
`$score$`为**定量变量**，消费群体主要肤色
`$race$`为**定性变量**。

>从变量的度量尺度来看：
- 年薪
`$salary$`和销售额
`$sale$`两个变量为**比率尺度**变量
- 客户评价
`$score$`变量为**序数尺度**变量
- 消费群体主要肤色
`$race$`为**名义尺度**变量

---
layout: false
class: center, middle, duke-softblue,hide_logo
name: rule

# 10.1 虚拟变量的设置规则

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## 定性变量对回归模型的影响

计量经济学建模分析中，我们常常需要把一些**定性变量**（Qualitative variables）（如性别、地区、党派等）作为自变量放入回归模型中。

从**变量层次**（Variable Scale）来看，这些变量没有具体的取值，只有特定属性类别。例如，性别变量的具体取值往往为男或女。显然，诸如此类的变量如果直接放到线性回归模型中，将会产生一系列的参数估计、模型解释等问题。

`$$\begin{align}
salary_i = \beta_1 +\beta_2sale_i +\beta_3score_i + \beta_4 race_i + u_i
\end{align}$$`

- 一个**定性变量**的不同数据取值，称为该定性变量的**属性**。

- 定性变量的任一属性，都可以设置为一个**虚拟变量**。

- 我们可以用一套**虚拟变量体系**来完全表达一个定性变量。

- 按照一定规则构建虚拟变量回归模型，避免参数估计、模型解释等问题的出现。

---

## 虚拟变量的定义

**虚拟变量**（dummy variable）：将取值为0和1的**人造变量**称为虚拟变量。

- 对定性变量的量化可采用虚拟变量的方式实现。

- 一般而言，1表示出现（或具备）某种属性，0表示没有（或不具备）某种属性。

对于某定性变量的任一特定属性，可以构造出一个虚拟变量（记为D），使得该虚拟变量能够表达这一属性。同时，给该虚拟变量D赋值为1，记为具备这一属性；给该虚拟变量赋值为0，记为不具备该属性。

正式地，假设定性变量
`$X$`具有
`$m$`个属性
`$a_1,a_2,\cdots,a_m$`，对于任意属性
`$k,(k\in{1,2,\cdots,m})$`，可以定义如下的虚拟变量
`$D_k$`：

`$$\begin{align}
D_k=
  \begin{cases}
  1, & \text{if } a_k \\
  0, & \text{if not }  a_k
  \end{cases}
\end{align}$$`

---

### 区域经理年薪案例（虚拟变量）

**区域经理年薪**案例中，**定性变量**`race`（人种，其取值为黄种人/白种人/黑种人），可以构造出3个虚拟变量

`$$\begin{align}
race\{a_1=\text{yellow},a_2=\text{white},a_3=\text{black}\} \\
dummy \Longrightarrow  
  \begin{cases}
    D_1  =
    \begin{cases}
    1, & \text{yellow}\\
    0, & \text{not yellow}
    \end{cases} \\
    D_2 = 
    \begin{cases}
    1, & \text{white}\\
    0, & \text{not white}
    \end{cases} \\
    D_3  =
    \begin{cases}
    1, & \text{black}\\
    0, & \text{not black}
    \end{cases}
  \end{cases}
\end{align}$$`

---

## 虚拟变量体系

**虚拟变量体系**：完整表达某个定性变量全部信息的一组虚拟变量。

正式地，假设定性变量
`$X$`具有
`$m$`个属性
`$a_1,a_2,\cdots,a_m$`，可以用如下一组虚拟变量
`$D_1,\cdots,D_k,\cdots,D_m$`完全表达该定性变量：

`$$\begin{align}
X\{a_1,a_2,\cdots,a_m\} \Rightarrow  
  \begin{cases}
    D_1  =
    \begin{cases}
    1, & \text{if } a_1\\
    0, & \text{if not }  a_1
    \end{cases} \\
     \vdots \\
    D_k = 
    \begin{cases}
    1, & \text{if } a_k\\
    0, & \text{if not }  a_k
    \end{cases} \\
    \vdots\\
    D_m  =
    \begin{cases}
    1, & \text{if } a_m\\
    0, & \text{if not }  a_m
    \end{cases}
  \end{cases}
\end{align}$$`

---

### 区域经理年薪案例（虚拟变量体系）

实际数据操作中，一般需要对定性变量
`$race$`进行重新编码（recode），生成三个对应的虚拟变量。

<div id="htmlwidget-452f946373ec4359ae1c" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-452f946373ec4359ae1c">{"x":{"filter":"none","vertical":false,"caption":"<caption>把定性变量转变为虚拟变量体系<\/caption>","data":[["Zhang","Tom","Lee","Jeff","Kevin"],[25,35,27,42,38],[140,195,184,256,207],[2,3,1,5,4],["yellow","white","yellow","white","black"],[0,0,0,0,1],[0,1,0,1,0],[1,0,1,0,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>staff<\/th>\n      <th>salary<\/th>\n      <th>sale<\/th>\n      <th>score<\/th>\n      <th>race<\/th>\n      <th>race_black<\/th>\n      <th>race_white<\/th>\n      <th>race_yellow<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

---
layout: false
class: center, middle, duke-softblue,hide_logo
name: dummy-mod

# 10.2 方差分析模型模型（ANOVA model）

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## 定义

**区域经理薪水案例**中，如果不区分变量类型和特征，做如下的回归模型，则回归分析结果将会带来严重的问题。

`$$\begin{align}
salary_i = \beta_1 +\beta_2sale_i +\beta_3score_i + \beta_4 race_i + u_i
\end{align}$$`

事实上，应该将上述模型转换为**虚拟变量回归模型**（Dummy model）。

`$$\begin{align}
salary_i & ={\beta}_1 + &&\beta_2sale_i +\beta_3score_i+{\beta}_4race\_yellow_i+{\beta}_5race\_white_i+u_i \\
salary_i & = &&\beta_2sale_i +\beta_3score_i+{\beta}_4race\_yellow_i+{\beta}_5race\_white_i +{\beta}_6race\_black_i+u_i \\
\end{align}$$`

---

## 定义

一个线性回归模型，只要回归元中包含了虚拟变量，这种模型就被称为**虚拟变量回归模型**，也可以称为**方差分析模型** （Analysis of variance, ANOVA）。

**方差分析模型**（Analysis of variance, ANOVA）常用来分析定量化的因变量
`$Y$`与定性回归元或虚拟变量之间的统计显著性关系。

一般是通过比较不同类别或不同组的均值差，例如采用t检验可以判断两组均值是否有显著的差异。

> **提问**：你还能不能设置成其他类型的模型形式？怎样设置才是正确的**方差分析模型**？

---

## 方差分析模型：本质

`$$\begin{align}
salary_i & =\beta_1 + \beta_2sale_i +\beta_3score_i+\beta_4race\_yellow_i+\beta_5race\_white_i+u_i
\end{align}$$`

很显然，在上述总体回归模型下，可以得到所有3类“分组”情形下的期望年薪水平：

`$$\begin{align}
E(Y| & race\_yellow=1,race\_white=0,sale,score) \\
&=\beta_1+\beta_2sale+\beta_3score + \beta_4 &&\text{(market yellow)} \\
E(Y| & race\_yellow=0,race\_white=1,sale,score) \\
&=\beta_1+\beta_2sale+\beta_3score  + \beta_5 &&\text{(market white)} \\
E(Y| & race\_yellow=0,race\_white=0,sale,score) \\
&=\beta_1 +\beta_2sale+\beta_3score &&\text{(market black)}
\end{align}$$`

---

## 方差分析模型：内涵

`$$\begin{align}
salary_i & =\beta_1 + \beta_2sale_i +\beta_3score_i+\beta_4race\_yellow_i+\beta_5race\_white_i+u_i
\end{align}$$`

上述模型被称其为**有截距**的含有虚拟变量的、加法形式的回归模型。显然，虚拟变量
`$race\_{black}$`没有进入模型中；模型设置有截距项
`$\beta_1$`。在这种设定下，我们称：

