08-hetero-scadeciticity-slide.knit

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# 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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# 第8章：放宽基本假设：异方差性

[8.1 异方差性的定义和来源](#definition)

[8.2 异方差性的影响和后果](#effect)

[8.3 广义最小二乘法 (GLS)](#GLS)

[8.4 异方差性问题的诊断](#diagnose)

[8.5 异方差性问题的矫正](#adjust)

[8.6 案例展示（异方差问题的诊断和矫正）](#example)

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# 8.1 异方差性的定义和来源

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<div class="my-footer"><span>huhuaping@   <a href="#chapter08">第08章 放宽基本假设：异方差性 </a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#effect">8.1 异方差性的定义和来源</a> </span></div>

---

## 异方差的概念与内涵

k变量总体回归模型(PRM)

`$$\begin{align}
Y_i&=\beta_1+\beta_2X_{2i}+\beta_3X_{3i}+\cdots+\beta_kX_{ki}+u_i 
&& \text{(PRM)} \\
Y_i&=\hat{\beta}_1+\hat{\beta}_2X_{2i}+\hat{\beta}_3X_{3i}+\cdots+\hat{\beta}_kX_{ki}+e_i 
&& \text{(SRM)} \\
\mathbf{y} &= \mathbf{X}\mathbf{\beta}+\mathbf{u} && \text{(PRM-matrix)}  \\
\mathbf{y} &= \mathbf{X}\mathbf{\hat{\beta}}+\mathbf{e}  && \text{(SRM-matrix)} \\
\end{align}$$`

则随机干扰项的方差协方差矩阵可以写成：

`$$\begin{align}
var-cov(\boldsymbol{u})&=E(\boldsymbol{uu'})
=
  \begin{bmatrix}
  \sigma_1^2 & \sigma_{12}^2  &\cdots &\sigma_{1n}^2\\
  \sigma_{21}^2 & \sigma_2^2  &\cdots &\sigma_{2n}^2\\
  \vdots & \vdots &\vdots &\vdots \\
  \sigma_{n1}^2 & \sigma_{n2}^2  &\cdots &\sigma_n^2\\
  \end{bmatrix}
\end{align}$$`

---

## 异方差的概念与内涵

在**CLRM假设**下，随机干扰项为**同方差**（homoscedasticity），且随机干扰项之间**不相关**。也即：

`$$\begin{align}
E(u_i^2) &=\sigma^2;\quad  (i \in 1,2,\cdots,n)\\
cov(u_iu_j) &=0;\quad (i \neq j)
\end{align}$$`

则，随机干扰项的方差协方差矩阵进一步可以写成：

`$$\begin{align}
var-cov(\boldsymbol{u})=E(\boldsymbol{uu'})
\equiv \sigma^2
  \begin{bmatrix}
  1 & 0  &\cdots &0\\
  0 & 1  &\cdots &0\\
  \vdots & \vdots &\vdots &\vdots \\
  0 & 0  &\cdots &1\\
  \end{bmatrix}
\equiv \sigma^2\boldsymbol{I}
\end{align}$$`

---

## 异方差的概念与内涵

考虑违背了**CLRM假设**的一种情形：随机干扰项具有**异方差**（hetroscadasticity），但仍假设随机干扰项之间不相关。也即：

`$$\begin{align}
E(u_i^2) &=\sigma_i^2;\quad i ( \in 1,2,\cdots,n)\\
cov(u_iuj) & = 0;\quad (i \neq j)
\end{align}$$`

随机干扰项的方差协方差矩阵可以表达为：

`$$\begin{align}
var-cov(\boldsymbol{u})&=E(\boldsymbol{uu'})
=
  \begin{bmatrix}
  \sigma_1^2 & 0  &\cdots & 0\\
  0 & \sigma_2^2  &\cdots 0\\
  \vdots & \vdots &\vdots & \vdots \\
  0 & 0  &\cdots &\sigma_n^2\\
  \end{bmatrix}
\end{align}$$`

---

### 异方差的概念与内涵（图示1）

<div class="figure" style="text-align: center">
<img src="../pic/chpt3-CLRM-homoscedasticity.png" alt="随机干扰项的方差处处相等" width="608" />
<p class="caption">随机干扰项的方差处处相等</p>
</div>

---

### 异方差的概念与内涵（图示2）

<div class="figure" style="text-align: center">
<img src="../pic/chpt3-CLRM-heteroscedasiticity.png" alt="随机干扰项的方差随X取值不同而不同" width="599" />
<p class="caption">随机干扰项的方差随X取值不同而不同</p>
</div>

---

## 数值模拟比较1

---

## 数值模拟比较2

---

## 数值模拟比较3

---

## 异方差的来源

**异方差的来源1**：**边错边改**误差学习模型（error-learning models）的普遍存在。

- 人们的行为误差随时间而减少。

<div class="figure" style="text-align: center">
<img src="../pic/chpt8-hetero-scedasiticity-source1.png" alt="打字出错与打字练习时长的关系" width="662" />
<p class="caption">打字出错与打字练习时长的关系</p>
</div>

---

## 异方差的来源

**异方差的来源2**：随着收入的增长，人们在支出和储蓄中有更大的灵活性。

- 在做储蓄对收入的回归中，随机干扰项的方差
`$\sigma^2$`与收入俱增。

- 随着收入的增长，人们在支出和储蓄中有更大的灵活性。

**异方差的来源3**：随着数据采集技术的改进，随机干扰项的方差
`$\sigma^2$`可能会逐渐减小。

- 有成熟的数据处理设备的银行，在为客户提供的月度或季度报表中，相对于没有这种设备的银行，会出现更少的差错。

---

## 异方差的来源

**异方差的来源4**：**异常值**（outliers）的出现可能导致异方差性增大。

- 异常观测是来自于与产生其余观测值的总体不同的另一个总体。

- 对于小样本数据，问题会更大。

<div class="figure" style="text-align: center">
<img src="../pic/chpt8-hetero-scedasiticity-source4.png" alt="股票价格与消费价格的关系" width="396" />
<p class="caption">股票价格与消费价格的关系</p>
</div>

---

## 异方差的来源

**异方差的来源5**：回归模型的**设定偏误**。

- 比如忽略了重要的解释变量。

- 例如，做商品的需求量对价格的回归时，没有将互补品或替代品的价格包括进来，会引起异方差问题

<div class="figure" style="text-align: center">
<img src="../pic/chpt8-hetero-scedasiticity-source5.png" alt="广告印象Y与广告支出X的两种回归建模结果" width="697" />
<p class="caption">广告印象Y与广告支出X的两种回归建模结果</p>
</div>

---

## 异方差的来源

**异方差的来源6**：数据分布、数据变换和模型函数形式的原因。

- 一个或多个回归元的分布偏态(skewness)。诸如收入、财富和教育等经济变量都是很好的例子。众所周知，大多数社会中收入和财富的分配都是不匀称的，处在顶端的少数几人拥有大部分的收入和财富。

- 不正确的数据变换：(如比率或一阶差分变换等〉

- 不正确的函数形式(如线性与对数线性模型的变换)

