SEM-slide-part1-why.knit

background-image: url("../pic/slide-front-page.jpg")
class: center,middle
count: false

# 计量经济学II
# (Econometrics II)

### 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

### 2022-09-25

<div>
<style type="text/css">.xaringan-extra-logo {
width: 110px;
height: 70px;
z-index: 0;
background-image: url(../pic/logo/nwafu-logo-circle-wb.png);
background-size: contain;
background-repeat: no-repeat;
position: absolute;
top:0.2em;left:1em;
}
</style>
<script>(function () {
  let tries = 0
  function addLogo () {
    if (typeof slideshow === 'undefined') {
      tries += 1
      if (tries < 10) {
        setTimeout(addLogo, 100)
      }
    } else {
      document.querySelectorAll('.remark-slide-content:not(.title-slide):not(.inverse):not(.hide_logo)')
        .forEach(function (slide) {
          const logo = document.createElement('div')
          logo.classList = 'xaringan-extra-logo'
          logo.href = null
          slide.appendChild(logo)
        })
    }
  }
  document.addEventListener('DOMContentLoaded', addLogo)
})()</script>
</div>

---
layout: false
class: center, middle, duke-orange,hide_logo
name: chapter-navi
count: true

# 模块04：联立方程模型（SEM）

.large[

[Chapter 17. 内生性问题与工具变量法](#chapter17)

[.red[Chapter 18. 为什么要关心联立方程模型？]](#chapter18)

[Chapter 19. 联立方程模型的识别问题](#chapter19)

[Chapter 20. 联立方程模型的估计方法](#chapter20)

]

---
layout: false
class: center, middle, duke-softblue,hide_logo
name: chapter18

# Chapter 18. 为什么要关心联立方程模型？

[18.1 联立方程模型的本质](#nature)

[18.2 联立方程模型的表达和定义](#expression)

[18.3 OLS估计方法还合适么？](#OLS)

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

# 18.1 联立方程模型的本质

---
layout: true

<div class="my-footer"><span>huhuaping@ 
<a href="#chapter-navi">模块04 联立方程模型（SEM） |</a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#chapter18">第18章 为什么要关心联立方程模型？ |</a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#nature"> 18.1 联立方程模型的本质 |</a>
&emsp;&emsp;&emsp;&emsp;
</span></div>

---

### 联立方程模型的形态

**联立方程模型**（simultaneous equations model,SEM）：由多个方程组成的、方程之间存在相互关联的、联合影响的**内生变量**的方程系统。其基本形式如下：

`$$\begin{cases}
  \begin{align}
    Y_{1 i}=\beta_{10}+\gamma_{12} Y_{2 i}+\beta_{11} X_{1 i}+u_{i1} \\
    Y_{2 i}=\beta_{20}+\gamma_{21} Y_{1 i}+\beta_{21} X_{1 i}+u_{i2}
  \end{align}
\end{cases}$$`

---

### 常见的经济模型1：需求-供给模型

**需求-供给模型**：

`$$\begin{align}
\text { Demand function: } & {Q_{t}^{d}=\alpha_{0}+\alpha_{1} P_{t}+u_{ t1}, \quad \alpha_{1}<0} \\
{\text { supply function: }} & {Q_{t}^{s}=\beta_{0}+\beta_{1} P_{t}+u_{ t2}, \quad \beta_{1}>0} \\
{\text {Equilibrium condition:}} & {Q_{t}^{d}=Q_{t}^{s}}
\end{align}$$`

---

### 常见的经济模型2：凯恩斯收入决定模型

**凯恩斯收入决定模型**（Keynesian model of income determination）：

`$$\begin{cases}
  \begin{align}
    C_t &= \beta_0+\beta_1Y_t+\varepsilon_t &&\text{(消费函数)}\\
    Y_t &= C_t+I_t &&\text{(收入恒等式)}
  \end{align}
\end{cases}$$`

---

### 常见的经济模型3：IS模型

**投资储蓄的IS宏观经济模型**：

`$$\begin{align}
\text { Consumption function: } & C_{t}=\beta_{0}+\beta_{1} Y_{d t} & <\beta_{1}<1 \\ 
\text { Tax function: } & T_{t}=\alpha_{0}+\alpha_{1} Y_{t} & \quad 0<\alpha_{1}<1  \\ 
\text { Investment function: } & I_{t}=\gamma_{0}+\gamma_{1} r_{t} \\ 
\text { Definition: } & \gamma_{d t}=Y_{t}-T_{t} \\
\text { Government expenditure: } & G_{t}=\bar{G} \\ 
\text { National income identity: } & Y_{t}=C_{t}+I_{t}+G_{t}
\end{align}$$`

其中：
`$Y$`表示国民收入；
`$Y_d$`表示可支配收入；
`$r$`表示利率；
`$\bar{G}$`表示给定政府支出水平。

---

### 常见的经济模型4：LM模型

**货币市场均衡的LM宏观经济模型**：

`$$\begin{align}
{\text { Money demand function: }} & {M_{t}^{d}=a+b Y_{t}-c r_{t}} \\ 
{\text { Money supply function: }} & {M_{t}^{s}=\bar{M}} \\
{\text { Equilibrium condition: }} & {M_{t}^{d}=M_{t}^{s}}
\end{align}$$`

其中：
`$Y$`表示收入；
`$r$`表示利率;
`$\bar{M}$`表示给定货币供给量。

---

### 常见的经济模型5：克莱因模型

**克莱因模型**（Llein's model）：

`$$\begin{align}
\text { Consumption function: }  & C_{t}=\beta_{0}+\beta_{1} P_{t}+\beta_{2}\left(W+W^{\prime}\right)_{t}+\beta_{3} P_{t-1}+u_{ t1} \\ 
\text { Investment function: } & I_{t} =\beta_{4}+\beta_{5} P_{t}+\beta_{6} P_{t-1}+\beta_{7} K_{t-1}+u_{ t2} \\ 
\text { Demand for labor: } 
 & w_{t}= \beta_{8}+\beta_{9}\left(Y+T-W^{\prime}\right)_{t}  +\beta_{10}\left(Y+T-W^{\prime}\right)_{t-1}+\beta_{11} t+u_{ t3} \\ \text { Identity: } & Y_{t} = C_{t}+I_{t}+C_{t} \\ 
\text { Identity: }  & Y_{t}=W_{t}^{\prime}+W_{t}+P_{t} \\ 
\text { Identity: }  & K_{t}=K_{t-1}+I_{t} 
\end{align}$$`

其中：
`$C$`表示消费支出；
`$Y$`表示税后收入；
`$P$`表示利润；
`$W$`表示个人工资；
`$W^{\prime}$`表示政府工资；
`$K$`表示资本存货；
`$T$`表示税收。

---

### 生活中的模型1：凶杀犯罪率模型

`$$\begin{align}
\operatorname{murdpc} &=\alpha_{1} \operatorname{polpc}+\beta_{10}+\beta_{11} \text {incpc}+u_{1} \\
\text { polpc } &=\alpha_{2} \operatorname{murdpc}+\beta_{20}+\text { other factors. }
\end{align}$$`

其中：
`$murdpc$`表示人均凶杀犯罪数；
`$polpc$`表示人均警员数；
`$incpc$`表示人均收入。

---

### 生活中的模型2：住房支出-储蓄模型

`$$\begin{align}
\text {housing } & =\alpha_{1} \text {saving}+\beta_{10}+\beta_{11} \text {inc}+\beta_{12} e d u c+\beta_{13} \text {age}+u_{1} \\ 
\text {saving} &=\alpha_{2} \text {housing }+\beta_{20}+\beta_{21} \text {inc}+\beta_{22} e d u c+\beta_{23} \text {age}+u_{2}
\end{align}$$`

其中：
`$housing$`表示住房支出；
`$saving$`表示家庭储蓄；
`$inc$`表示家庭收入；
`$educ$`表示教育水平;
`$age$`表示年龄。

---

### 联立方程模型的特点

联立方程模型的本质是**内生变量**问题：

- 每一个方程都有其经济学因果关系（保持其他条件不变的）

- 样本数据只是各种变量的最终结果，但其中蕴含复杂相互因果互动关系

- 特定方程的参数估计，往往跟其他方程有联系

- 直接使用OLS方法估计参数将会不可靠

---

### 松露案例：案例背景

松露是美味的食材。它们是生长在地下的食用菌。我们考虑如下的松露供求模型:

`$$\begin{cases}
  \begin{align}
    \text { Demand: } & Q_{di}=\alpha_{1}+\alpha_{2} P_{i}+\alpha_{3} P S_{i}+\alpha_{4} D I_{i}+e_{d i} \\ 
    \text { Supply: } & Q_{si}=\beta_{1}+\beta_{2} P_{i}+\beta_{3} P F_{i}+e_{s i}\\
    \text { Equity: } & Q_{di}= Q_{si}
  \end{align}
\end{cases}$$`

>其中：
- 
`$Q_i=$`某一特定市场上松露的交易量;
- 
`$P_i=$`松露的市场价格;
- 
`$PS_i=$`松露替代物的市场价格;
- 
`$DI_i=$`当地居民每月人均可支配收入;
- 
`$PF_i=$`生产要素的价格，在本例中是搜索松露过程中租用小猪的每小时租金。

---

### 松露案例：变量说明

<div id="htmlwidget-ee8f7256fb40ab129753" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-ee8f7256fb40ab129753">{"x":{"filter":"none","vertical":false,"data":[["1","2","3","4","5"],["P","Q","PS","DI","PF"],["松露市场价格","松露产量或需求量","松露替代品的市场价格","当地居民可支配收入","生产成本(小猪的租金)"],["美元/盎司","盎司","美元/盎司","美元/人","美元/小时"]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>变量<\/th>\n      <th>含义<\/th>\n      <th>单位<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0}],"pageLength":6,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

???