- **黑色(black)**为模型的**基础组**

- **黄色(yellow)**和**白色(white)**分别为模型的**比较组**。

- 有序变量
`$score$`为**协变量(covariates)**或**控制变量(control variable)**

- 
`$\beta_1$`为**截距系数**，代表基础组的期望水平

- 
`$\beta_2;\beta_3$`为**平行斜率系数**，代表协变量的影响效应

- 
`$\beta_4,\beta_5$`为**极差系数**，代表的是比较组与基础组期望水平的差距

---

## 方差分析模型的类型：数量关系

根据回归元包含**定量变量**和**虚拟变量**的数量关系，可以将虚拟变量回归模型分为：

- 只含有虚拟变量的回归模型：全部解释变量都是由虚拟变量构成

- 同时含有虚拟变量和定量变量的回归模型：解释变量同时含有虚拟变量和定量变量

---

## 方差分析模型的类型：引入方式

根据模型中虚拟变量**引入方式**的不同，可以划分为：

- 加法模型：虚拟变量以独立项的形式出现在方程中

- 乘法模型：虚拟变量以交叉项的形式出现在方程中

- 混合模型：虚拟变量以独立项和/或交叉项的形式出现在方程中^[有时候模型设置中，某个虚拟变量体系（用来表达某个定性变量）的独立项可以完全不出现在方程中（也即没有它们的加法形式），而却可以出现它们与其他变量的交叉项（也即可以出现它们与其他变量的乘法形式）。]
    
    -  完全混合模型
    - 部分混合模型

---

## 方差分析模型的类型：基础组

根据虚拟变量模型是否参照**基础组**，可以划分为：

- 有截距模型：此时模型解释中将有明确的**基础组**，其他组可以直接与之参照对比。

- 无截距模型：此时模型解释中将没有明确的**基础组**，各组间将不直接参照对比。

---

## 方差分析模型的类型：函数形式

根据模型中的因变量
`$Y$`是否取对数，可以划分为（半对数或对数模型将蕴含着弹性和斜率的经济学含义，在解释虚拟变量回归模型中往往很有现实意义）：

- 经典线性模型：因变量为
`$Y$`

- 半对数模型：因变量为
`$ln(Y)$`

---

## 方差分析模型的类型：应用情景

根据虚拟变量模型**应用情景**的不同，可以划分为：

- 截面数据虚拟变量回归模型：此时虚拟变量用于表达回归元为定性变量的情形

- 时间序列季节虚拟变量回归模型：此时虚拟变量用于表达季节周期

- 分段线性虚拟变量回归模型：此时虚拟变量用于表达**阀值**分段

---

## 方差分析模型的类型：综合

对于具体的实证分析案例，我们往往需要根据变量的属性和特征，构建不同类型的虚拟变量回归模型，比较不同模型的回归分析结果，甄选并得到其中相对理想的模型。

例如，仅是考虑**基础组**的有截距模型，可能用到的各类备选组合模型至少包括：

- 只含有虚拟变量的、加法形式的**经典回归模型**
- 只含有虚拟变量的、加法形式的**半对数回归模型**
- 只含有虚拟变量的、乘法形式的**经典回归模型**
- 只含有虚拟变量的、乘法形式的**半对数回归模型**
- 
`$\cdots$`
- 同时含有虚拟变量和定量变量的、加法形式的**经典回归模型**
- 同时含有虚拟变量和定量变量的、加法形式的**半对数回归模型**
- 同时含有虚拟变量和定量变量的、乘法形式的**经典回归模型**
- 同时含有虚拟变量和定量变量的、乘法形式的**半对数回归模型**
- 
`$\cdots$`

---
layout: false
class: center, middle, duke-softblue,hide_logo
name: one

# 10.3 只含有一个定性变量的ANOVA模型

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<div class="my-footer"><span>huhuaping@   <a href="#chapter10">第10章 虚拟变量回归模型</a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#one">10.3 只含有一个定性变量的ANOVA模型</a> </span></div>

---

## 公立学校教师薪水案例

下面我们以公立学校教师薪水案例进行说明。

---

### 变量

一项研究关注于对美国51个州公立学校教师薪水的分析：

- 变量
`$Salary$`表示公立学校教师平均薪水；变量
`$Spend$`表示公立学校教师平均支出

- 
`$state$`表示公立学校所在州名称；
`$Region$`表示州所属的区位（`West`表示**西部**州；`M.E.N`表示**中东北部**州；`South`表示**南部**州）。

---

### 数据

<div id="htmlwidget-a9a7ea7fa3e9f4c0a7ce" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-a9a7ea7fa3e9f4c0a7ce">{"x":{"filter":"none","vertical":false,"caption":"<caption>美国三个区域公立学校教师的薪水n=(51)<\/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"],["Connecticut","Illinois","Indiana","Iowa","Kansas","Maine","Massachusetts","Michigan","Minnesota","Missouri","Nebraska","New Hampshire","New Jersey","New York","North Dakota","Ohio","Pennsylvania","Rhode Island","South Dakota","Vermont","Wisconsin","Alabama","Arkansas","Delaware","District of Columbia","Florida","Georgia","Kentucky","Louisiana","Maryland","Mississippi","North Carolina","Oklahoma","South Carolina","Tennessee","Texas","Virginia","West Virginia","Alaska","Arizona","California","Colorado","Hawaii","Idaho","Montana","Nevada","New Mexico","Oregon","Utah","Washington","Wyoming"],["M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","South","South","South","South","South","South","South","South","South","South","South","South","South","South","South","South","South","West","West","West","West","West","West","West","West","West","West","West","West","West"],[60822,58246,47831,43130,43334,41596,58624,54895,49634,41839,42044,46527,59920,58537,38822,51937,54970,55956,35378,48370,47901,43389,44245,54680,59000,45308,49905,43646,42816,56927,40182,46410,42379,44133,43816,44897,44727,40531,54658,45941,63640,45833,51922,42798,41225,45342,42780,50911,40566,47882,50692],[12436,9275,8935,7807,8373,11285,12596,9880,9675,7840,7900,10206,13781,13551,7807,10034,10711,11089,7911,12475,9965,7706,8402,12036,15508,7762,8534,8300,8519,9771,7215,7675,6944,8377,6979,7547,9275,9886,10171,5585,8486,8861,9879,7042,8361,6755,8622,8649,5347,7958,11596]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>state<\/th>\n      <th>Region<\/th>\n      <th>Salary<\/th>\n      <th>Spend<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0}],"pageLength":9,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[9,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

### 描述统计

<div id="htmlwidget-5addc0d8d9dbc43e760e" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-5addc0d8d9dbc43e760e">{"x":{"filter":"none","vertical":false,"caption":"<caption>三个区域公立学校教师的平均薪水n=(51)<\/caption>","data":[["M.E.N","South","West"],[21,17,13],[49538.7142857143,46293.5882352941,48014.6153846154],[60822,59000,63640],[35378,40182,40566],[7645.47168030107,5543.64696137416,6400.04621778184]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>Region<\/th>\n      <th>N.state<\/th>\n      <th>Mean.Salary<\/th>\n      <th>Max.Salary<\/th>\n      <th>Min.Salary<\/th>\n      <th>SD.Salary<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","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  }"},{"targets":4,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 2, 3, \",\", \".\", null);\n  }"},{"targets":5,"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}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render","options.columnDefs.2.render","options.columnDefs.3.render"],"jsHooks":[]}</script>

根据以上简单的汇总计算：

- 教师的平均薪水：**中东北部**为49 538.71 美元；**南部**为46 293.59 美元；
西部为48 104.62美元。

- 那么，三个地区的平均薪水在**统计上**也彼此不同吗?