---

## 异方差的来源

**异方差的来源7**：**截面数据**中更容易出现异方差性问题。

- 请看下面重点案例——10个行业员工薪水案例

---

### 案例：行业薪水数据（原始）

<div class="figure" style="text-align: center">
<div id="htmlwidget-f0115d6c8e65aa63cb15" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-f0115d6c8e65aa63cb15">{"x":{"filter":"none","vertical":false,"data":[[1,2,3,4,5,6,7,8,9,10],["食品干果","烟草产品","纺织品","器皿用具","纸张类","印刷出版","化工产品","石油与煤炭","相较于塑料","皮革及其制品"],[2994,1721,3600,3494,3498,3611,3875,4616,3538,3016],[3295,2057,3657,3787,3847,4206,4660,5181,3984,3196],[3565,3336,3674,3533,3913,4695,4930,5317,4014,3149],[3907,3320,3437,3215,4135,5083,5005,5337,4287,3317],[4189,2980,3340,3030,4445,5301,5114,5421,4221,3414],[4486,2848,3334,2834,4885,5269,5248,5710,4539,3254],[4676,3072,3225,2750,5132,5182,5630,6316,4721,3177],[4968,2969,3163,2967,5342,5395,5870,6455,4905,3346],[5342,3822,3168,3453,5326,5552,5876,6347,5481,4067]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>ID<\/th>\n      <th>Ind_chn<\/th>\n      <th>A<\/th>\n      <th>B<\/th>\n      <th>C<\/th>\n      <th>D<\/th>\n      <th>E<\/th>\n      <th>F<\/th>\n      <th>G<\/th>\n      <th>H<\/th>\n      <th>I<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"pageLength":10,"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>
<p class="caption">10个行业、不同规模企业下员工薪水数据</p>
</div>

其中，**1-10行**表示不同行业。**A~I列**表示不同企业规模（员工人数）：A=1-4人；B=5-9人；C=10-19人；D=20-49人；E=50-99人；F=100-249人；G=250-499人；H=500-999人；I=1000人及以上。

---

### 案例：行业薪水数据（绘图）

根据以上不同企业规模下员工薪水分布图，可以发现：

- 企业规模（人数）越大，员工薪水
`$Y$`的分布趋向于更加分散，也即薪水分布的标准差
`$\sigma_Y$`倾向于更大。

---

### 案例：行业薪水数据（汇总数据1）

根据前述原始数据，我们可以分别计算出10个行业不同企业规模下（A~I）员工的平均薪水
`$Y$`以及不同企业规模下（A~I）员工薪水的**标准差**
`$\sigma_Y$`。

同时，假定我们也得到了10个行业不同企业规模下员工的平均生产力
`$X$`数据。那么，我们就可以获得如下的**汇总数据表**：

<div class="figure" style="text-align: center">
<div id="htmlwidget-0a56020bb2400338727c" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-0a56020bb2400338727c">{"x":{"filter":"none","vertical":false,"data":[[11,12,13],["平均薪水","薪水的标准差","平均生产力"],[3396,743.7,9355],[3787,851.4,8584],[4013,727.8,7962],[4014,805.06,8275],[4146,929.9,8389],[4241,1080.6,9418],[4387,1243.2,9795],[4538,1307.7,10281],[4843,1112.5,11750]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>ID<\/th>\n      <th>Ind_chn<\/th>\n      <th>A<\/th>\n      <th>B<\/th>\n      <th>C<\/th>\n      <th>D<\/th>\n      <th>E<\/th>\n      <th>F<\/th>\n      <th>G<\/th>\n      <th>H<\/th>\n      <th>I<\/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>
<p class="caption">10个行业、不同规模企业下员工薪水数据</p>
</div>

其中，**A~I列**表示不同企业规模（员工人数）：A=1-4人；B=5-9人；C=10-19人；D=20-49人；E=50-99人；F=100-249人；G=250-499人；H=500-999人；I=1000人及以上。

---

### 案例：行业薪水数据（汇总数据2）

我们将汇总数据表进一步整理为**标准表**：

<div class="figure" style="text-align: center">
<div id="htmlwidget-ebc1b6281934c06f7a88" style="width:100%;height:auto;" class="datatables html-widget"></div>
<script type="application/json" data-for="htmlwidget-ebc1b6281934c06f7a88">{"x":{"filter":"none","vertical":false,"data":[["A","B","C","D","E","F","G","H","I"],["1-4","5-9","10-19","20-49","50-99","100-249","250-499","500-999","1000以上"],[3396.3,3787,4012.6,4104.3,4145.5,4240.7,4388.1,4538,4843.4],[742.180129378606,851.435389340978,727.80327012181,805.055146489288,929.995131409001,1080.60035268466,1241.48400893626,1307.70613416522,1110.72850968282],[9355,8584,7962,8275,8389,9418,9795,10281,11750]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>scales<\/th>\n      <th>企业规模<\/th>\n      <th>薪水\$ Y \$<\/th>\n      <th>薪水标准差\$ \\sigma_{Y} \$<\/th>\n      <th>生产力\$ X \$<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[{"targets":3,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 2, 3, \",\", \".\", null);\n  }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"order":[],"autoWidth":false,"orderClasses":false}},"evals":["options.columnDefs.0.render"],"jsHooks":[]}</script>
<p class="caption">不同规模企业员工薪水与生产力数据</p>
</div>

---

### 案例：行业薪水数据（汇总制图）

员工生产力
`$X$`与员工薪水
`$Y$`的散点图如下：

---

### 案例：行业薪水数据（汇总制图）

员工生产力
`$X$`与员工薪水标准差
`$\sigma_Y$`的散点图如下：

<img src="08-hetero-scadeciticity-slide_files/figure-html/unnamed-chunk-15-1.png" style="display: block; margin: auto;" />
> 员工薪水标准差
`$\sigma_Y$`不是处处相等，而呈现明显增大（随
`$X$`增大而增大），这有违CLRM假设。

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name:effect

# 8.2 异方差性的影响和后果

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## OLS估计量不再是BLUE

对于双变量模型(一元回归模型)：

`$$\begin {align} 
 Y_{i} &=\beta_{1}+\beta_{2} X_{2 i}+u_{i} 
\end {align}$$`

`$\beta_2$`的OLS估计量
`$\hat{\beta}_2$`为：

`$$\begin {align} 
\hat{\beta}_{2}=\frac{\sum x_{i} y_{i}}{\sum x_{i}^{2}}
=\frac{n \sum X_{i} Y_{i}-\sum X_{i} \sum Y_{i}}{n \sum X_{i}^{2}-\left(\sum X_{i}\right)^{2}}
\end {align}$$`

.pull-left[

可以证明，在**CLRM假设**下OLS估计量
`$\hat{\beta}_2$`的方差
`$\sigma^2_{\hat{\beta}_2}$`为：

`$$\begin {align} 
\operatorname{var}\left(\hat{\beta}_{2}\right)=\frac{\sigma^{2}}{\sum x_{i}^{2}}
\end {align}$$`

]

.pull-right[

若违背**CLRM假设**，随机干扰项
`$u_i$`为**异方差**（保证满足其他CLRM条件），则OLS估计量
`$\hat{\beta}_2$`的方差
`$\sigma^2_{\hat{\beta}_2}$`为：

`$$\begin {align} 
\operatorname{var}\left(\hat{\beta}_{2}\right)=\frac{\sum x_{i}^{2} \sigma_{i}^{2}}{\left(\sum x_{i}^{2}\right)^{2}}
\end {align}$$`

]

---

## OLS估计量不再是BLUE

异方差对OLS估计带来的后果：

- 若其他CLRM假定不变，**同方差性**假定不成立时（也即随机干扰项
`$u_i$`为**异方差**），则OLS估计量通常不再是.blue[**BLUE**]。

>也有例外，G-M定理只为OLS的有效性提供了充分条件（非必要），而OLS有效性的充要条件则由kruskal定理给出

- 此时，OLS估计量仍然是线性的和无偏的（记为**LUE**，因这两条性质都与方差无关）。但是，不再是“最优的”或“有效的”。

- 那么，在出现异方差性时，什么才是BLUE呢?