案例来源：Hill, R. C., W. E. Griffiths and G. C. Lim. Principles of Econometrics 4th Edition  [M], Wiley, 2011. chpt 11。实例参考：[PoE with R](https://bookdown.org/ccolonescu/RPoE4/simultaneous-equations-models.html)

---

### 松露案例：数据表

<div id="htmlwidget-dd3df885f72c6ca91ecc" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-dd3df885f72c6ca91ecc">{"x":{"filter":"none","vertical":false,"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"],[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],[29.64,40.23,34.71,41.43,53.37,38.52,54.33,40.56,67.35,49.65,58.17,66.87,49.95,64.95,52.68,61.2,80.55,89.94,70.77,57.33,46.23,77.43,83.01,70.71,66.75,76.8,83.7,81,88.44,105.45],[19.89,13.04,19.61,17.13,22.55,6.37,15.02,10.22,23.64,16.12,24.55,18.92,11.94,18.93,12.6,20.49,22.94,21.08,16.68,17.61,16.62,20.99,24.53,19.67,23.29,16.64,20.81,14.95,26.27,20.65],[19.97,18.04,22.36,20.87,19.79,15.98,17.94,17.09,22.72,15.74,24.64,23.7,15.93,23.34,15.21,26.04,22.95,27.1,23.65,20.06,26.38,24.28,26.64,22.65,19.68,23.82,28.98,18.52,28.16,28.43],[2.103,2.043,1.87,1.525,2.709,2.489,2.294,2.196,3.885,3.169,2.623,3.007,3.367,3.29,3.746,3.518,4.381,4.121,3.82,4.398,3.764,4.524,4.815,3.67,4.392,4.603,4.632,4.894,5.125,4.836],[10.52,19.67,13.74,17.95,13.71,24.95,24.17,23.61,19.52,20.03,15.38,22.98,25.76,25.17,25.82,19.31,26.02,29.65,27.45,18,18.87,24.58,25.25,24.24,22.63,27.35,27.8,30.34,24.12,34.01]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>id<\/th>\n      <th>P<\/th>\n      <th>Q<\/th>\n      <th>PS<\/th>\n      <th>DI<\/th>\n      <th>PF<\/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":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

### 松露案例：散点图(P VS Q)

---

### 松露案例：矩阵散点图

```
Warning in par(usr): argument 1 does not name a graphical
parameter

Warning in par(usr): argument 1 does not name a graphical
parameter

Warning in par(usr): argument 1 does not name a graphical
parameter
```

---

### 松露案例：简单线性回归

从最简单线性回归模型开始。通常我们会使用价格(P)和产量(Q)数据直接做简单线性回归建模：

`$$\begin{align}
P & = \hat{\beta}_1+\hat{\beta}_2Q +e_1 && \text{(simple P model)}\\
Q & = \hat{\beta}_1+\hat{\beta}_2P +e_2  && \text{(simple Q model)}
\end{align}$$`

---

### 松露案例：简单线性回归

我们都知道，两个变量的线性回归是不对称的，因此有：

.pull-left[

- 简单的价格(P)模型回归结果如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{P}=&&+23.23&&+2.14Q\\ &\text{(t)}&&(1.8748)&&(3.2831)\\&\text{(se)}&&(12.3885)&&(0.6518)\\&\text{(fitness)}&& R^2=0.2780;&& \bar{R^2}=0.2522\\& && F^{\ast}=10.78;&& p=0.0028 \end{alignedat} \end{equation}$$`

]

.pull-right[

- 简单的数量(Q)模型回归结果如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Q}=&&+10.31&&+0.13P\\ &\text{(t)}&&(3.9866)&&(3.2831)\\&\text{(se)}&&(2.5863)&&(0.0396)\\&\text{(fitness)}&& R^2=0.2780;&& \bar{R^2}=0.2522\\& && F^{\ast}=10.78;&& p=0.0028 \end{alignedat} \end{equation}$$`

]

---

### 松露案例：样本回归线

.pull-left[
<img src="SEM-slide-part1-why_files/figure-html/scatter-truffles-left-1.png" style="display: block; margin: auto;" />

]

.pull-right[

]

---

### 松露案例：多元线性回归

当然，我们也可以继续使用更多的变量作为自变量X，构建如下的回归模型：

`$$\begin{align}
P & = \hat{\beta}_1+\hat{\beta}_2Q +\hat{\beta}_3DI+\hat{\beta}_2PS +e_1 && \text{(added P model)}\\
Q & = \hat{\beta}_1+\hat{\beta}_2P +\hat{\beta}_2PF+e_2  && \text{(added Q model)}
\end{align}$$`

- 我们努力地想让这些方程更加“合理”、“可信”。

---

### 松露案例：多元线性回归

- 增加变量的价格(P)模型回归结果如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{P}=&&-13.62&&+0.15Q&&+12.36DI&&+1.36PS\\ &\text{(t)}&&(-1.4987)&&(0.3032)&&(6.7701)&&(2.2909)\\&\text{(se)}&&(9.0872)&&(0.4988)&&(1.8254)&&(0.5940)\\&\text{(fitness)}&& R^2=0.8013;&& \bar{R^2}=0.7784\\& && F^{\ast}=34.95;&& p=0.0000 \end{alignedat} \end{equation}$$`

- 增加变量的数量(Q)模型回归结果如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Q}=&&+20.03&&+0.34P&&-1.00PF\\ &\text{(t)}&&(16.3938)&&(15.5436)&&(-13.1028)\\&\text{(se)}&&(1.2220)&&(0.0217)&&(0.0764)\\&\text{(fitness)}&& R^2=0.9019;&& \bar{R^2}=0.8946\\& && F^{\ast}=124.08;&& p=0.0000 \end{alignedat} \end{equation}$$`

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

# 18.2 联立方程模型的表达和定义

---
layout: true

<div class="my-footer"><span>huhuaping@ 
<a href="#chapter-navi">模块04 联立方程模型（SEM） |</a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#chapter18">第18章 为什么要关心联立方程模型？ |</a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#expression"> 18.2 联立方程模型的表达和定义 |</a>
&emsp;&emsp;&emsp;&emsp;
</span></div>

---

### 结构化SEM (1)：代数表达式A

**结构化的SEM**(Structural SEM system)：是直接描述经济结构或行为的方程组。

**结构化的SEM**代数表达式（形式A）可以表达为:

`$$\begin{cases}
  \begin{alignat}{5}
  Y_{t1}&= & &+\gamma_{21}Y_{t2}+&\cdots  &+\gamma_{m1}Y_{tm} & +\beta_{11}X_{1t}+\beta_{21}X_{t2} &+\cdots+\beta_{k1}X_{tk} &+\varepsilon_{t1} \\
  Y_{t2}&=&\gamma_{12}Y_{t1} &+  & \cdots &+\gamma_{m2}Y_{tm}&+\beta_{12}X_{t1}+\beta_{22}X_{t2} &+\cdots+\beta_{k2}X_{tk} &+\varepsilon_{t2}\\
  & \vdots &\vdots &&\vdots &&\vdots  \\
  Y_{tm}&=&\gamma_{1m}Y_{t1}&+\gamma_{2m}Y_{t2}+& \cdots  & &+\beta_{1m}X_{t1} +\beta_{2m}X_{t2} &+\cdots+\beta_{km}X_{tk} &+\varepsilon_{tm} 
  \end{alignat}
\end{cases}$$`