---

### 虚拟变量体系

根据前述定义，我们可以将**定性变量**
`$Region$`设置为如下的**虚拟变量体系**：

`$$\begin{align}
Region\{West; \quad M.E.N; \quad South \} \\
\Rightarrow  
  \begin{cases}
    D_1  =
    \begin{cases}
    1, & \text{if } West\\
    0, & \text{if not }  West
    \end{cases} \\
    D_2 = 
    \begin{cases}
    1, & \text{if } M.E.N\\
    0, & \text{if not }  M.E.N
    \end{cases} \\
    D_3  =
    \begin{cases}
    1, & \text{if } South\\
    0, & \text{if not }  South
    \end{cases}
  \end{cases}
\end{align}$$`

]

]

---

### 虚拟变量变换

实际建模之前，我们需要把定性变量
`$Region$`进行**数据变换**，得到虚拟变量的数据：

<div id="htmlwidget-3c8f3009aeae6a919671" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-3c8f3009aeae6a919671">{"x":{"filter":"none","vertical":false,"caption":"<caption>把定性变量Region处理成虚拟变量n=(51)<\/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"],["Connecticut","Illinois","Indiana","Iowa","Kansas","Maine","Massachusetts","Michigan","Minnesota","Missouri","Nebraska","New Hampshire","New Jersey","New York","North Dakota","Ohio","Pennsylvania","Rhode Island","South Dakota","Vermont","Wisconsin","Alabama","Arkansas","Delaware","District of Columbia","Florida","Georgia","Kentucky","Louisiana","Maryland","Mississippi","North Carolina","Oklahoma","South Carolina","Tennessee","Texas","Virginia","West Virginia","Alaska","Arizona","California","Colorado","Hawaii","Idaho","Montana","Nevada","New Mexico","Oregon","Utah","Washington","Wyoming"],["M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","South","South","South","South","South","South","South","South","South","South","South","South","South","South","South","South","South","West","West","West","West","West","West","West","West","West","West","West","West","West"],[60822,58246,47831,43130,43334,41596,58624,54895,49634,41839,42044,46527,59920,58537,38822,51937,54970,55956,35378,48370,47901,43389,44245,54680,59000,45308,49905,43646,42816,56927,40182,46410,42379,44133,43816,44897,44727,40531,54658,45941,63640,45833,51922,42798,41225,45342,42780,50911,40566,47882,50692],[12436,9275,8935,7807,8373,11285,12596,9880,9675,7840,7900,10206,13781,13551,7807,10034,10711,11089,7911,12475,9965,7706,8402,12036,15508,7762,8534,8300,8519,9771,7215,7675,6944,8377,6979,7547,9275,9886,10171,5585,8486,8861,9879,7042,8361,6755,8622,8649,5347,7958,11596],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>state<\/th>\n      <th>Region<\/th>\n      <th>Salary<\/th>\n      <th>Spend<\/th>\n      <th>D1<\/th>\n      <th>D2<\/th>\n      <th>D3<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0}],"pageLength":5,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

### 有截距虚拟变量模型：PRM1

我们可以构建薪水（
`$Salary$`）对区域虚拟变量（
`$D2;D3$`）的**有截距**总体回归模型PRM：

`$$\begin {align} 
Salary_{i}=\beta_{1}+\beta_{2} D2_{ i}+\beta_{3} D3_{ i}+u_{i}
\end {align}$$`

理论上，我们可以得到三个区域教师薪水的期望值：

`$$\begin{align}
E(Salary|  D2=1,D3=0) 
&=\beta_1+\beta_2 &&\text{(M.E.N)} \\
E(Salary|  D2=0,D3=1) 
&=\beta_1+\beta_3 &&\text{(South)} \\
E(Salary|  D2=0,D3=0) 
&=\beta_1  &&\text{(West)}
\end{align}$$`

]

]

---

### 有截距虚拟变量模型：OLS估计

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Salary}=&&+48014.62&&+1524.10D2&&-1721.03D3\\ &\text{(t)}&&(25.8532)&&(0.6449)&&(-0.6976)\\&\text{(se)}&&(1857.2037)&&(2363.1394)&&(2467.1508)\\&\text{(fitness)}&& R^2=0.0440;&& \bar{R^2}=0.0041\\& && F^{\ast}=1.10;&& p=0.3399 \end{alignedat} \end{equation}$$`

> - 基础组是谁？极差系数的含义？
> - 三个区域的平均薪水具有统计上的显著差异吗？

---

### 有截距虚拟变量模型：OLS估计

]

]

---

### 无截距虚拟变量模型：PRM2

我们也可以构建薪水（
`$Salary$`）对区域虚拟变量（
`$D1;D2;D3$`）的**无截距**总体回归模型PRM：

`$$\begin {align} 
Salary_{i}=\alpha_{1} D1_{ i}+\alpha_{2} D2_{ i}+\alpha_{3} D3_{ i}+u_{i}
\end {align}$$`

理论上，我们可以得到三个区域教师薪水的期望值：

`$$\begin{align}
E(Salary|  D1=0,D2=1,D3=0) 
&=\alpha_2 &&\text{(M.E.N)} \\
E(Salary|  D1=0,D2=0,D3=1) 
&=\alpha_3 &&\text{(South)} \\
E(Salary|  D1=1,D2=0,D3=0) 
&=\alpha_1  &&\text{(West)}
\end{align}$$`

]

]

---

### 无截距虚拟变量模型：OLS估计

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Salary}=&&+48014.62D1&&+49538.71D2&&+46293.59D3\\ &\text{(t)}&&(25.8532)&&(33.9018)&&(28.5045)\\&\text{(se)}&&(1857.2037)&&(1461.2400)&&(1624.0775)\\&\text{(fitness)}&& R^2=0.9821;&& \bar{R^2}=0.9810\\& && F^{\ast}=876.74;&& p=0.0000 \end{alignedat} \end{equation}$$`

---

### 无截距虚拟变量模型：OLS估计

]

]

---

## 虚拟变量模型的构建规则

若定性因素具有m个相互排斥属性(或几个水平)：

- **规则1**：当回归模型有截距项时，只能设(m-1)个虚拟变量；

- **规则2**：当回归模型无截距项时，则可引入m个虚拟变量。否则，就会陷入“虚拟变量陷阱”。（为什么？）

- **规则3**：在虚拟变量的设置中：基础类型、肯定类型取值为1；比较类型、否定类型取值为0。

> **思考**：规则1和规则2分别建立虚拟变量回归模型，哪种更好呢？

---

## 虚拟变量模型的构建规则：示例

**建模1**: (正确模型)使用m-1个虚拟变量，并设定为**有截距**：

`$$\begin {align} 
Salary_{i}=\beta_{1}+\beta_{2} D2_{ i}+\beta_{3} D3_{ i}+u_{i}
\end {align}$$`

**建模2**: (正确模型)使用m个虚拟变量，并设定为**无截距**：

`$$\begin {align} 
Salary_{i}=\alpha_{1} D1_{ i}+\alpha_{2} D2_{ i}+\alpha_{3} D3_{ i}+u_{i}
\end {align}$$`

**建模3**: (错误模型)使用m个虚拟变量，并设定为**有截距**：

`$$\begin {align} 
Salary_{i}=\gamma_{0}+\gamma_{1} D1_{ i}+\gamma_{2} D2_{ i}+\gamma_{3} D3_{ i}+u_{i}
\end {align}$$`

> **提问**：
- 模型1和模型2的回归系数涵义是一样的么？
- 执意采用OLS方法估计模型3，会有什么后果？

---

## 虚拟变量模型的构建规则：R软件示例

```r
mod.main3 <- "Salary ~1+ D1+D2 +D3"
lm.main3 <- lm(mod.main3, data_demon)
summary(lm.main3)
```

```

Call:
lm(formula = mod.main3, data = data_demon)

Residuals:
   Min     1Q Median     3Q    Max 
-14161  -4566  -1638   4632  15625

Coefficients: (1 not defined because of singularities)
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    46294       1624  28.505   <2e-16 ***
D1              1721       2467   0.698    0.489    
D2              3245       2185   1.485    0.144    
D3                NA         NA      NA       NA    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 6696 on 48 degrees of freedom
Multiple R-squared:  0.04397,	Adjusted R-squared:  0.004134 
F-statistic: 1.104 on 2 and 48 DF,  p-value: 0.3399
```

]

对于错误的**建模3**，有些统计软件（如以上R软件）会自动去掉一个多于的虚拟变量。

`$$\begin {align} 
Salary_{i}=\gamma_{0}+\gamma_{1} D1_{ i}+\gamma_{2} D2_{ i}+\gamma_{3} D3_{ i}+u_{i}
\end {align}$$`

---

# 10.4 同时含有一个定性变量
# 和定量变量的ANOVA模型

---
layout: true
  
<div class="my-header-h2"></div>
<div class="watermark1"></div>
<div class="watermark2"></div>
<div class="watermark3"></div>

<div class="my-footer"><span>huhuaping@   <a href="#chapter10">第10章 虚拟变量回归模型</a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#one-one">10.4同时含有一个定性变量和定量变量的ANOVA模型</a> </span></div>