---

## 出现异方差性时的态度1

**态度1**：忽视异方差性，坚持错误地使用CLRM假设下OLS方法各种公式。

在**异方差性**存在的情形下，却坚持使用OLS方法下的方差公式：

`$$\begin {align} 
\hat{\beta}_{2} \parallel_{OLS}^{\sigma^2} 
=\frac{\sum x_{i} y_{i}}{\sum x_{i}^{2}}
=\frac{n \sum X_{i} Y_{i}-\sum X_{i} \sum Y_{i}}{n \sum X_{i}^{2}-\left(\sum X_{i}\right)^{2}}
\end {align}$$`

`$$\begin {align} 
\operatorname{var}\left(\hat{\beta}_{2}\right) \parallel_{OLS}^{\sigma^2} 
=\frac{\sigma^{2}}{\sum x_{i}^{2}}
\end {align}$$`

- 坚持使用方差公式
`$\operatorname{var}\left(\hat{\beta}_{2}\right) \parallel_{OLS}^{\sigma^2}$`是有偏的，可能高估或低估真实方差
`$\operatorname{var}\left(\hat{\beta}_{2}\right)$`。

- 坚持使用回归误差方差公式
`$\hat{\sigma}^{2} \parallel_{OLS}^{\sigma^2}=\frac{\sum \mathrm{e}_{i}^{2}}{n-2}$`，并不是真值
`$\sigma^2$`的**无偏估计量**。

- 进一步地，置信区间、t检验和F检验也将不准确。

> 如果我们忽视异方差性而执意使用惯常的检验程序，则无论我们得出什么结论或作出什么推断，都可能产生严重的误导。

---

## 出现异方差性时的态度2

**态度2**：承认“**异方差性**”这一事实，但仍旧使用OLS方法。

在**异方差性**存在的情形下，直接使用OLS方法，估计量及其方差公式写成：

`$$\begin {align} 
\hat{\beta}_{2} \parallel_{OLS}^{\sigma^2_i} 
=\frac{\sum x_{i} y_{i}}{\sum x_{i}^{2}}
=\frac{n \sum X_{i} Y_{i}-\sum X_{i} \sum Y_{i}}{n \sum X_{i}^{2}-\left(\sum X_{i}\right)^{2}}
\end {align}$$`

`$$\begin {align} 
\operatorname{var}\left(\hat{\beta}_{2}\right) \parallel_{OLS}^{\sigma^2_i} =\frac{\sum x_{i}^{2} \sigma_{i}^{2}}{\left(\sum x_{i}^{2}\right)^{2}}
\end {align}$$`

- 系数估计量仍是一致的

- 方差公式是有偏的，可能高估或低估其真实方差
`$\operatorname{var}\left(\hat{\beta}_{2}\right)$`。

---

## 出现异方差性时的态度3

**态度3**：在**异方差性**存在的情形下，首先想办法消除异方差性，再使用OLS方法。例如**加权最小二乘法**（WLS）下，估计量及其方差公式最终为写成：

`$$\begin {align} 
\hat{\beta}^*_{2} \parallel_{WLS}^{\sigma^2_i}
&=\frac{\left(\sum w_{i}\right)\left(\sum w_{i} X_{i} Y_{i}\right)-\left(\sum w_{i} X_{i}\right)\left(\sum w_{i} Y_{i}\right)}{\left(\sum w_{i}\right)\left(\sum w_{i} X_{i}^{2}\right)-\left(\sum w_{i} X_{i}\right)^{2}} 
&& \leftarrow \left[ w_i= \frac{1}{\sigma^2_i}\right]
\end {align}$$`

`$$\begin {align} 
\operatorname{var}\left(\hat{\beta}^*_{2}\right) \parallel_{WLS}^{\sigma^2_i}
& =\frac{\sum \omega_{i}}{\left(\sum \omega_{i}\right)\left(\sum \omega_{i} X_{i}^{2}\right)-\left(\sum \omega_{i} X_{i}\right)^{2}}
&& \leftarrow \left[ w_i= \frac{1}{\sigma^2_i}\right]
\end {align}$$`

- 系数估计量是一致的

- 方差公式是无偏的，其期望将等于真实方差
`$\operatorname{var}\left(\hat{\beta}_{2}\right)$`。

---

## 出现异方差性时的态度：总结

- **态度1**：“把头埋进沙堆的鸵鸟”

`$$\begin {align} 
\operatorname{var}\left(\hat{\beta}_{2}\right) \parallel_{OLS}^{\sigma^2} 
=\frac{\sigma^{2}}{\sum x_{i}^{2}}
\end {align}$$`

- **态度2**：“将错就错地走下去”

`$$\begin {align} 
\operatorname{var}\left(\hat{\beta}_{2}\right) \parallel_{OLS}^{\sigma^2_i} =\frac{\sum x_{i}^{2} \sigma_{i}^{2}}{\left(\sum x_{i}^{2}\right)^{2}}
\end {align}$$`

- **态度3**：“直面困难找出路”

---

## 出现异方差性时的态度：数据模拟

按照如下规则进行**蒙特卡罗模拟**设置，三种态度的差异见下表：

`$$\begin {align} 
Y_{i} &=\beta_{1}+\beta_{2} X_{i}+u_{i}  && \leftarrow 
\left[ 
\beta_{1} =1; \beta_{2}=1 ;
u_{i} \sim N\left(0, X_{i}^{k}\right); n=20000
\right]
\end {align}$$`

---

## 异方差性问题的一个小结

**总结**：

- 异方差是指模型中关键变量
`$u_i$`或
`$Y_i$`行为表现“不稳定”或“不一致”。

- 异方差的表现方式多种多样，会给建模分析带来一定困扰

**思考**：

- 实际数据中，异方差是否很容易就能被“识别”出来？

- 识别存在异方差与判明是何种形式的异方差，那项工作更轻松？

- 薪资与生产率案例中，你能看出传统回归分析报告有什么异常么？

- 异方差问题是不是一定带来“十分显眼”的回归异常？

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

# 8.3 广义最小二乘法 (GLS)

---
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="#chapter07">第08章 放宽基本假设：异方差性 </a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#GLS">8.3 广义最小二乘法 (GLS)</a> </span></div>

---

## 广义最小二乘法 (GLS)

**广义最小二乘法**（Generalized Least Square，GLS）：是对普通最小二乘法（OLS）的扩展。当**CLRM假设**不能满足时，OLS方法可能无法得到**BLUE估计量**，这时常用GLS来处理其中的问题，以得到BLUE估计量。

**加权最小二乘法**（Weighted Least Square，WLS），属于广义最小二乘法（GLS）的一种特殊情形。加权最小二乘法（WLS）是专门用来处理随机干扰项
`$u_i$`为异方差（违背CLRM假设，出现
`$var(u_i)=\sigma^2_i$`）情形的分析方法。

---

## 广义最小二乘法 (GLS)

**加权最小二乘法**通过额外的样本信息对异方差进行特定**权重**调节，从而保证随机干扰项
`$u_i$`为**同方差**，租后获得**BLUE估计量**。

- GLS比OLS更多地利用了样本数据所提供的信息!