.pull-left[
- **内生变量**或联合应变量(m个)：
`$Y_{1},Y_{2},\cdots,Y_{m}$`
- **前定变量**(K个):
`$X_{1},X_{2},\cdots,X_{k}$`
- **结构随机干扰项**(m个):
`$\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{m}$`
]

.pull-right[
- **内生变量系数**
`$\gamma$`
- **前定变量系数**
`$\beta$`
- **观测个数**
`$t=1,2,\cdots,T$`
]

---

### 结构化SEM (1)：结构系数

**结构系数**(Structural coefficients)：是结构化SEM中的参数 ，它们反映了经济结果或行为的关系。包括：

>- **.red[内生] 结构系数**:
`$\gamma_{11}, \gamma_{21},\cdots, \gamma_{m1}; \cdots; \gamma_{1m}, \gamma_{2m},\cdots, \gamma_{mm}$`

>- **.red[外生]结构系数**:
`$\beta_{11}, \beta_{21},\cdots, \beta_{m1}; \cdots; \beta_{1m}, \beta_{2m},\cdots, \beta_{mm};$`

---

### 结构化SEM (1)：结构变量

`$$\begin{cases}
  \begin{alignat}{5}
    Y_{t1}&= & &+\gamma_{21}Y_{t2}+&\cdots  &+\gamma_{m1}Y_{tm} & +\beta_{11}X_{1t}+\beta_{21}X_{t2} &+\cdots+\beta_{k1}X_{tk} &+\varepsilon_{t1} \\
    Y_{t2}&=&\gamma_{12}Y_{t1} &+  & \cdots &+\gamma_{m2}Y_{tm}&+\beta_{12}X_{t1}+\beta_{22}X_{t2} &+\cdots+\beta_{k2}X_{tk} &+\varepsilon_{t2}\\
    & \vdots &\vdots &&\vdots &&\vdots  \\
    Y_{tm}&=&\gamma_{1m}Y_{t1}&+\gamma_{2m}Y_{t2}+& \cdots  & &+\beta_{1m}X_{t1} +\beta_{2m}X_{t2} &+\cdots+\beta_{km}X_{tk} &+\varepsilon_{tm} 
  \end{alignat}
\end{cases}$$`

- **内生变量**(Endogenous variables)： 由SEM决定的那些变量。

- **前定变量**(Predetermined variables)：那些不由SEM本身在当期（**current time period**）决定的变量。

.pull-left[
> **内生变量**，例如:
`$Y_{t1};Y_{t2}; \cdots; Y_{tm}$`

]

.pull-right[

> **前定变量**，例如:
`$X_{..}$`

]

- **结构随机干扰项**(Structural disturbance)：也即结构化SEM系统中的**随机干扰项**。例如：
`$\varepsilon_{t1};\varepsilon_{t2}; \cdots; \varepsilon_{tm}$`

---

### 结构化SEM (1)：前定变量

**前定变量**(Predetermined variables)：那些不由SEM本身在当期（**current time period**）决定的变量。具体又包括：

- **外生变量**(exogenous variables)

- **滞后内生变量**(lagged endogenous variables)

`$$\begin{cases}
  \begin{alignat}{5}
    Y_{t1}&= & &+\gamma_{21}Y_{t2}+&\cdots  &+\gamma_{m1}Y_{tm} & +\beta_{11}X_{1t}+\beta_{21}X_{t2} &+\cdots+\beta_{k1}X_{tk} &+\varepsilon_{t1} \\
    Y_{t2}&=&\gamma_{12}Y_{t1} &+  & \cdots &+\gamma_{m2}Y_{tm}&+\beta_{12}X_{t1}+\beta_{22}X_{t2} &+\cdots+\beta_{k2}X_{tk} &+\varepsilon_{t2}\\
    & \vdots &\vdots &&\vdots &&\vdots  \\
    Y_{tm}&=&\gamma_{1m}Y_{t1}&+\gamma_{2m}Y_{t2}+& \cdots  & &+\beta_{1m}X_{t1} +\beta_{2m}X_{t2} &+\cdots+\beta_{km}X_{tk} &+\varepsilon_{tm} 
  \end{alignat}
\end{cases}$$`

???

You should remind that the predetermined variables  consist of both the exogenous variables and thelagged endogenous variables.

---

### 结构化SEM (1)：前定变量

- **.red[外生变量]**: 那些不被SEM本身决定的变量，它们既不会在**当期**（current period)被SEM决定，也不是在**滞后期**(lagged period)被SEM说决定。

- **.red[滞后内生变量]**(Lagged en]dogenous variables)：是当期内生变量的滞后变量。

.pull-left[

> **当期外生变量**:
- 
`$X_{t1}, X_{t2},\cdots, X_{tk}$`.

> **滞后外生变量**: 
- 变量
`$X_{t1}$`的滞后期变量:
`$X_{t-1,1}; X_{t-2,1};\cdots; X_{t-(T-1),1}$`
- 变量
`$X_{tk}$`的滞后期变量:
`$X_{t-1,k}; X_{t-2,k};\cdots; X_{t-(T-1),k}$`
- 
`$\cdots$`

]

.pull-right[
> **滞后内生变量**:
- 变量
`$Y_{t1}$`的滞后变量:
`$Y_{t-1,1}; Y_{t-2,1}; \cdots, Y_{t-(T-1),1}$`
- 变量
`$Y_{tm}$`的滞后变量：
`$Y_{t-1,m}; Y_{t-2,m}; \cdots;Y_{t-(T-1),m}$`
- 
`$\cdots$`
]

???

> So, you may get two types Exogenous variables, which are current period exogenous, such as all X_ts, and Lagged .red[en]dogenous variables, such as all X_t-s.

---

### 结构化SEM (1)：前定系数

`$$\begin{cases}
  \begin{alignat}{5}
    Y_{t1}&= & &+\gamma_{21}Y_{t2}+&\cdots  &+\gamma_{m1}Y_{tm} & +\beta_{11}X_{1t}+\beta_{21}X_{t2} &+\cdots+\beta_{k1}X_{tk} &+\varepsilon_{t1} \\
    Y_{t2}&=&\gamma_{12}Y_{t1} &+  & \cdots &+\gamma_{m2}Y_{tm}&+\beta_{12}X_{t1}+\beta_{22}X_{t2} &+\cdots+\beta_{k2}X_{tk} &+\varepsilon_{t2}\\
    & \vdots &\vdots &&\vdots &&\vdots  \\
    Y_{tm}&=&\gamma_{1m}Y_{t1}&+\gamma_{2m}Y_{t2}+& \cdots  & &+\beta_{1m}X_{t1} +\beta_{2m}X_{t2} &+\cdots+\beta_{km}X_{tk} &+\varepsilon_{tm} 
  \end{alignat}
\end{cases}$$`

**前定系数**(Predetermined coefficients): 是指前定变量前的系数.

>例如:
- 所有的
`$\beta_{..}$`

???

Of course, we may denote coefficients before predetermined variables as **Predetermined coefficients**.

---

### 结构化SEM (1)：代数表达式B

通过简单的变形，结构化SEM的**代数表达式A**，可以转换为如下的**代数表达式B** :

`$$A: \begin{cases}
  \begin{alignat}{5}
    Y_{t1}&= & &+\gamma_{21}Y_{t2}+&\cdots  &+\gamma_{m1}Y_{tm} & +\beta_{11}X_{1t}+\beta_{21}X_{t2} &+\cdots+\beta_{k1}X_{tk} &+\varepsilon_{t1} \\
    Y_{t2}&=&\gamma_{12}Y_{t1} &+  & \cdots &+\gamma_{m2}Y_{tm}&+\beta_{12}X_{t1}+\beta_{22}X_{t2} &+\cdots+\beta_{k2}X_{tk} &+\varepsilon_{t2}\\
    & \vdots &\vdots &&\vdots &&\vdots  \\
    Y_{tm}&=&\gamma_{1m}Y_{t1}&+\gamma_{2m}Y_{t2}+& \cdots  & &+\beta_{1m}X_{t1} +\beta_{2m}X_{t2} &+\cdots+\beta_{km}X_{tk} &+\varepsilon_{tm} 
  \end{alignat}
\end{cases}$$`