---

## 公立学校教师薪水案例

我们继续来看公立学校教师薪水案例的建模分析。

---

### 数据

把定性变量
`$Region$`进行**数据变换**，得到数据：

<div id="htmlwidget-dd77ffe7ebe98749bb18" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-dd77ffe7ebe98749bb18">{"x":{"filter":"none","vertical":false,"caption":"<caption>把定性变量Region处理成虚拟变量n=(51)<\/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"],["Connecticut","Illinois","Indiana","Iowa","Kansas","Maine","Massachusetts","Michigan","Minnesota","Missouri","Nebraska","New Hampshire","New Jersey","New York","North Dakota","Ohio","Pennsylvania","Rhode Island","South Dakota","Vermont","Wisconsin","Alabama","Arkansas","Delaware","District of Columbia","Florida","Georgia","Kentucky","Louisiana","Maryland","Mississippi","North Carolina","Oklahoma","South Carolina","Tennessee","Texas","Virginia","West Virginia","Alaska","Arizona","California","Colorado","Hawaii","Idaho","Montana","Nevada","New Mexico","Oregon","Utah","Washington","Wyoming"],["M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","M.E.N","South","South","South","South","South","South","South","South","South","South","South","South","South","South","South","South","South","West","West","West","West","West","West","West","West","West","West","West","West","West"],[60822,58246,47831,43130,43334,41596,58624,54895,49634,41839,42044,46527,59920,58537,38822,51937,54970,55956,35378,48370,47901,43389,44245,54680,59000,45308,49905,43646,42816,56927,40182,46410,42379,44133,43816,44897,44727,40531,54658,45941,63640,45833,51922,42798,41225,45342,42780,50911,40566,47882,50692],[12436,9275,8935,7807,8373,11285,12596,9880,9675,7840,7900,10206,13781,13551,7807,10034,10711,11089,7911,12475,9965,7706,8402,12036,15508,7762,8534,8300,8519,9771,7215,7675,6944,8377,6979,7547,9275,9886,10171,5585,8486,8861,9879,7042,8361,6755,8622,8649,5347,7958,11596],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>state<\/th>\n      <th>Region<\/th>\n      <th>Salary<\/th>\n      <th>Spend<\/th>\n      <th>D1<\/th>\n      <th>D2<\/th>\n      <th>D3<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0}],"pageLength":7,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[7,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

### 有截距虚拟变量模型：PRM1

我们可以构建薪水（
`$Salary$`）对区域虚拟变量（
`$D2;D3$`）和定量变量
`$Spend$`的**有截距**总体回归模型PRM：

`$$\begin {align} 
Salary_{i}=\beta_{1}+\beta_{2} D2_{ i}+\beta_{3} D3_{ i}+\lambda Spend_i+u_{i}
\end {align}$$`

理论上，我们可以得到三个区域教师薪水的期望值：

`$$\begin{align}
E(Salary|  D2=1,D3=0) 
&=\beta_1+\beta_2 +\lambda Spend &&\text{(M.E.N)} \\
E(Salary|  D2=0,D3=1) 
&=\beta_1+\beta_3 +\lambda Spend &&\text{(South)} \\
E(Salary|  D2=0,D3=0) 
&=\beta_1 +\lambda Spend &&\text{(West)}
\end{align}$$`

]

<img src="../pic/chpt-10-region-show.png" width="656" style="display: block; margin: auto;" />
]

---

### 有截距虚拟变量模型：OLS估计

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Salary}=&&+28694.92&&-2954.13D2&&-3112.19D3&&+2.34Spend\\ &\text{(t)}&&(8.7953)&&(-1.5860)&&(-1.7101)&&(6.5152)\\&\text{(se)}&&(3262.5213)&&(1862.5756)&&(1819.8725)&&(0.3592)\\&\text{(fitness)}&& R^2=0.4977;&& \bar{R^2}=0.4656\\& && F^{\ast}=15.52;&& p=0.0000 \end{alignedat} \end{equation}$$`

.pull-left[
 **提问1**：大白话解释上述回归函数！
 
 **思考1**：基准组是什么？谁是协变量？
 
 **思考2**：三条线为什么是平行的？
 
 **思考3**：统计上来看，南部线和西部线是不一样的么？
 
]

<img src="../pic/chpt-10-region-show.png" width="454" style="display: block; margin: auto;" />
]

---

### 有截距虚拟变量模型：OLS估计

---

### 无截距虚拟变量模型：PRM2

我们可以构建薪水（
`$Salary$`）对区域虚拟变量（
`$D1; D2;D3$`）和定量变量
`$Spend$`的**无截距**总体回归模型PRM：

`$$\begin {align} 
Salary_{i}=\alpha_{1} D1_{ i}+\alpha_{2} D2_{ i}+\alpha_{3} D3_{ i}+\lambda Spend_i+u_{i}
\end {align}$$`

理论上，我们可以得到三个区域教师薪水的期望值：

`$$\begin{align}
E(Salary|  D1=0,D2=1,D3=0;Spend) 
&=\alpha_2 +\lambda Spend &&\text{(M.E.N)} \\
E(Salary|  D1=0,D2=0,D3=1;Spend) 
&=\alpha_3 +\lambda Spend &&\text{(South)} \\
E(Salary|  D1=1,D2=0,D3=0;Spend) 
&=\alpha_1 +\lambda Spend &&\text{(West)}
\end{align}$$`

]

<img src="../pic/chpt-10-region-show.png" width="656" style="display: block; margin: auto;" />
]

---

### 无截距虚拟变量模型：OLS估计

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Salary}=&&+28694.92D1&&+25740.79D2&&+25582.72D3&&+2.34Spend\\ &\text{(t)}&&(8.7953)&&(6.7627)&&(7.5372)&&(6.5152)\\&\text{(se)}&&(3262.5213)&&(3806.2835)&&(3394.1819)&&(0.3592)\\&\text{(fitness)}&& R^2=0.9906;&& \bar{R^2}=0.9898\\& && F^{\ast}=1235.97;&& p=0.0000 \end{alignedat} \end{equation}$$`

- **提问1**：大白话解释上述回归函数！

- **思考1**：基准组是什么？谁是协变量？

- **思考2**：三条线为什么是平行的？

- **思考3**：统计上来看，南部线和西部线是不一样的么？

- **思考4**：**有截距模型**和**无截距模型**的图形为什么是一样的？

]

<img src="../pic/chpt-10-region-show.png" width="454" style="display: block; margin: auto;" />
]

---

### 无截距虚拟变量模型：OLS估计

---

# 10.5 同时含有多个定性变量
# 和定量变量的ANOVA模型

---

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<div class="my-footer"><span>huhuaping@   <a href="#chapter10">第10章 虚拟变量回归模型</a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#multiple">10.5 同时含有多个定性变量和定量变量的ANOVA模型</a> </span></div>

---

## 虚拟变量的引入方式（定义）

如果自变量中存在k个定性变量（
`$X_{1i},X_{2i},\cdots,X_{ki}$`），而且每个定性变量还有自己的属性个数（
`$X_{ki} \quad (a_1,a_2,\cdots, a_m)$`）。那么把这些**定性变量**转换成各自的**虚拟变量体系**后，虚拟变量在模型中出现的关系则可以有多种形式：

1. 虚拟变量以**加法形式**引入的回归模型：是指各个**定性变量**的虚拟变量体系，**各自**以**独立项**的形式出现在模型中。

2. 虚拟变量以**乘法形式**引入的回归模型：是指各个**定性变量**的虚拟变量体系，存在**相互**以**交叉项**的形式出现在模型中。

---

## 虚拟变量的引入方式（定义）

3. 虚拟变量以**混合形式**（既有加法形式也有乘法形式）引入的回归模型：是指各个**定性变量**的虚拟变量体系，及有**各自**以**独立项**的形式，也有**相互**以**交叉项**的形式出现在模型中。又具体分为两种情形：

- 完全混合模型：两个定性变量的虚拟变量体系，既有各自**独立项**，又有它们相互间完全**交叉项**。
    
    - 部分混合模型：两个定性变量的虚拟变量体系，既有各自**独立项**，又有它们相互间不完全的**交叉项**（也即部分交叉）。

---

## 虚拟变量的引入方式（示例）

为了研究工人工资的影响因素，我们可以考虑如下变量：

- 定量变量：工资
`$wage$`；年龄
`$age$`

- 定性变量：教育程度
`$edu\{a_1=\text{ill},a_2=\text{pri},a_3=\text{mid},a_4=\text{hig}\}$`；工作部门
`$dpt\{a_1=\text{tem},a_2=\text{per}\}$`；性别
`$sex\{a_1=\text{f},a_2=\text{m}\}$`。