- GLS是更一般的方法，它包括了加权最小二乘法（WLS）、工具变量法（IV）等。

- WLS是GLS的一种特殊方法，专门用来解决异方差问题（前提是能获得**权重**信息）。

- WLS的基本思路是：对来自变异较大的总体的观测值赋予较小的权重，而对来自较小变异的总体现测值赋予较大的权重。

---

## 广义最小二乘法 (GLS)

WLS与OLS的区别：

- OLS的思想实质是最小化:

`$$\begin {align} 
\sum e_{i}^{2}=\sum\left(Y_{i}-\hat{\beta}_{1}-\hat{\beta}_{2} X_{i}\right)^{2}
\end {align}$$`

- WLS的思想实质是最小化:

`$$\begin {align} 
\sum w_{i} e_{i}^{2}=\sum w_{i}\left(Y_{i}-\hat{\beta}_{1}-\hat{\beta}_{2} X_{i}\right)^{2}
\end {align}$$`

其中，加权最小二乘法（WLS）需要确定**权重**向量
`$\boldsymbol{\omega}=[w_1,w_2,\cdots, w_n]^t$`

---

## 加权最小二乘法（WLS）：原理

对于k变量总体回归模型(PRM)：

加权最小二乘法（WLS）需要确定**权重**向量
`$\boldsymbol{\omega}=[w_1,w_2,\cdots, w_n]^t$`，并将异方差修正为同方差
`$\boldsymbol{\frac{\sigma^2_{i}}{\omega}=\sigma I}$`。

`$$\begin{align}
var-cov(\boldsymbol{u})&=E(\boldsymbol{uu'})\\
&=
  \begin{bmatrix}
  \sigma_1^2 & \sigma_{12}^2  &\cdots &\sigma_{1n}^2\\
  \sigma_{21}^2 & \sigma_2^2  &\cdots &\sigma_{2n}^2\\
  \vdots & \vdots &\vdots &\vdots \\
  \sigma_{n1}^2 & \sigma_{n2}^2  &\cdots &\sigma_n^2\\
  \end{bmatrix}
=
  \begin{bmatrix}
  \sigma_1^2 & 0  &\cdots & 0\\
  0 & \sigma_2^2  &\cdots & 0\\
  \vdots & \vdots &\vdots &\vdots \\
  0 & 0  &\cdots &\sigma_n^2\\
  \end{bmatrix}
=
  \begin{bmatrix}
  w_1\sigma^2 & 0  &\cdots & 0\\
  0 & w_2\sigma^2  &\cdots & 0\\
  \vdots & \vdots &\vdots &\vdots \\
  0 & 0  &\cdots & w_n\sigma^2\\
  \end{bmatrix}\\
\end{align}$$`

---

## 加权最小二乘法（WLS）：加权处理

对于双变量模型(一元回归模型)：

`$$\begin {align} 
 Y_{i} &=\beta_{1}+\beta_{2} X_{2 i}+u_{i} 
\end {align}$$`

`$$\begin {align} 
Y_{i} & =\beta_{1} X_{0 i}+\beta_{2} X_{i}+u_{i} && \leftarrow \left[ X_{0i} \equiv 1 \right] \\
\frac{Y_{i}}{\sigma_{i}} &=\beta_{1}\left(\frac{X_{0 i}}{\sigma_{i}}\right)+\beta_{2}\left(\frac{X_{i}}{\sigma_{i}}\right)+\left(\frac{u_{i}}{\sigma_{i}}\right) && \leftarrow \left[ \text{if know} \quad \sigma_i \right] \\
Y_{i}^{*} &=\beta_{1}^{*} X_{0 i}^{*}+\beta_{2}^{*} X_{i}^{*}+u_{i}^{*} && \leftarrow \left[ \text{both divided} \quad \sigma_i \right]
\end {align}$$`

`$$\begin {align} 
\operatorname{var}\left(u_{i}^{*}\right)=E\left(u_{i}^{*}\right)^{2}=E\left(\frac{u_{i}}{\sigma_{i}}\right)^{2}=\frac{1}{\sigma_{i}^{2}} E\left(u_{i}^{2}\right)=\frac{1}{\sigma_{i}^{2}}\left(\sigma_{i}^{2}\right)=1
\end {align}$$`

- 使用加权最小二乘法（WLS），转换后模型的随机干扰项满足同方差性假定，再用OLS方法，就可以得到BLUE估计量。

- 使用加权最小二乘法（WLS），得到估计量称为WLS估计量（记为
`$\boldsymbol{\hat{\beta}}_{WLS}$`）！

---

## 加权最小二乘法（WLS）：估计量

利用样本数据，可以进行如下的WLS估计过程：

`$$\begin {align} 
Y_{i} & =\hat{\beta}_{1} X_{0 i}+\hat{\beta}_{2} X_{i}+e_{i} && \leftarrow \left[ X_{0i} \equiv 1 \right] \\
\frac{Y_{i}}{\sigma_{i}} &=\hat{\beta}_{1}\left(\frac{X_{0 i}}{\sigma_{i}}\right)+\hat{\beta}_{2}\left(\frac{X_{i}}{\sigma_{i}}\right)+\left(\frac{u_{i}}{\sigma_{i}}\right) && \leftarrow \left[ \text{if know} \quad \sigma_i \right] \\
Y_{i}^{*} &=\hat{\beta}_{1}^{*} X_{0 i}^{*}+\hat{\beta}_{2}^{*} X_{i}^{*}+e_{i}^{*} && \leftarrow \left[ \text{both divided} \quad \sigma_i \right]
\end {align}$$`

`$$\begin {align} 
\sum e_{i}^{*^{2}} 
& =\sum\left(Y_{i}^{*}-\hat{\beta}_{1} X_{0 i}^{*}-\hat{\beta}_{2} X_{i}^{*}\right)^{2} \\ \sum\left(\frac{e_{i}}{\sigma_{i}}\right)^{2} 
& =\sum\left[\left(\frac{Y_{i}}{\sigma_{i}}\right)-\hat{\beta}_{1}\left(\frac{X_{0 i}}{\sigma_{i}}\right)-\hat{\beta}_{2}\left(\frac{X_{i}}{\sigma_{i}}\right)\right]^{2}
\end {align}$$`

---

## 加权最小二乘法（WLS）：估计量

最小化并求偏导，得到斜率系数
`$\hat{\beta}_2^*$`的GLS估计量和方差：

`$$\begin {align} 
\hat{\beta}^*_{2} 
&=\frac{\left(\sum w_{i}\right)\left(\sum w_{i} X_{i} Y_{i}\right)-\left(\sum w_{i} X_{i}\right)\left(\sum w_{i} Y_{i}\right)}{\left(\sum w_{i}\right)\left(\sum w_{i} X_{i}^{2}\right)-\left(\sum w_{i} X_{i}\right)^{2}} 
&& \leftarrow \left[ w_i= \frac{1}{\sigma^2_i}\right]
\end {align}$$`

`$$\begin {align} 
\operatorname{var}\left(\hat{\beta}^*_{2}\right) 
& =\frac{\sum \omega_{i}}{\left(\sum \omega_{i}\right)\left(\sum \omega_{i} X_{i}^{2}\right)-\left(\sum \omega_{i} X_{i}\right)^{2}}
&& \leftarrow \left[ w_i= \frac{1}{\sigma^2_i}\right]
\end {align}$$`

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

# 8.4 异方差性问题的诊断

---
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="#chapter08">第08章 放宽基本假设：异方差性 </a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#diagnose">8.4 异方差性问题的诊断</a> </span></div>

---

## 数据类型诊断法

根据数据性质做判断（截面数据经常有）

- 在涉及不均匀、异质性（heterogeneous）单元（国家、省份、企业、家庭）的横截面数据中，异方差性可能是一种常规，而不是例外!