.small[

`$$\Rightarrow B: \begin{cases}
  \begin{alignat}{5}
  \gamma_{11}Y_{t1} &+ \gamma_{21}Y_{t2}&+\cdots &+\gamma_{m-1,1}Y_{t,m-1} &+\gamma_{m1}Y_{tm} & +\beta_{11}X_{t1}+\beta_{21}X_{t2} &+\cdots+\beta_{k1}X_{tk} &=\varepsilon_{t1} \\
  \gamma_{12}Y_{t1} &+\gamma_{22}Y_{t2} &+   \cdots&+\gamma_{m-1,2}Y_{t,m-1} &+\gamma_{m2}Y_{tm}&+\beta_{12}X_{t1}+\beta_{22}X_{t2} &+\cdots+\beta_{k2}X_{tk} &= \varepsilon_{t2}\\
  & \vdots &\vdots &&\vdots &&\vdots  \\
  \gamma_{1m}Y_{t1}&+\gamma_{2m}Y_{t2}&+ \cdots &+\gamma_{m-1,m}Y_{t,m-1} & +\gamma_{mm}Y_{tm}  &+\beta_{1m}X_{t1} +\beta_{2m}X_{t2} &+\cdots+\beta_{km}X_{tk} &=\varepsilon_{tm} 
  \end{alignat}
\end{cases}$$`

]

---

### 结构化SEM (2)：矩阵表达式

采用矩阵语言(Matrix language)，前述结构化SEM还可以进一步表达为 **矩阵形式**(matrix expression):

`$$\begin{equation}
    \begin{bmatrix}
    Y_1 & Y_2 & \cdots & Y_m \\
  \end{bmatrix} _t
  \begin{bmatrix}
    \gamma_{11} & \gamma_{12} & \cdots & \gamma_{1m} \\
    \gamma_{21} & \gamma_{22} & \cdots & \gamma_{2m} \\
    \cdots & \cdots & \cdots  & \cdots \\
    \gamma_{m1} & \gamma_{m2} & \cdots & \gamma_{mm} \\ 
  \end{bmatrix} + \\
  \begin{bmatrix}
    X_1 & X_2 & \cdots & X_m \\
  \end{bmatrix} _t
  \begin{bmatrix}
    \beta_{11} & \beta_{12} & \cdots & \beta_{1m} \\
    \beta_{21} & \beta_{22} & \cdots & \beta_{2m} \\
    \cdots & \cdots & \cdots & \cdots\\
    \beta_{k1} & \beta_{k2} & \cdots & \beta_{km} \\
  \end{bmatrix} \\ =
  \begin{bmatrix}
    \varepsilon_1 & \varepsilon_2 & \cdots & \varepsilon_m \\
  \end{bmatrix} _t
\end{equation}$$`

???

Here, we arranged the matrix expression of SEM based on former algebraic expression B.

---

### 结构化SEM (2)：矩阵表达式

简单起见，我们可以将结构化SEM的**矩阵形式**一般化记为：

`$$\begin{aligned}
& \boldsymbol{y^{\prime}_t} \boldsymbol{\Gamma} &+ & \boldsymbol{x^{\prime}_t} \boldsymbol{B} &= & \boldsymbol{{\varepsilon^{\prime}_t}} \\
&(1 \ast m)(m \ast m) & & (1 \ast k)(k \ast m) & & (1 \ast m)
\end{aligned}$$`

> **其中**：
- 粗体大写英文字母和希腊字母表示 **矩阵**(matrix)
>
- 粗体小写英文字母和希腊字母表示**列向量**(column vector)

???

For Simplicity, we can generized the **matrix expression** of SEM as follow.

> we should remind that：
- the dimension of the vector and matrix is important;
- and matrix compatibility  /kəmˌpætəˈbɪləti/ is needed in matrix calculation.

---

### 结构化SEM (2)：内生系数矩阵

对于 **.red[内生系数矩阵]**
`$\boldsymbol{\Gamma}$`：

- 为了确保每一个方程起码有1个 **因变量**，则矩阵
`$\boldsymbol{\Gamma}$` 的每1列起码要有1个元素含有常数1
- 如果矩阵
`$\boldsymbol{\Gamma}$` 是一个**上三角矩阵**(upper triangular matrix)，那么SEM将会是一个**递归模型**(recursive model)系统。
- 同时，为了保证SEM的参数估计解存在，矩阵
`$\boldsymbol{\Gamma}$` 必须是**非奇异矩阵**(nonsingular matrix).

.pull-left[

`$$\begin{equation}
\boldsymbol{\Gamma} =
  \begin{bmatrix}
      \gamma_{11} & \gamma_{12} & \cdots & \gamma_{1m} \\
      \gamma_{21} & \gamma_{22} & \cdots & \gamma_{2m} \\
      \cdots & \cdots & \cdots  & \cdots \\
      \gamma_{m1} & \gamma_{m2} & \cdots & \gamma_{mm} \\ 
  \end{bmatrix} \\
  \text{if }\Rightarrow 
  \begin{bmatrix}
      \gamma_{11} & \gamma_{12} & \cdots & \gamma_{1m} \\
      0 & \gamma_{22} & \cdots & \gamma_{2m} \\
      \cdots & \cdots & \cdots  & \cdots \\
      0 & 0 & \cdots & \gamma_{mm} \\ 
  \end{bmatrix}
\end{equation}$$`

]

.pull-right[

`$$\begin{cases}
  \begin{aligned}
  y_{1t} &=& f_{1}\left(\mathbf{x}_{t}\right)+\varepsilon_{t1} \\ 
  y_{2t} &=& f_{2}\left(y_{t1}, \mathbf{x}_{t}\right)+\varepsilon_{t2} \\ 
  & \vdots & \vdots \\ 
  y_{mt} &=& f_{m}\left(y_{t1}, y_{t2}, \ldots, \mathbf{x}_{t}\right)+\varepsilon_{mt} 
  \end{aligned}
\end{cases}$$`

]

???

Now, let's focus the **.red[En]dogenous parameter matrix** firstly.

---

### 结构化SEM (2)：内生系数矩阵

**.red[外生]系数矩阵**
`$\boldsymbol{B}$`：

`$$\begin{equation}
\boldsymbol{B} =
  \begin{bmatrix}
    \beta_{11} & \beta_{12} & \cdots & \beta_{1m} \\
    \beta_{21} & \beta_{22} & \cdots & \beta_{2m} \\
    \cdots & \cdots & \cdots & \cdots\\
    \beta_{k1} & \beta_{k2} & \cdots & \beta_{km} \\
  \end{bmatrix}
\end{equation}$$`

> 需要注意的是：SEM系统是有截距的，因此我们要记住外生系数矩阵的**第1列**都是截距系数。

---

### 约简化SEM (1):代数表达式

**约简方程**(Reduced equations):  将1个**内生变量**(endogenous variable)，表达成只包含**前定变量**(predetermined variables)和**随机干扰项**的方程。

`$$\begin{cases}
  \begin{alignat}{5}
  Y_{t1}&=  & +\pi_{11}X_{t1}+\pi_{21}X_{t2} &+\cdots+\pi_{k1}X_{tk} & + v_{t1} \\
  Y_{t2}&=&+\pi_{12}X_{t1}+\pi_{22}X_{t2} &+\cdots+\pi_{k2}X_{tk} & + v_{t2}\\
  & \vdots &\vdots &&\vdots &  \\
  Y_{tm}&=&+\pi_{1m}X_{t1} +\pi_{2m}X_{t2} &+\cdots+\pi_{km}X_{tk} & + v_{tm}
  \end{alignat}
\end{cases}$$`

上面，我们把结构化方程里的每1个内生变量，都表达为了约简方程，这样的方程系统被称为**约简化SEM**系统。

???

As you see, we know well with the structural SEMs because we have learned thess models in the economic textbooks.

Now, we will go ahead to  the important concept of reduced SEM.

___

So, a reduced equation is one that expresses an **endogenous variable** solely in terms of the **predetermined variables** and the **stochastic disturbances**.

We denoted the reduced SEM as follows.

---

### 约简化SEM (1): 约简系数和随机干扰项

- **约简系数**(Reduced coefficients): 也即约简化SEM系统里的所有**参数**.

- **约简随机干扰项**(Reduced disturbance): 也即约简化SEM系统里**随机干扰项**

.pull-left[

> 约简系数:
- 
`$\pi_{11},\pi_{21},\cdots, \pi_{k1}$`
- 
`$\pi_{1m},\pi_{2m},\cdots, \pi_{km}$`.

]

.pull-right[

> 约简随机干扰项
- 
`$v_{1},v_2,\cdots, v_m$`。

]

???

Hence, we induce the concepts of Reduced coefficients and Reduced disturbance.

- Reduced coefficients are parameters in the reduced SEM, such as all `$\pi_{km}$`s.

- Reduced disturbance: stochastic disturbance terms in the reduced SEM, such as all `$v_{m}$`s.