> 教育程度edu：ill表示文盲；pri表示初等教育；mid表示中等教育；hig表示高等教育。

> 工作类型dpt：tem表示临时工；per表示合同工。

> 性别sex：f表示女性；m表示男性。

---
class: page-font-21
## 虚拟变量的引入方式（示例）

因为涉及到三个定性变量，我们可以将他们转换为各自的**虚拟变量体系**：

**教育程度**定性变量`edu`：

`$$\begin{align}
&edu\{a_1=\text{ill},a_2=\text{pri},a_3=\text{mid} ,a_4=\text{hig}\} \\
&dummy \Longrightarrow  
  \begin{cases}
    edu\_ill  =
    \begin{cases}
    1, & \text{ill}\\
    0, & \text{not ill}
    \end{cases} \\
    edu\_pri = 
    \begin{cases}
    1, & \text{pri}\\
    0, & \text{not pri}
    \end{cases} \\
    edu\_mid  =
    \begin{cases}
    1, & \text{mid}\\
    0, & \text{not mid}
    \end{cases}\\
     edu\_hig  =
    \begin{cases}
    1, & \text{hig}\\
    0, & \text{not hig}
    \end{cases}
  \end{cases}
\end{align}$$`

]

**工作类型**定性变量`dpt`：

`$$\begin{align}
&dpt\{a_1=\text{tem},a_2=\text{per}\} \\
&dummy \Longrightarrow  
  \begin{cases}
    dpt\_tem  =
    \begin{cases}
    1, & \text{tem}\\
    0, & \text{not tem}
    \end{cases} \\
    dpt\_per = 
    \begin{cases}
    1, & \text{per}\\
    0, & \text{not per}
    \end{cases} 
  \end{cases}
\end{align}$$`

**性别**定性变量`sex`：

`$$\begin{align}
&sex\{a_1=\text{f},a_2=\text{m}\} \\
&dummy \Longrightarrow  
  \begin{cases}
    sex\_f  =
    \begin{cases}
    1, & \text{f}\\
    0, & \text{not f}
    \end{cases} \\
    sex\_m = 
    \begin{cases}
    1, & \text{m}\\
    0, & \text{not m}
    \end{cases} 
  \end{cases}
\end{align}$$`

]

---

## 虚拟变量的引入方式（示例:加法模型）

`$$\begin{equation} \begin{alignedat}{999} &wage=&& + \beta_{1} && + \beta_{2} edu_{pri}&& + \beta_{3} edu_{mid}&& + \beta_{4} edu_{hig}&& + \beta_{5} dpt_{per}&& + \beta_{6} sex_{m}&& + \beta_{7} age&&+u_i\\ \end{alignedat} \end{equation}$$`

假定我们关注这样两个群体：

**A群体**：年龄为30岁的、女性（
`$sex_m=0$`）、**受过高等教育**（
`$edu_pri=0,edu_mid=0,edu_hig=1$`）的、拥有一份**合同工**的（
`$dpt_per=1$`）。

`$$\begin{align}
E(wage&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 1;sex_{m}= 0)\\
=&+\beta_{1}+\beta_{2}(0)+\beta_{3}(0)+\beta_{4}(1)+\beta_{5}(1)+\beta_{6}(0)+\beta_{7}(30)\\
=&+\beta_{1}+\beta_{4}+\beta_{5}+30\beta_{7}\\
\end{align}$$`

---

## 虚拟变量的引入方式（示例:加法模型）

假定我们关注这样两个群体：

**B群体**：年龄为30岁的、女性（
`$sex_m=0$`）、**受过高等教育**（
`$edu_pri=0,edu_mid=0,edu_hig=1$`）的、拥有一份**临时工**的（
`$dpt_per=0$`）。

`$$\begin{align}
E(wage&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 0;sex_{m}= 0)\\
=&+\beta_{1}+\beta_{2}(0)+\beta_{3}(0)+\beta_{4}(1)+\beta_{5}(0)+\beta_{6}(0)+\beta_{7}(30)\\
=&+\beta_{1}+\beta_{4}+30\beta_{7}\\
\end{align}$$`

---

## 虚拟变量的引入方式（示例:加法模型）

如果
`$\beta_5>0$`，这将意味着：

- 只要拥有一份**合同工**（
`$dpt_{per}=1$`）。那么，**在其他同等情况下**，这个人的工资都要高于拥有一份**临时工**（
`$dpt_{per}=0$`）的人。——无论是高学历的同等条件（
`$edu_{pri}=0,edu_{mid}=0,edu_{hig}=1$`），还是中等学历的同等条件（
`$edu_{pri}=1,edu_{mid}=0,edu_{hig}=0$`），还是文盲学历的同等条件（
`$edu_{pri}=0,edu_{mid}=0,edu_{hig}=0$`）。

- 换言之，**工作类型**（dpt）与**受教育程度**（edu）是**独立地**作用于**工资**（wage）的！

---

## 虚拟变量的引入方式（示例:乘法模型）

`$$\begin{equation} \begin{alignedat}{999} &wage=&& + \beta_{1} && + \beta_{2} sex_{m}&& + \beta_{3} age&& + \beta_{4} edu_{pri} \ast dpt_{per}\\ &\text{(cont.)}&& + \beta_{5} dpt_{per} \ast edu_{mid}&& + \beta_{6} dpt_{per} \ast edu_{hig}&&+u_i\\ \end{alignedat} \end{equation}$$`

假定我们关注这样两个群体：

**A群体**：年龄为30岁的、女性（
`$sex_m=0$`）、**受过高等教育**（
`$edu_pri=0,edu_mid=0,edu_hig=1$`）的、拥有一份**合同工**的（
`$dpt_per=1$`）。

`$$\begin{align}
E(wage&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 1;sex_{m}= 0)\\
=&+\beta_{1}+\beta_{2}(0)+\beta_{3}(30)+\beta_{4}(0)\cdot(1)+\beta_{5}(1)\cdot(0)+\beta_{6}(1)\cdot(1)\\
=&+\beta_{1}+30\beta_{3}+\beta_{6}\\
\end{align}$$`

---

## 虚拟变量的引入方式（示例:乘法模型）

假定我们关注这样两个群体：

**B群体**：年龄为30岁的、女性（
`$sex_m=0$`）、**受过高等教育**（
`$edu_pri=0,edu_mid=0,edu_hig=1$`）的、拥有一份**临时工**的（
`$dpt_per=0$`）。

`$$\begin{align}
E(wage&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 0;sex_{m}= 0)\\
=&+\beta_{1}+\beta_{2}(0)+\beta_{3}(30)+\beta_{4}(0)\cdot(0)+\beta_{5}(0)\cdot(0)+\beta_{6}(0)\cdot(1)\\
=&+\beta_{1}+30\beta_{3}\\
\end{align}$$`

---

## 虚拟变量的引入方式（示例:乘法模型）

如果
`$\beta_6>0$`且显著，这将意味着：

- **在其他同等情况下**，一个拥有一份**合同工**（
`$dpt_{per}=1$`）且拥有**高学历**（
`$edu_{hig}=1$`）的人。这个人的工资都要**高于**拥有一份**临时工**（
`$dpt_{per}=0$`）或**没有**受过高学历教育（
`$edu_{hig}=0$`）的人。——包括：临时工&文盲；临时工&初等学历；临时工&中等学历；临时工&高等学历；合同工&文盲；合同工&初等学历；合同工&中等学历。

- 换言之，**工作类型**（dpt）与**受教育程度**（edu）是**交互地**作用于**工资**（wage）的！——此处重点针对拥有高学历还是不拥有有高学历。

---

## 虚拟变量的引入方式（示例:混合模型）

`$$\begin{equation} \begin{alignedat}{999} &wage=&& + \beta_{1} && + \beta_{2} sex_{m}&& + \beta_{3} dpt_{per}&& + \beta_{4} age&& + \beta_{5} edu_{pri} \ast dpt_{per}\\ &\text{(cont.)}&& + \beta_{6} edu_{mid} \ast dpt_{per}&& + \beta_{7} edu_{hig} \ast dpt_{per}&&+u_i\\ \end{alignedat} \end{equation}$$`

假定我们关注这样两个群体：

**A群体**：年龄为30岁的、女性（
`$sex_m=0$`）、**受过高等教育**（
`$edu_pri=0,edu_mid=0,edu_hig=1$`）的、拥有一份**合同工**的（
`$dpt_per=1$`）。

`$$\begin{align}
E(wage&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 1;sex_{m}= 0)\\
=&+\beta_{1}+\beta_{2}(0)+\beta_{3}(1)+\beta_{4}(30)+\beta_{5}(0)\cdot(1)+\beta_{6}(0)\cdot(1)+\beta_{7}(1)\cdot(1)\\
=&+\beta_{1}+\beta_{3}+30\beta_{4}+\beta_{7}\\
\end{align}$$`