- 例如，投资(Y)与销售量(X2)、利率(X3)等变量之间关系的横截面分析中，如果样本同时包含小、中和大型厂家，一般都预期有异方差性。

---

## 图示法

**图示法**重点关注模型**残差平方**序列（
`$e_i^2$`）是否存在某种系统化模式。

- 图形1：残差平方序列
`$e_i^2$`**描点图**（dot plot）。

- 图形2：残差平方序列
`$e_i^2$`与因变量序列
`$Y_i$`的**散点图**（scatter plot）。

- 图形3：残差平方序列
`$e_i^2$`与自变量序列
`$X_{pi},(p \in2,3,\cdots,k)$`的**散点图**（scatter plot）。

- 图形4：残差平方序列
`$e_i^2$`与自变量序列平方项
`$X^2_{pi},(p \in2,3,\cdots,k)$`的**散点图**（scatter plot）。

---

### 图示法：模拟演示

残差平方序列
`$e_i^2$`与因变量拟合序列
`$\hat{Y}_i$`的**散点图**的若干假想分布模式：

---

### 图示法：案例演示（主回归）

首先构建如下**主回归模型**：

`$$\begin{equation} \begin{alignedat}{999} &Y=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} X&&+e_i\\ \end{alignedat} \end{equation}$$`

主回归模型回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Y}=&&+1990.67&&+0.23X\\ &\text{(t)}&&(2.1262)&&(2.3358)\\&\text{(se)}&&(936.2559)&&(0.0998)\\&\text{(fitness)}&& R^2=0.4380;&& \bar{R^2}=0.3577\\& && F^{\ast}=5.46;&& p=0.0522 \end{alignedat} \end{equation}$$`

---

### 图示法：案例演示（主回归EViews报告）

作为对照，下面给出的是主模型的EViews报告：

---

### 图示法：案例演示（残差数据）

得到主回归模型的残差序列
`$e_i$`、残差平方和序列
`$e_i^2$`（及其他变换数据）：

<div id="htmlwidget-a6606204240a08946960" style="width:100%;height:auto;" class="datatables html-widget"></div>
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> **提示**：后面BPJ检验中需要用到的数据
`$P_{i}=\frac{{e_i^2}} {\tilde{\sigma}^2};\quad \tilde{\sigma}^2=\frac{\sum{e_i^2}} {n}$`

---

### 图示法：案例演示（残差分布模式1）

**对数化**残差平方序列
`$log(e_i^2)$`的**描点图**（dot plot）为：

---

### 图示法：案例演示（残差分布模式2）

**对数化**残差平方序列
`$log(e_i^2)$`与
`$Y_i$`的**散点图**（scatter plot）为：

---

### 图示法：案例演示（残差分布模式3）

**对数化**残差平方序列
`$log(e_i^2)$`与
`$X_i$`的**散点图**（scatter plot）为：

---

### 图示法：案例演示（残差分布模式4）

**对数化**残差平方序列
`$log(e_i^2)$`与
`$X_i^2$`的**散点图**（scatter plot）为：

---

## 帕克检验（Park heteroscedastic test）

**原理**：若
`$\sigma^2_i$`随着
`$X_i$`而变化（
`$H_0:$`同方差模式），则可以将其表达为：

`$$\begin {align} 
\sigma_{i}^{2} &=\sigma^{2} X_{i}^{\beta} e^{v_{i}} \\
\ln \sigma_{i}^{2} &=\ln \sigma^{2}+\beta \ln X_{i}+v_{i} \\
\ln e_{i}^{2} &=\ln \sigma^{2}+\beta \ln X_{i}+v_{i} && \leftarrow \left[ e_i^2 \simeq \sigma^2_i \right]\\ 
&=\alpha+\beta \ln X_{i}+v_{i} && \leftarrow \left[ \alpha \equiv ln(\sigma^2) \right]
\end {align}$$`

---

## 帕克检验（Park heteroscedastic test）

**步骤**：

- 先做主回归（不考虑异方差问题，直接OLS估计）

- 再利用主回归模型的残差序列，做如下的**帕克辅助回归**：

`$$\begin{equation}
ln(e^2_i)=\hat{\alpha}_1+\hat{\alpha}_2ln(X_{2i})+\cdots+\hat{\alpha}_kln(X_{ki})+v_i
\end{equation}$$`

**诊断标准**：

- 如果**帕克检验辅助方程**的F检验**不显著**（对应的概率值P>0.1），则表明主模型是同方差。

- 如果**帕克检验辅助方程**的F检验**显著**（对应的概率值P<0.1），则表明主模型是异方差。

---

### 帕克检验（案例检验）

薪水案例中，为验证**主模型**是否存在异方差问题。我们可以构建如下的**帕克检验辅助方程**

`$$\begin{equation} \begin{alignedat}{999} &log(ei^2)=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} log(X)&&+e_i\\ \end{alignedat} \end{equation}$$`

**帕克检验辅助方程**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{log(ei^2)}=&&+35.82&&-2.80log(X)\\ &\text{(t)}&&(0.9334)&&(-0.6667)\\&\text{(se)}&&(38.3770)&&(4.2021)\\&\text{(fitness)}&& R^2=0.0597;&& \bar{R^2}=-0.0746\\& && F^{\ast}=0.44;&& p=0.5263 \end{alignedat} \end{equation}$$`

从而可以得到**帕克检验**初步结论：F检验不显著，不能拒绝原假设
`$H_0$`，认为**主模型**为**同方差**。

---

### 帕克检验（案例检验EViews报告）

薪水案例中，**帕克检验**的EViews报告如下（注意：EViews软件中没有帕克检验菜单，但可以通过哈维（Harvey）检验方法来实现）：

---

## 格莱泽检验（Glejser heteroscedastic test）

**原理**：若随机干扰项
`$|u_i|$`随着
`$X_i$`而变化（
`$H_0:$`同方差模式），则将可能的异方差模式表达为：

`$$\begin {align} 
\left|u_{i}\right| &=\beta_{1}+\beta_{2} X_{i}+v_{i}  && \text{(G1)}\\
\left|u_{i}\right| &=\beta_{1}+\beta_{2} \sqrt{X}_{i}+v_{i}  && \text{(G2)}\\
\left|u_{i}\right| &=\beta_{1}+\beta_{2} \frac{1}{X_{i}}+v_{i}  && \text{(G3)}\\
\left|u_{i}\right| &=\beta_{1}+\beta_{2} \frac{1}{\sqrt{X}_{i}}+v_{i}  && \text{(G4)}\\
\left|u_{i}\right| &=\sqrt{\beta_{1}+\beta_{2} X_{i}}+v_{i}  && \text{(G5)}\\
\left|u_{1}\right| &=\sqrt{\beta_{1}+\beta_{2} X_{i}^{2}}+v_{i}  && \text{(G6)}
\end {align}$$`

---

## 格莱泽检验（Glejser heteroscedastic test）
 
**步骤**：

- 先做主回归（不考虑异方差问题，直接OLS估计）

- 再利用主回归模型的残差序列，做如下的**格莱泽辅助回归**：

.pull-left[

`$$\begin {align} 
\left|e_{i}\right| &=\beta_{1}+\beta_{2} X_{i}+v_{i}  && \text{(G1)}\\
\left|e_{i}\right| &=\beta_{1}+\beta_{2} \sqrt{X}_{i}+v_{i}  && \text{(G2)}\\
\left|e_{i}\right| &=\beta_{1}+\beta_{2} \frac{1}{X_{i}}+v_{i}  && \text{(G3)}
\end {align}$$`

]

.pull-right[

`$$\begin {align} 
\left|u_{i}\right| &=\beta_{1}+\beta_{2} \frac{1}{\sqrt{X}_{i}}+v_{i}  && \text{(G4)}\\
\left|e_{i}\right| &=\sqrt{\beta_{1}+\beta_{2} X_{i}}+v_{i}  && \text{(G5)}\\
\left|e_{1}\right| &=\sqrt{\beta_{1}+\beta_{2} X_{i}^{2}}+v_{i}  && \text{(G6)}
\end {align}$$`