---

### 约简化SEM (2):矩阵表达式

对于上述.red[代数]形式的**约简SEM**，我们也可以表达为如下.red[矩阵]形式的**约简SEM**:

`$$\begin{equation}
    \begin{bmatrix}
    Y_1 & Y_2 & \cdots & Y_m \\
  \end{bmatrix} _t = \\
  \begin{bmatrix}
    X_1 & X_2 & \cdots & X_m \\
  \end{bmatrix} _t
  \begin{bmatrix}
    \pi_{11} & \pi_{12} & \cdots & \pi_{1m} \\
    \pi_{21} & \pi_{22} & \cdots & \pi_{2m} \\
    \cdots & \cdots & \cdots  & \cdots \\
    \pi_{m1} & \pi_{m2} & \cdots & \pi_{mm} \\ 
  \end{bmatrix}  + 
  \begin{bmatrix}
    v_1 & v_2 & \cdots & v_m \\
  \end{bmatrix} _t
\end{equation}$$`

???

Also, we can denote the matrix form of the reduced SEM as below.

---

### 约简化SEM (2):矩阵表达式

简单起见，我们可以把.red[矩阵]形式的**约简SEM**，进一步记为：

`$$\begin{aligned}
& \boldsymbol{y^{\prime}_t} & =  &\boldsymbol{x^{\prime}_t} \boldsymbol{\Pi}  & + & \boldsymbol{{v^{\prime}_t}}  \\
&(1 \ast m) & & (1 \ast k)(k \ast m) & &  (1 \ast m) 
\end{aligned}$$`

.pull-left[

- **约简系数矩阵**记为:

`$$\begin{equation}
\boldsymbol{\Pi} = 
  \begin{bmatrix}
    \pi_{11} & \pi_{12} & \cdots & \pi_{1m} \\
    \pi_{21} & \pi_{22} & \cdots & \pi_{2m} \\
    \cdots & \cdots & \cdots  & \cdots \\
    \pi_{m1} & \pi_{m2} & \cdots & \pi_{mm} \\ 
  \end{bmatrix}
\end{equation}$$`

]

.pull-right[

- **约简随机干扰项**记为:

`$$\begin{equation}
\boldsymbol{{v^{\prime}_t}}=
  \begin{bmatrix}
  v_1 & v_2 & \cdots & v_m \\
  \end{bmatrix}_t
\end{equation}$$`

]

---

### 结构化SEM与约简化SEM的关系: 方程系统

显然，我们可以从**结构化SEM**推导得到**约简化SEM**:

`$$\Rightarrow\begin{cases}
  \begin{alignat}{5}
  Y_{t1}&=  & +\pi_{11}X_{t1}+\pi_{21}X_{t2} &+\cdots+\pi_{k1}X_{tk} & + v_{t1} \\
  Y_{t2}&=&+\pi_{12}X_{t1}+\pi_{22}X_{t2} &+\cdots+\pi_{k2}X_{tk} & + v_{t2}\\
  & \vdots &\vdots &&\vdots &  \\
  Y_{tm}&=&+\pi_{1m}X_{t1} +\pi_{2m}X_{t2} &+\cdots+\pi_{km}X_{tk} & + v_{tm}
  \end{alignat}
\end{cases}$$`

???

Untill now, we have two notation systems. one is the structural SEM, and the other is the reduced SEM.

So, Why we need these two notation systems ?

And what is the relationship between these two notation systems ?

A short answer is that we can deduce the structural parameters with the reduced coefficients.

---

### 结构化SEM与约简化SEM的关系:  系数关系

对于结构化SEM的矩阵形式：

`$$\begin{aligned}
 \boldsymbol{y^{\prime}_t} \boldsymbol{\Gamma} +  \boldsymbol{x^{\prime}_t} \boldsymbol{B} =  \boldsymbol{{\varepsilon^{\prime}_t}} 
\end{aligned}$$`

以及约简化SEM的矩阵形式：

`$$\begin{aligned}
 \boldsymbol{y^{\prime}_t}  =  \boldsymbol{x^{\prime}_t} \boldsymbol{\Pi}   +  \boldsymbol{{v^{\prime}_t}}  
\end{aligned}$$`

.pull-left[

- 不难发现二者系数矩阵存在如下关系:

`$$\begin{align}
\boldsymbol{\Pi} &= - \boldsymbol{B} \boldsymbol{\Gamma^{-1}}\\
\boldsymbol{{v^{\prime}_t}} &= \boldsymbol{{\varepsilon^{\prime}_t}} \boldsymbol{\Gamma}^{-1} 
\end{align}$$`

]

.pull-right[

- 其中:

]

???

This slide shows the relationship between the structural SEM and the reduced SEM.

And the reduced coefficients equals negetive B times the inverse matrix of structural pars.

___

`$$\begin{aligned}
 & \boldsymbol{y^{\prime}_t} & =  &-\boldsymbol{x^{\prime}_t} \boldsymbol{B} \boldsymbol{\Gamma^{-1}} & + & \boldsymbol{{\varepsilon^{\prime}_t}} \boldsymbol{\Gamma}^{-1}  \\
&(1 \ast m) & & (1 \ast k)(k \ast m)(m \ast m) & &  (1 \ast m)(m \ast m) 
\end{aligned}$$`

---

### 结构化SEM与约简化SEM的关系: 矩关系(Moments)

下面，我们来关注一下二者**随机干扰项**的**一阶矩**和**二阶矩**，以及它们之间的关系：

- 首先, 我们假定**结构随机干扰项**满足如下条件：

`$$\begin{align}
\mathbf{E[\varepsilon_t | x_t]} &= \mathbf{0} \\
\mathbf{E[\varepsilon_t \varepsilon^{\prime}_t |x_t]} &= \mathbf{\Sigma} \\
E\left[\boldsymbol{\varepsilon}_{t} \boldsymbol{\varepsilon}_{s}^{\prime} | \mathbf{x}_{t}, \mathbf{x}_{s}\right] &=\mathbf{0}, \quad \forall t, s
\end{align}$$`

- 随后，我们将能够证明**约简随机干扰项**将满足:

`$$\begin{align}
E\left[\mathbf{v}_{t} | \mathbf{x}_{t}\right] &=\left(\mathbf{\Gamma}^{-1}\right)^{\prime} \mathbf{0}=\mathbf{0} \\ E\left[\mathbf{v}_{t} \mathbf{v}_{t}^{\prime} | \mathbf{x}_{t}\right] &=\left(\mathbf{\Gamma}^{-1}\right)^{\prime} \mathbf{\Sigma} \mathbf{\Gamma}^{-1}=\mathbf{\Omega} \\
\text{where: }\mathbf{\Sigma} &=\mathbf{\Gamma}^{\prime} \mathbf{\Omega} \mathbf{\Gamma}
\end{align}$$`

???

This slide shows the first and second moments of the disturbance on both structural and reduced SEM.

First , let us assumed the moments of structural disturbances satisfy following moment conditions. which means that the structural disturbances are drawn from an M-variate distribution with zero conditional expectation and zero covariance.

Then, we can prove that the reduced disturbances will also be zero conditional expectation and its variance-covariance matrix equals omega.

Finaly, we know the relationship between these two variance-covariance matrix.

We denote the relationship as Sigma equals transpose Gamma times Omega times Gamma.

---

### 结构化SEM与约简化SEM的关系: 给定样本数据*

在给定的样本数据下，我们可以按行来得到数据矩阵(假设总共有
`$T$`个样本观测数):

`$$\begin{align}
\left[\begin{array}{lll}{\mathbf{Y}} & {\mathbf{X}} & {\mathbf{E}}\end{array}\right]=\left[\begin{array}{ccc}{\mathbf{y}_{1}^{\prime}} & {\mathbf{x}_{1}^{\prime}} & {\boldsymbol{\varepsilon}_{1}^{\prime}} \\ {\mathbf{y}_{2}^{\prime}} & {\mathbf{x}_{2}^{\prime}} & {\boldsymbol{\varepsilon}_{2}^{\prime}} \\ {\vdots} & {} \\ {\mathbf{y}_{T}^{\prime}} & {\mathbf{x}_{T}^{\prime}} & {\boldsymbol{\varepsilon}_{T}^{\prime}}\end{array}\right]
\end{align}$$`

那么，**结构化SEM**可以表达为：

`$$\begin{align}
\mathbf{Y} \mathbf{\Gamma}+\mathbf{X} \mathbf{B}=\mathbf{E}
\end{align}$$`

**结构随机干扰项**的1阶矩和2阶矩可以表达为:

`$$\begin{align}
E[\mathbf{E} | \mathbf{X}] &=\mathbf{0} \\ 
E\left[(1 / T) \mathbf{E}^{\prime} \mathbf{E} | \mathbf{X}\right] &=\mathbf{\Sigma}
\end{align}$$`

???