---

## 虚拟变量的引入方式（示例:混合模型）

假定我们关注这样两个群体：

**B群体**：年龄为30岁的、女性（
`$sex_m=0$`）、**受过高等教育**（
`$edu_pri=0,edu_mid=0,edu_hig=1$`）的、拥有一份**临时工**的（
`$dpt_per=0$`）。

`$$\begin{align}
E(wage&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 0;sex_{m}= 0)\\
=&+\beta_{1}+\beta_{2}(0)+\beta_{3}(0)+\beta_{4}(30)+\beta_{5}(0)\cdot(0)+\beta_{6}(0)\cdot(0)+\beta_{7}(1)\cdot(0)\\
=&+\beta_{1}+30\beta_{4}\\
\end{align}$$`

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# 10.6 印度工人工资案例

---
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<div class="watermark3"></div>

---

### 数据：原始

**印度工人工资**：114位印度工人工资方面的数据如下。

<div id="htmlwidget-5a8f4930778c36ac712e" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-5a8f4930778c36ac712e">{"x":{"filter":"none","vertical":false,"caption":"<caption>印度工人工资(n=114)<\/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],[117,375,175,100,162.5,300,175,287.5,350,385,395,224,91.8,119,377,400,139.5,139.5,54,342,77.5,156,350,427,40,65,162.5,162.5,162.5,325,18,127.6,30,189.5,50,112,82,87.5,600,116.5,75,127,324,150,135,62.5,97,54,120.5,950,120,132,30,24,75,42,100,135.5,107,90,162,49.16,81,105,200,80,125,500,100,105,100,140,150,131,120,122,79,110,370,261.15,129.5,107.7,20,18,120,40.5,120,75,52,115.25,102.5,272.5,94.25,30,30,25,47.5,175,150,50,175,50,85.32,111,47.5,40.5,47,25,15,25,25,75,53.84,50],[26,42,33,33,30,35,40,50,42,30,45,48,35,38,46,57,44,30,62,56,19,26,57,55,15,14,25,28,24,55,19,25,19,62,25,27,22,25,35,28,27,22,26,30,25,18,25,16,27,47,57,38,18,12,55,18,32,41,48,45,40,22,20,40,30,26,22,21,19,35,30,30,36,55,21,20,60,23,46,23,33,26,14,12,38,17,34,18,19,33,27,43,22,38,17,22,25,50,24,25,40,16,26,23,46,37,41,28,13,18,11,45,14,26],["pri","pri","pri","pri","pri","mid","mid","mid","mid","mid","mid","hig","hig","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","pri","pri","pri","pri","pri","pri","pri","pri","pri","pri","mid","mid","mid","mid","mid","mid","mid","mid","mid","hig","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill","mid","hig","hig","hig","ill","ill","ill","ill","ill","ill","ill","ill","pri","pri","mid","ill","ill","ill","ill","ill","ill","ill","ill","ill","ill"],["per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","per","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","per","per","per","per","per","per","per","per","per","per","per","per","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem","tem"],["f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","f","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m","m"]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>obs<\/th>\n      <th>wage<\/th>\n      <th>age<\/th>\n      <th>edu<\/th>\n      <th>dpt<\/th>\n      <th>sex<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

### 数据：变量定义

变量说明见下表：

<table class="table" style="font-size: 22px; margin-left: auto; margin-right: auto;">
<caption style="font-size: initial !important;">变量定义及说明</caption>
 <thead>
  <tr>
   <th style="text-align:center;"> variable </th>
   <th style="text-align:center;"> label </th>
   <th style="text-align:center;"> remark </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> obs </td>
   <td style="text-align:center;width: 6em; "> 工人编号 </td>
   <td style="text-align:center;"> 序号(observations) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> wage </td>
   <td style="text-align:center;width: 6em; "> 工人工资 </td>
   <td style="text-align:center;"> 美元/周(\$/week) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> age </td>
   <td style="text-align:center;width: 6em; "> 年龄 </td>
   <td style="text-align:center;"> 岁(year) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> edu </td>
   <td style="text-align:center;width: 6em; "> 教育水平 </td>
   <td style="text-align:center;"> ill=文盲(illiteracy)；pri=初等教育(primary)；mid=中等教育(middle)；hig=高等教育(higher) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> dpt </td>
   <td style="text-align:center;width: 6em; "> 合同类型 </td>
   <td style="text-align:center;"> tem=短期合同(temporary)；per=长期合同(permanent) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> sex </td>
   <td style="text-align:center;width: 6em; "> 性别 </td>
   <td style="text-align:center;"> f=女(female)；m=男(male) </td>
  </tr>
</tbody>
</table>

---

###数据：定性变量的属性统计

<div id="htmlwidget-1650833bf6a8d96ba36f" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-1650833bf6a8d96ba36f">{"x":{"filter":"none","vertical":false,"caption":"<caption>定性变量及其属性的统计表(n=114)<\/caption>","data":[["edu","edu","edu","edu","dpt","dpt","sex","sex"],["ill","pri","mid","hig","tem","per","f","m"],[74,17,17,6,72,42,89,25]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>定性变量<\/th>\n      <th>属性<\/th>\n      <th>频次<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

---
class: page-font-20
### 数据：定性变量的虚拟变量变换

将基础组设定为{文盲，临时工，女性}（也即{illiteracy，temporary，female}）。则可以将全部定性变量的基础组属性{illiteracy，temporary，female}分别设置为虚拟变量`edu_ill`、`dpt_tem`和`sex_f`。

**教育程度**定性变量`edu`：

]

**工作类型**定性变量`dpt`：

**性别**定性变量`sex`：

]

---

### 数据：教育变量的虚拟变量变换

<div id="htmlwidget-882a119628dd5f294c8d" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-882a119628dd5f294c8d">{"x":{"filter":"none","vertical":false,"caption":"<caption>用虚拟变量系统完全表达定性变量edu (n=114)<\/caption>","data":[[92,18,10,5],["hig","ill","mid","pri"],[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>obs<\/th>\n      <th>edu<\/th>\n      <th>edu_ill<\/th>\n      <th>edu_pri<\/th>\n      <th>edu_mid<\/th>\n      <th>edu_hig<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"pageLength":4,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[4,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

### 数据：工作部门变量的虚拟变量变换

<div id="htmlwidget-6233d4a4523cc14db4f1" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-6233d4a4523cc14db4f1">{"x":{"filter":"none","vertical":false,"caption":"<caption>用虚拟变量系统完全表达定性变量dpt (n=114)<\/caption>","data":[[5,35],["per","tem"],[0,1],[1,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>obs<\/th>\n      <th>dpt<\/th>\n      <th>dpt_tem<\/th>\n      <th>dpt_per<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"pageLength":3,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[3,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

### 数据：性别变量的虚拟变量变换

<div id="htmlwidget-70ec7c7660401f01eff8" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-70ec7c7660401f01eff8">{"x":{"filter":"none","vertical":false,"caption":"<caption>用虚拟变量系统完全表达定性变量sex (n=114)<\/caption>","data":[[2,91],["f","m"],[1,0],[0,1]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>obs<\/th>\n      <th>sex<\/th>\n      <th>sex_f<\/th>\n      <th>sex_m<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"pageLength":5,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

### 加法模型：总体回归模型PRM

同时含虚拟变量和定量变量的、加法形式的经典回归模型：

OLS估计的简要报告如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{wage}=&&+6.79&&+23.96edu_{pri}&&+61.59edu_{mid}&&+150.49edu_{hig}\\ &\text{(t)}&&(0.2130)&&(0.7734)&&(1.9867)&&(3.0054)\\&\text{(se)}&&(31.8931)&&(30.9789)&&(31.0035)&&(50.0725)\\&\text{(cont.)}&&+31.16dpt_{per}&&-83.20sex_{m}&&+3.99age\\&\text{(t)}&&(1.3141)&&(-3.0819)&&(4.5129)\\&\text{(se)}&&(23.7120)&&(26.9981)&&(0.8835)\\&\text{(fitness)}&& R^2=0.3450;&& \bar{R^2}=0.3083\\& && F^{\ast}=9.39;&& p=0.0000\\ \end{alignedat} \end{equation}$$`

---

### 加法模型：EViews报告

---

### 加法模型：基础组

* 基础组（**文盲 & 短期合同 & 女性**），也即(**illiteracy & temporary & female**)的期望工资收入为（给定**年龄**为30岁）：