]

**诊断标准**：

- 如果**格莱泽检验辅助方程**的F检验**不显著**（对应的概率值P>0.1），则表明主模型是同方差。

- 如果**格莱泽检验辅助方程**的F检验**显著**（对应的概率值P<0.1），则表明主模型是异方差。

---

### 格莱泽检验（案例检验1）

薪水案例中，为验证**主模型**是否存在异方差问题。我们可以构建如下的**格莱泽检验辅助方程1**

`$$\begin{equation} \begin{alignedat}{999} &abs(ei)=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} X&&+e_i\\ \end{alignedat} \end{equation}$$`

**格莱泽检验辅助方程1**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{abs(ei)}=&&+407.26&&-0.02X\\ &\text{(t)}&&(0.6434)&&(-0.3014)\\&\text{(se)}&&(632.9891)&&(0.0675)\\&\text{(fitness)}&& R^2=0.0128;&& \bar{R^2}=-0.1282\\& && F^{\ast}=0.09;&& p=0.7719 \end{alignedat} \end{equation}$$`

从而可以得到**格莱泽检验**初步结论：F检验不显著，不能拒绝原假设
`$H_0$`，认为**主模型**为**同方差**。

.footnote[ **提示**：.red[`abs(ei)`]表示残差的绝对值（absolute），也即
`$|e_i|$`。]

---

### 格莱泽检验（案例检验1EViews报告）

薪水案例中，**格莱泽检验**（辅助方程1）的EViews报告如下：

---

### 格莱泽检验（案例检验2）

薪水案例中，为验证**主模型**是否存在异方差问题。我们可以构建如下的**格莱泽检验辅助方程2**

`$$\begin{equation} \begin{alignedat}{999} &abs(ei)=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} sqrt(X)&&+e_i\\ \end{alignedat} \end{equation}$$`

**格莱泽克检验辅助方程2**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{abs(ei)}=&&+575.14&&-3.71sqrt(X)\\ &\text{(t)}&&(0.4480)&&(-0.2788)\\&\text{(se)}&&(1283.8425)&&(13.3042)\\&\text{(fitness)}&& R^2=0.0110;&& \bar{R^2}=-0.1303\\& && F^{\ast}=0.08;&& p=0.7885 \end{alignedat} \end{equation}$$`

从而可以得到**格莱泽检验**初步结论：F检验不显著，不能拒绝原假设
`$H_0$`，认为**主模型**为**同方差**。

.footnote[ **提示**： .red[`abs(ei)`]表示残差的绝对值（absolute），也即
`$|e_i|$`。.red[`sqrt(X)`]表示自变量开根号（square root），也即
`$\sqrt{X_i}$` ]

---

### 格莱泽检验（案例检验2EViews报告）

薪水案例中，**格莱泽检验**（辅助方程2）的EViews报告如下：

---

### 格莱泽检验（案例检验3）

薪水案例中，为验证**主模型**是否存在异方差问题。

.pull-left[
我们可以构建如下的**格莱泽检验辅助方程3**

`$$\begin{equation} \begin{alignedat}{999} &abs(ei)=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} I(1/X)&&+e_i\\ \end{alignedat} \end{equation}$$`

]

.pull-right[
**格莱泽克检验辅助方程3**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{abs(ei)}=&&+76.61&&+1297807.20I(1/X)\\ &\text{(t)}&&(0.1121)&&(0.2081)\\&\text{(se)}&&(683.2112)&&(6237225.2050)\\&\text{(fitness)}&& R^2=0.0061;&& \bar{R^2}=-0.1358\\& && F^{\ast}=0.04;&& p=0.8411 \end{alignedat} \end{equation}$$`

]

从而可以得到**格莱泽检验**初步结论：F检验不显著，不能拒绝原假设
`$H_0$`，认为**主模型**为**同方差**。

.footnote[ **提示**： .red[`abs(ei)`]表示残差的绝对值（absolute），也即
`$|e_i|$`。.red[`I(1/X)`]表示自变量的倒数（reciprocal），也即
`$\frac{1}{X_i}$` ]

---

### 格莱泽检验（案例检验3EViews报告）

薪水案例中，**格莱泽检验**（辅助方程3）的EViews报告如下：

---

### 格莱泽检验（案例检验4）

薪水案例中，为验证**主模型**是否存在异方差问题。我们可以构建如下的**格莱泽检验辅助方程4**

`$$\begin{equation} \begin{alignedat}{999} &abs(ei)=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} I(1/sqrt(X))&&+e_i\\ \end{alignedat} \end{equation}$$`

**格莱泽克检验辅助方程4**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{abs(ei)}=&&-91.15&&+29668.14I(1/sqrt(X))\\ &\text{(t)}&&(-0.0683)&&(0.2320)\\&\text{(se)}&&(1334.3962)&&(127897.8910)\\&\text{(fitness)}&& R^2=0.0076;&& \bar{R^2}=-0.1341\\& && F^{\ast}=0.05;&& p=0.8232 \end{alignedat} \end{equation}$$`

从而可以得到**格莱泽检验**初步结论：F检验不显著，不能拒绝原假设
`$H_0$`，认为**主模型**为**同方差**。

.footnote[ **提示**： .red[`abs(ei)`]表示残差的绝对值（absolute），也即
`$|e_i|$`。.red[`I(1/sqrt(X))`]表示自变量开根号的倒数，也即
`$\frac{1}{\sqrt{X_i}}$` ]

---

### 格莱泽检验（案例检验4EViews报告）

薪水案例中，**格莱泽检验**（辅助方程4）的EViews报告如下：

---

### 格莱泽检验（案例检验6）

薪水案例中，为验证**主模型**是否存在异方差问题。我们可以构建如下的**格莱泽检验辅助方程6**

`$$\begin{equation} \begin{alignedat}{999} &abs(ei)=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} I(X^2)&&+e_i\\ \end{alignedat} \end{equation}$$`

**格莱泽克检验辅助方程6**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{abs(ei)}=&&+321.92&&-0.00I(X^2)\\ &\text{(t)}&&(1.0331)&&(-0.3442)\\&\text{(se)}&&(311.6146)&&(0.0000)\\&\text{(fitness)}&& R^2=0.0166;&& \bar{R^2}=-0.1238\\& && F^{\ast}=0.12;&& p=0.7408 \end{alignedat} \end{equation}$$`

从而可以得到**格莱泽检验**初步结论：F检验不显著，不能拒绝原假设
`$H_0$`，认为**主模型**为**同方差**。

.footnote[ **提示**： .red[`abs(ei)`]表示残差的绝对值（absolute），也即
`$|e_i|$`。.red[`I(X^2)`]表示自变量的平方，也即
`$X_i^2$` ]

---

### 格莱泽检验（案例检验6EViews报告）

薪水案例中，**格莱泽检验**（辅助方程6）的EViews报告如下：

---

## BPG检验（Breusch-Pagan-Goldfrey heteroscedastic test）

**BPG检验**有时候又被称为**BP检验**（Breusch–Pagan test），或者**BP LM检验**（Breusch–Pagan Lagrange Multiplier test）。

**原理**：若
`$\sigma^2_i$`随着某些**非随机变量**
`$Z_{mi}$`而变化（
`$H_0:$`同方差模式），则可以将其表达为：

`$$\begin {align} 
Y_{i} &=\beta_{1}+\beta_{2} X_{2 i}+\cdots+\beta_{k} X_{k i}+u_{i} \\
\sigma_{i}^{2} &=f\left(\alpha_{1}+\alpha_{2} Z_{2 i}+\alpha_{3} Z_{3 i}+\cdots+\alpha_{m} Z_{m i}\right) \\ 
&=\alpha_{1}+\alpha_{2} Z_{2 i}+\alpha_{3} Z_{3 i}+\cdots+\alpha_{m} Z_{m i} 
\end {align}$$`