The next two slides will show us some useful relationships under sample data set.

I will not discuss them here, and you should try to learn this content by yourself.

---

### 结构化SEM与约简化SEM的关系: 给定样本数据*

假定:

`$$\begin{align}
(1 / T) \mathbf{X}^{\prime} \mathbf{X} & \rightarrow \mathbf{Q}  \text{ ( a  finite positive definite matrix)} \\
(1 / T) \mathbf{X}^{\prime} \mathbf{E} & \rightarrow \mathbf{0}
\end{align}$$`

那么**约简化SEM**可以表达为:

`$$\begin{align}
\mathbf{Y} & =\mathbf{X} \boldsymbol{\Pi}+\mathbf{V} && \leftarrow  \mathbf{V}=\mathbf{E} \mathbf{\Gamma}^{-1}
\end{align}$$`

此外，我们还可以得到如下一些有用的样本统计量：

`$$\begin{align}
\frac{1}{T}
  \begin{bmatrix}
    {\mathbf{Y}^{\prime}} \\ 
    {\mathbf{X}^{\prime}} \\ 
{\mathbf{V}^{\prime}}
  \end{bmatrix}
  \begin{bmatrix}
    {\mathbf{Y}} & {\mathbf{X}} & {\mathbf{V}}
  \end{bmatrix}
\quad \rightarrow  \quad
  \begin{bmatrix}
    {\mathbf{I}^{\prime} \mathbf{Q} \mathbf{I}+\mathbf{\Omega}} & {\mathbf{I} \mathbf{I}^{\prime} \mathbf{Q}} & {\mathbf{\Omega}} \\
    {\mathbf{Q} \mathbf{I}} & {\mathbf{Q}} & {\mathbf{0}^{\prime}} \\ {\mathbf{\Omega}} & {\mathbf{0}} & {\mathbf{\Omega}}
  \end{bmatrix}
\end{align}$$`
???

The next two slides will show us some useful relationships under sample data set.

I will not discuss them here, and you should try to learn this content by yourself.

---

### 案例1：凯恩斯收入决定模型（2方程模型）

以凯恩斯收入决定的联立方程为例（结构模型）：

`$$\begin{cases}
  \begin{align}
    C_t &= \beta_0+\beta_1Y_t+\varepsilon_t &&\text{(消费函数)}\\
    Y_t &= C_t+I_t &&\text{(收入恒等式)}
  \end{align}
\end{cases}$$`

上述结构模型中：

**内生变量**有2个：
`$c_t;Y_t$`

**前定变量**有1个，其中：

- **外生变量**有1个：
`$I_t$`

- **滞后内生变量**有0个

> **思考提问**：怎样把这个结构模型转变为约简模型？（考点）

---

### 案例1：凯恩斯收入决定模型（2方程模型）

上述**结构模型**，可以变换为为如下**约简方程**：

`$$\begin{cases}
  \begin{align}
    Y_t &=\frac{\beta_0}{1-\beta_1}+\frac{1}{1-\beta_1}I_t+\frac{\varepsilon_t}{1-\beta_1}
&&\text{(变换方程1)} \\
    C_t &=\frac{\beta_0}{1-\beta_1}+\frac{\beta_1}{1-\beta_1}I_t+\frac{\varepsilon_t}{1-\beta_1}
&&\text{(变换方程2)}
  \end{align}
\end{cases}$$`

进一步可记为如下约简方程形式：

`$$\begin{cases}
  \begin{align}
    Y_t &= \pi_{11}+\pi_{21}I_t+v_{t1} &&\text{(约简方程1)}\\
    C_t &= \pi_{12}+\pi_{22}I_t+v_{t2} &&\text{(约简方程2)}
  \end{align}
\end{cases}$$`

易知：**结构系数**共有2个
`$\beta_0;\beta_1$`；而**约简系数**共有4个
`$\pi_{11},\pi_{21};\pi_{12},\pi_{22}$`（实际上只有3个！）

---

### 案例1：凯恩斯收入决定模型（2方程模型）

其中**约简系数**和**结构系数**的关系为：

`$$\begin{cases}
  \begin{align}
    \pi_{11} & = \frac{\beta_0}{1-\beta_1}; 
    & \pi_{12} & = \frac{1}{1-\beta_1} & \\ 
    \pi_{21} & = \frac{\beta_0}{1-\beta_1}; 
    & \pi_{22} & = \frac{\beta_1}{1-\beta_1} \\
    v_{t1} & = \frac{\varepsilon_t}{1-\beta_1};
    & v_{t2} & = \frac{\varepsilon_t}{1-\beta_1}
  \end{align}
\end{cases}$$`

---

### 案例2：小型宏观经济模型（3方程联立模型）

考虑如下的**小型宏观经济模型**（结构方程）：

`$$\begin{cases}
  \begin{aligned} 
    \text { consumption: }  c_{t} &=\alpha_{0}+\alpha_{1} y_{t}+\alpha_{2} c_{t-1}+\varepsilon_{t, c} \\ 
    \text { investment: }  i_{t} &=\beta_{0}+\beta_{1} r_{t}+\beta_{2}\left(y_{t}-y_{t-1}\right)+\varepsilon_{t, j} \\ 
    \text { demand: }  y_{t} &=c_{t}+i_{t}+g_{t} 
  \end{aligned}
\end{cases}$$`

其中：
`$c_t$`表示消费；
`$y_t$`表示产出；
`$i_t$`表示投资；
`$r_t$`表示利率；
`$g_t$`表示政府支出。

**内生变量**有3个：
`$c_t;i_t;Y_t$`

**前定变量**有4个，其中：

- **外生变量**有2个：
`$r_t;g_t$`

- **滞后内生变量**有2个：
`$y_{t-1};c_{t-1}$`

结构系数共有6个：
`$\alpha_0,\alpha_1,\alpha_2;\beta_0,\beta_1,\beta_2;$`

---

### 案例2：小型宏观经济模型（3方程联立模型）

上述**结构方程**可以变换为如下**约简方程**：（HOW TO??）

`$$\begin{cases}
  \begin{align}
    c_{t}  = &  [{\alpha_{0}}{\left(1-\beta_{2}\right)}+\beta_{0} \alpha_{1}+\alpha_{1} \beta_{1}  r_{t}+\alpha_{1} g_{t}+\alpha_{2}\left(1-\beta_{2}\right) c_{t-1}-\alpha_{1} \beta_{2} y_{t-1} \\
    +&\left(1-\beta_{2}\right) \varepsilon_{t, c}+\alpha_{1} \varepsilon_{t, j}] /{\Lambda} \\ 
    i_{t}  = & [\alpha_{0} \beta_{2}+\beta_{0}\left(1-\alpha_{1}\right)+\beta_{1}\left(1-\alpha_{1}\right) r_{t}+\beta_{2} g_{t}+\alpha_{2} \beta_{2} c_{t-1}-\beta_{2}\left(1-\alpha_{1}\right) y_{t-1} \\
    +&\beta_{2} \varepsilon_{t, c}+\left(1-\alpha_{1}\right) \varepsilon_{t, j}]/{\Lambda} \\ 
    y_{t} = & [\alpha_{0}+\beta_{0}+\beta_{1} r_{t}+g_{t}+\alpha_{2} c_{t-1}-\beta_{2} y_{t-1}
    +\varepsilon_{t, c}+\varepsilon_{t, j}] /{\Lambda} 
  \end{align}
\end{cases}$$`

其中：
`$\Lambda = 1- \alpha_1 -\beta_2$`。或者将上述**约简方程**记为：

`$$\begin{cases}
  \begin{aligned}
    c_{t} & =  \pi_{11} +\pi_{21}r_t +\pi_{31}g_t +\pi_{41}c_{t-1} +\pi_{51}y_{t-1} +v_{t1} \\
    i_{t} & =  \pi_{12} +\pi_{22}r_t +\pi_{32}g_t +\pi_{42}c_{t-1} +\pi_{52}y_{t-1} +v_{t2} \\
    i_{t} & =  \pi_{13} +\pi_{23}r_t +\pi_{33}g_t +\pi_{43}c_{t-1} +\pi_{53}y_{t-1} +v_{t3}
  \end{aligned}
\end{cases}$$`