`$$\begin{align}
E(wage&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 0;dpt_{per}= 0;sex_{m}= 0)\\
=&+\beta_{1}+\beta_{2}(0)+\beta_{3}(0)+\beta_{4}(0)+\beta_{5}(0)+\beta_{6}(0)+\beta_{7}(30)\\
=&+\beta_{1}+30\beta_{7}\\
\end{align}$$`

`$$\begin{align}
(\widehat{wage}&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 0;dpt_{per}= 0;sex_{m}= 0)\\
=&+\hat{\beta}_{1}+\hat{\beta}_{2}(0)+\hat{\beta}_{3}(0)+\hat{\beta}_{4}(0)\\
&+\hat{\beta}_{5}(0)+\hat{\beta}_{6}(0)+\hat{\beta}_{7}(30)\\
=&+[6.79]+[23.96]\cdot(0)+[61.59]\cdot(0)+[150.49]\cdot(0)\\
&+[31.16]\cdot(0)+[-83.20]\cdot(0)+[3.99]\cdot(30)\\
=&126.4104\\
\end{align}$$`

---

### 加法模型：比较组1

比较组1（**高等学历 & 短期合同 & 女性**），也即(**high & temporary & female**)的期望工资收入为（给定**年龄**为30岁）：

`$$\begin{align}
(\widehat{wage}&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 0;sex_{m}= 0)\\
=&+\hat{\beta}_{1}+\hat{\beta}_{2}(0)+\hat{\beta}_{3}(0)+\hat{\beta}_{4}(1)\\
&+\hat{\beta}_{5}(0)+\hat{\beta}_{6}(0)+\hat{\beta}_{7}(30)\\
=&+[6.79]+[23.96]\cdot(0)+[61.59]\cdot(0)+[150.49]\cdot(1)\\
&+[31.16]\cdot(0)+[-83.20]\cdot(0)+[3.99]\cdot(30)\\
=&276.8995\\
\end{align}$$`

---

### 加法模型：比较组2

比较组2（**高等教育 & 长期合同 & 女性**），也即(**higher & permanent & female**)的期望工资收入为（给定**年龄**为30岁）：

`$$\begin{align}
(\widehat{wage}&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 1;sex_{m}= 0)\\
=&+\hat{\beta}_{1}+\hat{\beta}_{2}(0)+\hat{\beta}_{3}(0)+\hat{\beta}_{4}(1)\\
&+\hat{\beta}_{5}(1)+\hat{\beta}_{6}(0)+\hat{\beta}_{7}(30)\\
=&+[6.79]+[23.96]\cdot(0)+[61.59]\cdot(0)+[150.49]\cdot(1)\\
&+[31.16]\cdot(1)+[-83.20]\cdot(0)+[3.99]\cdot(30)\\
=&308.0604\\
\end{align}$$`

---

### 乘法模型：总体回归模型PRM

同时含虚拟变量和定量变量的、加法形式的经典回归模型：

`$$\begin{equation} \begin{alignedat}{999} &wage=&& + \beta_{1} && + \beta_{2} sex_{m}&& + \beta_{3} age&& + \beta_{4} edu_{pri} \ast dpt_{per}&& + \beta_{5} dpt_{per} \ast edu_{mid}&& + \beta_{6} dpt_{per} \ast edu_{hig}&&+u_i\\ \end{alignedat} \end{equation}$$`

OLS估计的简要报告如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{wage}=&&+22.92&&-71.41sex_{m}&&+4.11age&&+28.21edu_{pri} \ast dpt_{per}\\ &\text{(t)}&&(0.7366)&&(-2.5720)&&(4.6325)&&(0.5166)\\&\text{(se)}&&(31.1120)&&(27.7639)&&(0.8870)&&(54.5979)\\&\text{(cont.)}&&+112.68dpt_{per} \ast edu_{mid}&&+33.12dpt_{per} \ast edu_{hig}\\&\text{(t)}&&(2.4012)&&(0.5965)\\&\text{(se)}&&(46.9258)&&(55.5274)\\&\text{(fitness)}&& R^2=0.2872;&& \bar{R^2}=0.2542\\& && F^{\ast}=8.70;&& p=0.0000\\ \end{alignedat} \end{equation}$$`

---

### 乘法模型：EViews报告

---

### 乘法模型：基础组

* 基础组（**文盲 & 短期合同 & 女性**），也即(**illiteracy & temporary & female**)的期望工资收入为（给定**年龄**为30岁）：

`$$\begin{align}
E(wage&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 0;dpt_{per}= 0;sex_{m}= 0)\\
=&+\beta_{1}+\beta_{2}(0)+\beta_{3}(30)+\beta_{4}(0)\cdot(0)+\beta_{5}(0)\cdot(0)+\beta_{6}(0)\cdot(0)\\
=&+\beta_{1}+30\beta_{3}\\
\end{align}$$`

`$$\begin{align}
(\widehat{wage}&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 0;dpt_{per}= 0;sex_{m}= 0)\\
=&+\hat{\beta}_{1}+\hat{\beta}_{2}(0)+\hat{\beta}_{3}(30)+\hat{\beta}_{4}(0)\\
&+\hat{\beta}_{5}(0)+\hat{\beta}_{6}(0)\\
=&+[22.92]+[-71.41]\cdot(0)+[4.11]\cdot(30)+[28.21]\cdot(0)\\
&+[112.68]\cdot(0)+[33.12]\cdot(0)\\
=&146.1879\\
\end{align}$$`

---

### 乘法模型：比较组1

比较组1（**高等学历 & 短期合同 & 女性**），也即(**high & temporary & female**)的期望工资收入为（给定**年龄**为30岁）：

`$$\begin{align}
(\widehat{wage}&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 0;sex_{m}= 0)\\
=&+\hat{\beta}_{1}+\hat{\beta}_{2}(0)+\hat{\beta}_{3}(30)+\hat{\beta}_{4}(0)\\
&+\hat{\beta}_{5}(0)+\hat{\beta}_{6}(0)\\
=&+[22.92]+[-71.41]\cdot(0)+[4.11]\cdot(30)+[28.21]\cdot(0)\\
&+[112.68]\cdot(0)+[33.12]\cdot(0)\\
=&146.1879\\
\end{align}$$`

---

### 乘法模型：比较组2

比较组2（**高等教育 & 长期合同 & 女性**），也即(**higher & permanent & female**)的期望工资收入为（给定**年龄**为30岁）：

`$$\begin{align}
(\widehat{wage}&|age=30;edu_{pri}= 0;edu_{mid}= 0;edu_{hig}= 1;dpt_{per}= 1;sex_{m}= 0)\\
=&+\hat{\beta}_{1}+\hat{\beta}_{2}(0)+\hat{\beta}_{3}(30)+\hat{\beta}_{4}(0)\\
&+\hat{\beta}_{5}(0)+\hat{\beta}_{6}(1)\\
=&+[22.92]+[-71.41]\cdot(0)+[4.11]\cdot(30)+[28.21]\cdot(0)\\
&+[112.68]\cdot(0)+[33.12]\cdot(1)\\
=&179.3103\\
\end{align}$$`

---

# 10.7 时间序列季节虚拟变量模型

---

layout: true
  
<div class="my-header-h2"></div>
<div class="watermark1"></div>
<div class="watermark2"></div>
<div class="watermark3"></div>

<div class="my-footer"><span>huhuaping@   <a href="#chapter10">第10章 虚拟变量回归模型</a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#series">10.7 时间序列季节虚拟变量模型</a></span></div>

---

## 时间序列季节虚拟变量模型

**时间序列季节虚拟变量模型**：是指时间变量以虚拟变量形式进入回归方程的模型，它是虚拟变量回归模型的一种特定形式及应用。

- 季节模式（seasonal pattern）：大多数时间序列经济变量，通常表现出来的季节性往复行为或现象。

- 季节调整（seasonal adjusted）：将时间序列经济变量的季节性变化成分去除，从而得到一个新的变量序列的处理过程。

事实上，一个时间序列经济变量往往同时存在四个成分，分别是：

- **季节成分**（seasonal component）

- **周期成分**（cyclical component）

- **趋势成分**（trend component）

- **严格随机成分**（strictly random component）

---

## 时间序列季节虚拟变量模型

如果把定性变量季节（season：Q1；Q2；Q3；Q4）变换为**虚拟变量体系**，则分别可以构建以第一季度为**基础组**的时间序列季节虚拟变量模型和**无基础组**的时间序列季节虚拟变量模型（
`$X_t$`为定量变量）。