其中，部分或全部的自变量
`$X_{pi},p \in (2,3,\cdots,k)$`可以作为
`$Z_{mi}$`。

---

## BPG检验（Breusch-Pagan-Goldfrey heteroscedastic test）

**步骤**：

- 先做主回归（不考虑异方差问题，直接OLS估计）

- 再利用主回归模型的残差序列，做如下的**BPG辅助回归**：

`$$\begin {align} 
P_{i}
&=\alpha_{1}+\alpha_{2} Z_{2 i}+\alpha_{3} Z_{3 i}+\cdots+\alpha_{m} Z_{m i} && \leftarrow \left[ P_{i}=\frac{{e_i^2}} {\tilde{\sigma}^2};\quad \tilde{\sigma}^2=\frac{\sum{e_i^2}} {n} \right]
\end {align}$$`

- 计算**BPG辅助回归**方程的ESS(回归平方和)，并得到如下**LM统计量**（卡方统计量）：

`$$\begin {align} 
LM \equiv {\chi^2}^{\ast}=\frac{ESS}{2} \sim \chi^{2}(m-1)
\end {align}$$`

---

## BPG检验（Breusch-Pagan-Goldfrey heteroscedastic test）

**诊断标准**：

- 如果**BPG辅助回归方程**的卡方检验**不显著**，也即
`$LM \equiv {\chi^2}^{\ast}< \chi^2_{1-\alpha}(m-1)$`（对应的概率值P>0.1），则表明主模型是同方差。

- 如果**BPG辅助回归方程**的卡方检验**显著**，也即
`$LM \equiv {\chi^2}^{\ast} > \chi^2_{1-\alpha}(m-1)$`（对应的概率值P<0.1），则表明主模型是异方差。

---

### BPG检验（案例检验）

薪水案例中，为验证**主模型**是否存在异方差问题。我们可以构建如下的**BPG检验辅助方程**

`$$\begin{equation} \begin{alignedat}{999} &Pi=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} X&&+e_i\\ \end{alignedat} \end{equation}$$`

**BPG检验辅助方程**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Pi}=&&+1.50&&-0.00X\\ &\text{(t)}&&(0.2314)&&(-0.0774)\\&\text{(se)}&&(6.4714)&&(0.0007)\\&\text{(fitness)}&& R^2=0.0009;&& \bar{R^2}=-0.1419\\& && F^{\ast}=0.01;&& p=0.9404 \end{alignedat} \end{equation}$$`

**BPG检验辅助方程**的回归平方和为
`$ESS=$` -5.3979；**LM统计量**（绝对值）为
`$LM \equiv {\chi^2}^{\ast}=ESS/2=$` 2.6990。给定显著性水平
`$\alpha=0.05$`时卡方分布的理论查表值为
`$\chi^2_{1-\alpha}(m-1)=\chi^2_{0.95}(1)=$` 3.841459。从而可以得到**BPG检验**初步结论：卡方检验不显著，不能拒绝原假设
`$H_0$`，认为**主模型**为**同方差**。

>**提示**：
`$P_{i}=\frac{{e_i^2}} {\tilde{\sigma}^2};\quad \tilde{\sigma}^2=\frac{\sum{e_i^2}} {n}$`

---

### BPG检验（案例检验EViews报告）

薪水案例中，**BPG检验**的EViews报告如下（**注意**：EViews采用了BPG不同分析方法，EViews结果（本页）与手动计算结果（上一页））：

---

## GQ检验（Goldfeld–Quandt heteroscedastic test）

**原理**：若
`$\sigma^2_i$`随着某一个自变量的平方项
`$(X_i^2$`而变化（
`$H_0:$`同方差模式），则可以将其表达为：

`$$\begin {align} 
Y_{i} &=\beta_{1}+\beta_{2} X_2+u_{i} \\
\sigma^2_i &=\sigma^2 X_i^2
\end {align}$$`

---

## GQ检验（Goldfeld–Quandt heteroscedastic test）

**步骤**：

- 把数据按
`$X_i$`由小到大排序，从小到大；去掉中间大约c个数（预先确定的）

- 把数据分成两份样本：前一半样本数
`$(n-c)/2$`，后一半样本数
`$(n-c)/2$`

- 对两份**分段样本**数据，做如下的**GQ辅助回归**，并分别进行OLS回归

`$$\begin {align} 
\overleftarrow{Y_{i}} &=\overleftarrow{\alpha_{1}}+\overleftarrow{\alpha_{2}} \overleftarrow{X}_i+\overleftarrow{e_{i}} \\
\overrightarrow{Y_{i}} &=\overrightarrow{\alpha_{1}}+\overrightarrow{\alpha_{2}} \overrightarrow{X}_i+\overrightarrow{e_{i}} 
\end {align}$$`

- 分别得到两个辅助回归方程的残差平方和：
`$RSS_1$`和
`$RSS_2$`，其分别对应的自由度为
`$df_1=df_2=(n-c-2k)/2$`。

- 计算得到如下F统计量：

`$$\begin {align} 
F^{*}=\frac{\mathrm{RSS}_{1} / df_1}{\mathrm{RSS}_{2} / df_2} \sim F((n-c-2k)/2,(n-c-2k)/2)
\end {align}$$`

---

## GQ检验（Goldfeld–Quandt heteroscedastic test）

**诊断标准**：

- 如果**GQ辅助回归方程**的F检验**不显著**，也即
`$F^{*} < F_{1-\alpha}((n-c-2k)/2,(n-c-2k)/2)$`（对应的概率值P>0.1），则表明主模型是同方差。

- 如果**GQ辅助回归方程**的F检验**显著**，也即
`$F^{*} > F_{1-\alpha}((n-c-2k)/2,(n-c-2k)/2)$`（对应的概率值P<0.1），则表明主模型是异方差。

---

## 怀特检验（White heteroscedastic test）

**步骤**：

- 先做主回归（不考虑异方差问题，直接OLS估计）

- 再利用主回归模型的残差序列，做如下的**怀特辅助回归**：

`$$\begin {align} 
e_i^2 &=\alpha_{1}+\alpha_{2} X_{2 i}+\alpha_{3} X_{3 i}+\alpha_{4} X_{2 i}^{2}+\alpha_{5} X_{3 i}^{2}+\alpha_{6} X_{2 i} X_{3 i}+v_{i}
\end {align}$$`

- 计算**怀特辅助回归**方程的判定系数
`$R^2$`，并得到如下卡方统计量（
`$\tilde{k}$`为辅助方程中回归系数个数）：

`$$\begin {align} 
\chi^{*}=n \cdot R^2 \sim \chi^{2}(\tilde{k}-1)
\end {align}$$`

---

## 怀特检验（White heteroscedastic test）

**诊断标准**：

- 如果**怀特辅助回归方程**的卡方检验**不显著**，也即
`$\chi^{*} < \chi^2_{1-\alpha}(m-1)$`（对应的概率值P>0.1），则表明主模型是同方差。

- 如果**怀特辅助回归方程**的卡方检验**显著**，也即
`$\chi^{*} > \chi^2_{1-\alpha}(m-1)$`（对应的概率值P<0.1），则表明主模型存在异方差问题。

---

### 怀特检验（案例检验）

薪水案例中，为验证**主模型**是否存在异方差问题。我们可以构建如下的**怀特检验辅助方程**

`$$\begin{equation} \begin{alignedat}{999} &I(ei^2)=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} X&& + \hat{\beta}_{3} I(X^2)&&+e_i\\ \end{alignedat} \end{equation}$$`