易知**约简系数**共有15个！！

---

### 案例2：小型宏观经济模型（3方程联立模型）

**思考**：

- 结构方程和约简方程有什么用？

- （结构方程中的）消费函数，利率
`$i_t$`不会对消费
`$c_t$`产生影响？

- 从约简方程中则可以很快得到答案
    `$\frac{\Delta c_t}{\Delta r_t} = \frac{\alpha_1 \beta_1}{\Lambda}$`

- （结构方程中的）消费函数，收入
`$y_t$`对消费
`$c_t$`产生影响，具体是来自什么原因？

- 进行中介变换也容易得到答案
    `$\frac{\Delta c_t}{ \Delta y_t} = \frac{\Delta c_t / \Delta r_t}{\Delta y_t / \Delta r_t} = \frac{\alpha_1 \beta_1 / \Lambda}{ \beta_1 / \Lambda} = \alpha_1$`

---

### 案例2：小型宏观经济模型（3方程联立模型）

因为约简方程的矩阵形式可以表达为：

`$$\begin{aligned}
 \boldsymbol{y^{\prime}_t}  =  &-\boldsymbol{x^{\prime}_t} \boldsymbol{\Pi}   +  \boldsymbol{{v^{\prime}_t}}  
 =  -\boldsymbol{x^{\prime}_t} \boldsymbol{B} \boldsymbol{\Gamma^{-1}}  +  \boldsymbol{{\varepsilon^{\prime}_t}} \boldsymbol{\Gamma}^{-1} 
\end{aligned}$$`

容易得到如下相关矩阵：

.pull-left[

`$$\begin{align}
\mathbf{y}^{\prime} & =
  \begin{bmatrix}
    c & i & y
  \end{bmatrix}\\
\mathbf{x}^{\prime} & =
  \begin{bmatrix}
    1 & r & g & c_{-1} & y_{-1}
  \end{bmatrix}
\end{align}$$`

`$$\begin{align}
\mathbf{B}=
  \begin{bmatrix}
    {-\alpha_{0}} & {-\beta_{0}} & {0} \\
    {0} & {-\beta_{1}} & {0} \\ 
    {0} & {0} & {-1} \\ {-\alpha_{2}} & {0} & {0} \\ 
    {0} & {\beta_{2}} & {0}
  \end{bmatrix}
\end{align}$$`

]

.pull-right[

`$$\begin{align}
\Gamma &= 
  \begin{bmatrix}
    {1} & {0} & {-1} \\ 
   {0} & {1} & {-1} \\
    {-\alpha_{1}} & {-\beta_{2}} & {1}
  \end{bmatrix} \\
  \mathbf{\Gamma}^{-1} &=\frac{1}{\Lambda}
  \begin{bmatrix}
    {1-\beta_{2}} & {\beta_{2}} & {1} \\ 
    {\alpha_{1}} & {1-\alpha_{1}} & {1} \\ 
    {\alpha_{1}} & {\beta_{2}} & {1}
  \end{bmatrix}
\end{align}$$`

]

---

### 案例2：小型宏观经济模型（3方程联立模型）

根据约简方程的矩阵表达式：

根据前述计算结果，则可以得到约简系数与结构系数的关系为：

`$$\begin{align}
\boldsymbol{\Pi}^{\prime}=\frac{1}{\Lambda}
  \begin{bmatrix}
    {\alpha_{0}\left(1-\beta_{2}\right)+\beta_{0} \alpha_{1}} & {\alpha_{1} \beta_{1}} & {\alpha_{1}} & {\alpha_{2}\left(1-\beta_{2}\right)} & {-\beta_{2} \alpha_{1}} \\ 
    {\alpha_{0} \beta_{2}+\beta_{0}\left(1-\alpha_{1}\right)} & {\beta_{1}\left(1-\alpha_{1}\right)} & {\beta_{2}} & {\alpha_{2} \beta_{2}} & {-\beta_{2}\left(1-\alpha_{1}\right)} \\ 
    {\alpha_{0}+\beta_{0}} & {\beta_{1}} & {1} & {\alpha_{2}} & {-\beta_{2}}
  \end{bmatrix}
\end{align}$$`

其中：
`$\Lambda = 1- \alpha_1 -\beta_2$`。

> **难点**：矩阵的逆的计算。

> 掌握了就是快刀一把，手起刀落，麻利干脆！

---

### 附录：逆矩阵求解方法和步骤

A.用初等行运算（高斯－若尔当）来求逆矩阵：

1. 构造**增广矩阵**

2. 对增广矩阵进行多次变换，直至达到目标。

<br>

B.用余子式、代数余子式和伴随来求逆矩阵

1. 计算**余子式矩阵**和**代数余子式矩阵**

2. 计算**伴随矩阵**：就是代数余子式矩阵的**转置**

3. 计算原矩阵**行列式**：原矩阵**顶行**的每个元素乘以其对应"代数余子式矩阵"**顶行**元素。

4. 计算得出逆矩阵：**1/行列式** X **伴随矩阵**

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

# 18.3 OLS估计方法还合适么？

---
layout: true

<div class="my-footer"><span>huhuaping@ 
<a href="#chapter-navi">模块04 联立方程模型（SEM） |</a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#chapter18">第18章 为什么要关心联立方程模型？ |</a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#OLS"> 18.3 OLS估计方法还合适么？ |</a>
&emsp;&emsp;&emsp;&emsp;
</span></div>

---

### 内生变量问题

以凯恩斯收入决定模型为例，我们将可以证明
`$Y_t$`
和
`$u_t$`
将会出现相关，从而违背CLRM假设。

`$$\begin{cases}
  \begin{align}
    C_t &= \beta_0+\beta_1Y_t+u_t   &(0<\beta_1<1)  &&\text{(消费函数)}\\
    Y_t &= C_t+I_t  & &&\text{(收入恒等式)}
  \end{align}
\end{cases}$$`

将上述结构方程进行变换，得到：

`$$\begin{align}
Y_t &= \beta_0+\beta_1Y_t+ I_t +u_t   \\
Y_t &= \frac{\beta_0}{1-\beta_1}+\frac{1}{1-\beta_1}I_t+\frac{1}{1-\beta_1}u_t   && \text{(式1：约简方程)}\\
E(Y_t)&=\frac{\beta_0}{1-\beta_1}+\frac{1}{1-\beta_1}I_t && \text{(式2：两边取期望)} 
\end{align}$$`

---

### 内生变量问题

进一步地：

`$$\begin{align}
Y_t - E(Y_t)& = \frac{u_t}{1-\beta_1} && \text{(式1 - 式2)}\\
u_t-E(u_t) &= u_t && \text{(式3：期望等于0)} \\
cov(Y_t,u_t) &= E([Y_t-E(Y_t)][u_t-E(u_t)])  && \text{(式4：协方差定义式)}\\ 
&=\frac{E(u^2_t)}{1-\beta_1} && \text{(式5：方差定义式)}\\
&=\frac{\sigma^2}{1-\beta_1}\neq 0 && \text{(式6：方差不为0)}
\end{align}$$`

因此，凯恩斯模型的需求方程，将会不满足CLRM假设中
`$Y_t$`
与
`$u_t$`
不相关的假设。从而使用OLS方法对需求方程不能得到**最优线性无偏估计量**（BLUE）。

---

### 系数的OLS估计量是有偏的

下面将进一步证明，使用OLS方法估计
`$\beta_1$`
是有偏的，也即
`$E(\hat{\beta}_1) \neq \beta_1$`。证明过程如下：

`$$\begin{cases}
  \begin{align}
    C_t &= \beta_0+\beta_1Y_t+u_t   &(0<\beta_1<1)  &&\text{(消费函数)}\\
    Y_t &= C_t+I_t  & &&\text{(收入恒等式)}
  \end{align}
\end{cases}$$`

`$$\begin{align}
\hat{\beta}_1 
= \frac{\sum{c_ty_t}}{\sum{y^2_t}}
= \frac{\sum{C_ty_t}}{\sum{y^2_t}}
= \frac{\sum{\left [ (\beta_0+\beta_1Y_t+u_t)y_t \right ]}}{\sum{y^2_t}} 
 = \beta_1 + \frac{\sum{u_ty_t}}{\sum{y^2_t}} 
&& \text{(式1)}
\end{align}$$`

对式1两边取期望，因此有：

`$$\begin{align}
E(\hat{\beta}_1) &= \beta_1 + E \left ( \frac{\sum{u_ty_t}}{\sum{y^2_t}} \right )
\end{align}$$`

问题是：
`$E \left ( \frac{\sum{u_ty_t}}{\sum{y^2_t}} \right )$`
是否等于0？我们可以证明它将不等于0（证明过程见后）。