`$$\begin{align}
Y_t=\beta_1+\beta_2X_t+\lambda_2D2_t+\lambda_3D3_t+\lambda_4D4_t+u_t \\
Y_t=\beta_2X_t+\lambda_1D1_t+\lambda_2D2_t+\lambda_3D3_t+\lambda_4D4_t+u_t 
\end{align}$$`

---

## 交通事故案例

下面我们对交通事故案例进行分析讨论。

---

### 数据

<div id="htmlwidget-6f55aa61ef3a2bb0166a" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-6f55aa61ef3a2bb0166a">{"x":{"filter":"none","vertical":false,"caption":"<caption>交通事故数据 (n=108)<\/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],[1981,1981,1981,1981,1981,1981,1981,1981,1981,1981,1981,1981,1982,1982,1982,1982,1982,1982,1982,1982,1982,1982,1982,1982,1983,1983,1983,1983,1983,1983,1983,1983,1983,1983,1983,1983,1984,1984,1984,1984,1984,1984,1984,1984,1984,1984,1984,1984,1985,1985,1985,1985,1985,1985,1985,1985,1985,1985,1985,1985,1986,1986,1986,1986,1986,1986,1986,1986,1986,1986,1986,1986,1987,1987,1987,1987,1987,1987,1987,1987,1987,1987,1987,1987,1988,1988,1988,1988,1988,1988,1988,1988,1988,1988,1988,1988,1989,1989,1989,1989,1989,1989,1989,1989,1989,1989,1989,1989],["1","2","3","4","5","6","7","8","9","10","11","12","1","2","3","4","5","6","7","8","9","10","11","12","1","2","3","4","5","6","7","8","9","10","11","12","1","2","3","4","5","6","7","8","9","10","11","12","1","2","3","4","5","6","7","8","9","10","11","12","1","2","3","4","5","6","7","8","9","10","11","12","1","2","3","4","5","6","7","8","9","10","11","12","1","2","3","4","5","6","7","8","9","10","11","12","1","2","3","4","5","6","7","8","9","10","11","12"],[40511,36034,40328,37699,38816,38900,38625,39539,38070,40676,39270,39734,36672,32699,41283,36455,36690,35837,36850,36284,36791,38366,37741,38331,37201,35901,40354,38276,36544,36127,37446,38977,39443,40098,41483,43493,35242,35215,39151,37808,39706,38857,40083,41797,42034,43836,44638,45369,38642,38616,44493,41134,42369,42430,41219,43359,41422,43961,45906,44165,43522,42104,47128,44892,46833,46349,48098,49329,48899,49025,46975,50266,45158,44303,47529,45364,47072,47090,48880,48357,46768,52326,48475,51924,43569,42585,45944,47003,46066,45719,47161,47558,47052,48297,49136,52971,43142,42475,49753,45161,46083,45540,46325,46742,47932,48678,47971,47251]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>obs<\/th>\n      <th>year<\/th>\n      <th>month<\/th>\n      <th>totacc<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"pageLength":9,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[9,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

### 散点图

---

### 虚拟变量

<div id="htmlwidget-89cf405fe1645c04aa29" style="width:100%;height:auto;" class="datatables html-widget"></div>
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---

### 模型设定PRM

我们以1月份为基础组，构建如下的有截距虚拟变量回归模型：

`$$\begin{equation} \begin{alignedat}{999} &log(totacc)=&& + \beta_{1} && + \beta_{2} feb&& + \beta_{3} mar&& + \beta_{4} apr\\ &\text{(cont.)}&& + \beta_{5} may&& + \beta_{6} jun&& + \beta_{7} aug&& + \beta_{8} sep\\ &\text{(cont.)}&& + \beta_{9} oct&& + \beta_{10} nov&& + \beta_{11} dec&&+u_i\\ \end{alignedat} \end{equation}$$`

---

### OLS估计

以上模型的OLS估计结果如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{log(totacc)}=&&+10.63&&-0.07feb&&+0.06mar&&-0.00apr\\ &\text{(t)}&&(433.2562)&&(-1.5759)&&(1.3715)&&(-0.0072)\\&\text{(se)}&&(0.0245)&&(0.0425)&&(0.0425)&&(0.0425)\\&\text{(cont.)}&&+0.02may&&+0.01jun&&+0.05aug&&+0.04sep\\&\text{(t)}&&(0.3778)&&(0.1622)&&(1.0867)&&(0.8778)\\&\text{(se)}&&(0.0425)&&(0.0425)&&(0.0425)&&(0.0425)\\&\text{(cont.)}&&+0.08oct&&+0.07nov&&+0.10dec\\&\text{(t)}&&(1.8779)&&(1.6876)&&(2.3376)\\&\text{(se)}&&(0.0425)&&(0.0425)&&(0.0425)\\&\text{(fitness)}&& R^2=0.1644;&& \bar{R^2}=0.0783\\& && F^{\ast}=1.91;&& p=0.0529\\ \end{alignedat} \end{equation}$$`

---

# 10.8 分段线性回归模型

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<div class="my-footer"><span>huhuaping@   <a href="#chapter10">第10章 虚拟变量回归模型</a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#piecewise">10.8 分段线性回归模型</a> </span></div>

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## 分段线性回归模型

**分段现象**：在经济关系中，当解释变量
`$X$`的值达到某一水平/阀值
`$X^\ast$`之前，与被解释变量之间存在某种线性关系；当解释变量X的值达到或者超过水平/阀值
`$X^\ast$`以后，与被解释变量的关系就会发生变化。因而总体看来，似乎被明显“分段”了。

**分段线性回归模型**（piecewise linear regression）：是指用虚拟变量估计不同水平/阀值的解释变量
`$X$`对被解释变量
`$Y$`的影响的一类线性回归模型。它是虚拟变量回归模型的一种特定形式及应用。

- 一个阀值的分段线性回归模型：

`$$\begin{align}
Y_i & =\beta_1+\beta_2X_i+\lambda(X_i-X^{\ast})D_i+u_i 
\end{align}$$`

- 两个阀值的分段线性回归模型：

`$$\begin{align}
Y_i & =\beta_1+\beta_2X_i+\lambda_1(X_i-X^{\ast}_1)D1_i+\lambda_2(X_i-X^{\ast}_2)D2_i+u_i 
\end{align}$$`

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## 供给和需求分析案例

下面我们对供给和需求分析案例进行分析讨论。

---

### 数据

<div id="htmlwidget-f26f03d2f5db4b0cad21" style="width:100%;height:auto;" class="datatables html-widget"></div>
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???
案例参考(https://stackoverflow.com/questions/8758646/piecewise-regression-with-r-plotting-the-segments)

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### 散点图

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### 虚拟变量体系

使用某些特定的方法，我们可以把数据的阀值设置为22.4，并对数据进行虚拟变量变换：

`$$\begin{align}
&level \quad \{a_1: \quad <22.4,a_2: \quad >22.4\} \\
&dummy \Longrightarrow  
  \begin{cases}
    D1  =
    \begin{cases}
    1, & \text{offer < 22.4}\\
    0, & \text{other}
    \end{cases} \\
    D2 = 
    \begin{cases}
    1, & \text{offer > 22.4}\\
    0, & \text{other}
    \end{cases} 
  \end{cases}
\end{align}$$`

---

### 虚拟变量变换

<div id="htmlwidget-6fcf3070a420d7fbd083" style="width:100%;height:auto;" class="datatables html-widget"></div>
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---

### 模型估计

我们以offer=22.4为门阀值，构建如下的分段线性回归模型：

`$$\begin{equation} \begin{alignedat}{999} &demand=&& + \beta_{1} && + \beta_{2} offer&& + \beta_{3} I(offer - 22.4) \ast D1&&+u_i\\ \end{alignedat} \end{equation}$$`

以上模型的OLS估计结果如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{demand}=&&-434.61&&+33.19offer&&-19.88I(offer - 22.4) \ast D1\\ &\text{(t)}&&(-4.8602)&&(11.1818)&&(-4.0721)\\&\text{(se)}&&(89.4227)&&(2.9685)&&(4.8828)\\&\text{(fitness)}&& R^2=0.8630;&& \bar{R^2}=0.8576\\& && F^{\ast}=160.60;&& p=0.0000 \end{alignedat} \end{equation}$$`

---

### 回归线

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# 本章结束