---

### 证明附录1

`$$\begin{align}
 \frac{\sum{c_ty_t}}{\sum{y^2_t}}
&= \frac{\sum{(C_t-\bar{C})(Y_t - \bar{Y})}}{\sum{y^2_t}}
= \frac{\sum{(C_t-\bar{C})y_t}}{\sum{y^2_t}} \\
& =\frac{\sum{C_ty_t}-\sum{\bar{C}y_t}}{\sum{y^2_t}}
 =\frac{\sum{C_ty_t}-\sum{\bar{C}(Y_t- \bar{Y})}}{\sum{y^2_t}} \\
& = \frac{\sum{C_ty_t}-\bar{C}\sum{Y_t}- \sum{\bar{C}\bar{Y}}}{\sum{y^2_t}} 
= \frac{\sum{C_ty_t}-\bar{C}\sum{Y_t}- n{\bar{C}\bar{Y}}}{\sum{y^2_t}}
= \frac{\sum{C_ty_t}}{\sum{y^2_t}}
\end{align}$$`

`$$\begin{align}
\hat{\beta_1} & = \frac{\sum\left(\beta_{0}+\beta_{1} Y_{t}+u_{t}\right) y_{t}}{\sum y_{t}^{2}} 
= \frac{\sum{\beta_{0}y_t} +\sum{\beta_1Y_ty_t}+\sum{u_{t}y_t} }{\sum y_{t}^{2}} \\
& = \frac{\beta_1\sum{(y_t+\bar{Y})y_t}+\sum{u_{t}y_t} }{\sum y_{t}^{2}}
=\beta_{1}+\frac{\sum y_{t} u_{t}}{\sum y_{t}^{2}}
\end{align}$$`

`$$\begin{align}
 \Leftarrow &\sum{y_t} =0  ; &&  \frac{\sum{Y_ty_t}}{y^2_t} = 1
\end{align}$$`

---

### 证明附录2

依概率取极限：

`$$\begin{align}
\operatorname{plim}\left(\hat{\beta}_{1}\right) &=\operatorname{plim}\left(\beta_{1}\right)+\operatorname{plim}\left(\frac{\sum y_{t} u_{t}}{\sum y_{t}^{2}}\right) \\
&=\operatorname{plim}\left(\beta_{1}\right)+\operatorname{plim}\left(\frac{\sum y_{t} u_{t} / n}{\sum y_{t}^{2} / n}\right) =\beta_{1}+\frac{\operatorname{plim}\left(\sum y_{t} u_{t} / n\right)}{\operatorname{plim}\left(\sum y_{t}^{2} / n\right)} 
\end{align}$$`

而我们已经证明过：

`$$\begin{align}
cov(Y_t,u_t) &= E([Y_t-E(Y_t)][u_t-E(u_t)])  =\frac{E(u^2_t)}{1-\beta_1} 
=\frac{\sigma^2}{1-\beta_1}\neq 0 
\end{align}$$`

因此，
`$E \left ( \frac{\sum{u_ty_t}}{\sum{y^2_t}} \right ) \neq 0$`得证。

---

### 数值模拟：人为控制的总体

`$$\begin{align}
C_t &= \beta_0+\beta_1Y_t+u_t   &(0<\beta_1<1)  &&\text{(消费函数)}\\
Y_t &= C_t+I_t  & &&\text{(收入恒等式)}
\end{align}$$`

`$$\begin{align}
C_t &= 2+ 0.8Y_t+u_t   &(0<\beta_1<1)  &&\text{(消费函数)}\\
Y_t &= C_t+I_t  & &&\text{(收入恒等式)}
\end{align}$$`

人为控制的总体被设置为：

- `$\beta_0=2, \beta_1=0.8, I_t \leftarrow \text{给定值}$`

- `$E(u_t)=0, var(u_t)=\sigma^2=0.04$`

- `$E(u_tu_{t+j})=0,j \neq 0$`

- `$cov(u_t,I_t)=0$`

---

### 数值模拟：模拟数据集

给定条件下的模拟数据为：

<div id="htmlwidget-8408141cafb56999af4f" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-8408141cafb56999af4f">{"x":{"filter":"none","vertical":false,"caption":"<caption><\/caption>","data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20"],[18.15697,19.5998,21.93468,21.55145,21.88427,22.42648,25.4094,22.69523,24.36465,24.39334,24.09215,24.8745,25.3158,26.30465,25.78235,26.08018,27.2444,28.00963,30.89301,28.98706],[16.15697,17.5998,19.73468,19.35145,19.48427,20.02648,22.8094,20.09523,21.56465,21.59334,21.09215,21.8745,22.1158,23.10465,22.38235,22.68018,23.6444,24.40963,27.09301,25.18706],[2,2,2.2,2.2,2.4,2.4,2.6,2.6,2.8,2.8,3,3,3.2,3.2,3.4,3.4,3.6,3.6,3.8,3.8],[-0.368606,-0.0800400000000003,0.186935999999999,0.110289999999999,-0.0231460000000041,0.085295999999996,0.481879999999997,-0.0609539999999988,0.0729299999999995,0.0786680000000004,-0.181570000000001,-0.0251000000000019,-0.136839999999999,0.060929999999999,-0.243530000000003,-0.183964,-0.151119999999999,0.00192599999999743,0.378601999999997,-0.00258800000000292]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>Y<\/th>\n      <th>C<\/th>\n      <th>I<\/th>\n      <th>u<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"targets":1,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 4, 3, \",\", \".\", null);\n  }"},{"targets":2,"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, 4, 3, \",\", \".\", null);\n  }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0},{"orderable":false,"targets":0}],"pageLength":8,"digits":4,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":["options.columnDefs.0.render","options.columnDefs.1.render","options.columnDefs.2.render"],"jsHooks":[]}</script>

---

### 数值模拟：手工计算

根据前述公式，可以计算得到回归系数：

容易计算出：
`$\sum{u_ty_t}$`
=3.8000

以及:
`$\sum{y^2_t}$`
=184.0000

以及：
`$\frac{\sum{u_ty_t}}{\sum{y^2_t}}$`
=0.0207

因此：
`$\hat{\beta}_1 = \beta_1 + \frac{\sum{u_ty_t}}{\sum{y^2_t}}$`
=0.8+0.0207= 0.8207

这也意味着：
`$\hat{\beta_1}$` 
比真值
`$\beta_1=0.8$`
有0.0207的偏差。

---

### 数值模拟：散点图

---

### 数值模拟：回归报告1

下面我们利用模拟的数据，进行回归分析，得到原始报告：

```

Call:
lm(formula = mod_monte$mod.C, data = monte)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.27001 -0.15855 -0.00126  0.09268  0.46310

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.49402    0.35413   4.219 0.000516 ***
Y            0.82065    0.01434  57.209  < 2e-16 ***
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.1946 on 18 degrees of freedom
Multiple R-squared:  0.9945,	Adjusted R-squared:  0.9942 
F-statistic:  3273 on 1 and 18 DF,  p-value: < 2.2e-16
```

---

### 数值模拟：回归报告2

下面我们对原始报告进行整理，得到精简报告：

`$$\begin{alignedat}{999}
&\widehat{C}=&&+1.49&&+0.82Y\\
&\text{(t)}&&(4.2188)&&(57.2090)\\
&\text{(se)}&&(0.3541)&&(0.0143)\\
&\text{(fitness)}&& n=20;&& R^2=0.9945;&& \bar{R^2}=0.9942\\
& && F^{\ast}=3272.87;&& p=0.0000\\
\end{alignedat}$$`

这样直接OLS回归的结果也表明是有偏的。

---

### 数值模拟：样本回归线

这是样本回归线。

---

### 结论和要点

- 与单方程模型对比，联立方程模型涉及多于一个因变量或内生变量，从而有多少个内生变量就需要有多少个方程。

- 联立方程模裂的一个特有性质是，一个方程中的内生变量(即回归子)作为解释变量而出现在方程组的另一个方程之中。

- 这使得内生解释变量变成了随机的，而且常常和它作为解释变量所在方程中的误差项有相关关系。

- 在这种情况下，经典OLS未必适用，因为这样得到的估计量是不一致的。就是说，不管样本容量有多大，这些估计量都不会收敛于其真实总体值

- 凯恩斯模型的蒙特卡洛模拟，说明了当一个回归方程中的回归元与干扰项相关时(这正是联立方程模型的典型情况) ，用OLS方法估计其参数会内在地导致偏误。

---
layout:false
background-image: url("../pic/thank-you-gif-funny-little-yellow.gif")
class: inverse,center
# 本章结束