**怀特检验辅助方程**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{I(ei^2)}=&&-3793851.07&&+810.57X&&-0.04I(X^2)\\ &\text{(t)}&&(-0.7835)&&(0.8105)&&(-0.8169)\\&\text{(se)}&&(4842167.3561)&&(1000.0435)&&(0.0510)\\&\text{(fitness)}&& R^2=0.1008;&& \bar{R^2}=-0.1989\\& && F^{\ast}=0.34;&& p=0.7269 \end{alignedat} \end{equation}$$`

---

### 怀特检验（案例检验）

**怀特检验辅助方程**的判定系数为
`$R^2=$` 0.1008；样本卡方统计量（绝对值）为
`${\chi^2}^{\ast}=nR^2=$` 0.9076。给定显著性水平
`$\alpha=0.05$`时卡方分布的理论查表值为
`$\chi^2_{1-\alpha}(m-1)=\chi^2_{0.95}(1)=$` 5.991465。从而可以得到**怀特检验**初步结论：卡方检验不显著，不能拒绝原假设
`$H_0$`，认为**主模型**为**同方差**。

>**提示**： .red[`I(ei^2)`]表示残差的平方，也即
`$e_i^2$`。.red[`I(X^2)`]表示自变量平方，也即
`$X_i^2$`

---

### 怀特检验（案例检验EViews报告）

薪水案例中，**怀特检验**的EViews报告如下：

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

# 8.5 异方差性问题的矫正

---
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="#chapter08">第08章 放宽基本假设：异方差性 </a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#adjust">8.5 异方差性问题的矫正</a> </span></div>

---

## WLS矫正1（方差初步已知）

**矫正情形1**：方差
`$\sigma^2_i$`已知且等于样本方差
`$S^2_{Y_i}$`

如果主模型存在异方差问题，且假设方差正比于样本方差
`$S^2_{Y_i}$`，则有：

`$$\begin{equation}
var(u_i)=E(u^2_i)=\sigma^2_i=\sigma^2S^2_{Y_i}
\end{equation}$$`

此时，加权最小二乘法（WLS）矫正可以采用权重
`$w_i=\frac{1}{S_i}$`

也即可以对主模型的变量进行如下加权变换：

`$Y^{\ast}=w_iY_i=\frac{Y_i}{S_i}$`；
`$X^{\ast}=w_iX_i=\frac{X_i}{S_i}$`；
`$v_i=w_iu_i=\frac{u_i}{S_i}$`。

---

## WLS矫正1（方差初步已知）

进而得到WLS变换后的新模型：

`$$\begin{align}
Y_{i} &=\beta_{1}+\beta_{2} X_{2 i}+\cdots+\beta_{k} X_{k i}+u_{i} \\
\frac{Y_i}{S_i} &=\frac{\beta_1}{S_i}+\beta_2\frac{X_{2i}}{S_i}+ \cdots +\beta_k\frac{X_{ki}}{S_i}+\frac{u_{i}}{S_i}\\
Y^{\ast}_i &=\beta^{\ast}_1+\beta^{\ast}_2X^{\ast}_{2i}+ \cdots +\beta^{\ast}_kX^{\ast}_{ki}+v_{i}
\end{align}$$`

容易证明新模型将消除异方差性：

`$$\begin {align} 
E\left(v_{i}^{2}\right)=E\left(\frac{u_{i}}{S_{Y_i}}\right)^{2}=\frac{1}{S_{Y_i}^{2}} E\left(u_{i}^{2}\right)=\sigma^{2}
\end {align}$$`

---

## WLS矫正2（方差正比于自变量平方项）

**矫正情形2**：方差
`$\sigma^2_i$`正比于
`$X^2_i$`

如果主模型存在异方差问题，且假设方差正比于
`$X^2_{2i}$`，则有：

`$$\begin{equation}
var(u_i)=E(u^2_i)=\sigma^2_i=\sigma^2X^2_{2i} 
\end{equation}$$`

---

## WLS矫正2（方差正比于自变量平方项）

此时，加权最小二乘法（WLS）矫正可以采用权重
`$w_i=\frac{1}{X_{2i}}$`

也即可以对主模型的变量进行如下加权变换：

`$Y^{\ast}=w_iY_i=\frac{Y_i}{X_{2i}}$`；
`$X^{\ast}=w_iX_i=\frac{X_i}{X_{2i}}$`；
`$v_i=w_iu_i=\frac{u_i}{X_{2i}}$`。

进而得到WLS变换后的新模型：

`$$\begin{align}
Y_{i} &=\beta_{1}+\beta_{2} X_{2 i}+\cdots+\beta_{k} X_{k i}+u_{i}  \\
\frac{Y_i}{X_{2i}} &=\frac{{\beta}_1}{X_{2i}}+\beta_2+\beta_3\frac{X_{3i}}{X_{2i}}+{\beta}_4\frac{X_{4i}}{X_{2i}}+{\beta}_5\frac{X_{5i}}{X_{2i}}+\frac{u_{i}}{X_{2i}} \\
Y^{\ast}_i &=\beta^{\ast}_2X^{\ast}_{2i}+\beta^{\ast}_1+ \beta^{\ast}_3X^{\ast}_{3i}\cdots +\beta^{\ast}_kX^{\ast}_{ki}+v_{i}
\end{align}$$`

容易证明新模型将消除异方差性：

`$$\begin {align} 
E\left(v_{i}^{2}\right)=E\left(\frac{u_{i}}{X_{2i}}\right)^{2}=\frac{1}{X_{2i}^{2}} E\left(u_{i}^{2}\right)=\sigma^{2}
\end {align}$$`

---

## WLS矫正3（方差正比于自变量）

**矫正情形3**：方差
`$\sigma^2_i$`正比于
`$X_i$`

如果主模型存在异方差问题，且假设方差正比于
`$X_{2i}$`，则有：

`$$\begin{equation}
var(u_i)=E(u^2_i)=\sigma^2_i=\sigma^2X_{2i}
\end{equation}$$`

---

## WLS矫正3（方差正比于自变量）

此时，加权最小二乘法（WLS）矫正可以采用权重
`$w_i=\frac{1}{\sqrt{X_{2i}}}$`

也即可以对主模型的变量进行如下加权变换：

`$Y^{\ast}=w_iY_i=\frac{Y_i}{\sqrt{X_{2i}}}$`；
`$X^{\ast}=w_iX_i=\frac{X_i}{\sqrt{X_{2i}}}$`；
`$v_i=w_iu_i=\frac{u_i}{\sqrt{X_{2i}}}$`。

进而得到WLS变换后的新模型：

`$$\begin{align}
Y_{i} &=\beta_{1}+\beta_{2} X_{2 i}+\cdots+\beta_{k} X_{k i}+u_{i}  \\
\frac{Y_i}{\sqrt{X_{2i}}} &=\frac{{\beta}_1}{\sqrt{X_{2i}}}+{\beta}_2\sqrt{X_{2i}}+{\beta}_3\frac{X_{3i}}{\sqrt{X_{2i}}}+\cdots+{\beta}_5\frac{X_{5i}}{\sqrt{X_{2i}}}+\frac{u_{i}}{\sqrt{X_{2i}}}\\
Y^{\ast}_i &=\beta^{\ast}_1+\beta^{\ast}_2X^{\ast}_{2i}+ \cdots +\beta^{\ast}_kX^{\ast}_{ki}+v_{i}
\end{align}$$`

容易证明新模型将消除异方差性：

`$$\begin {align} 
E\left(v_{i}^{2}\right)=E\left(\frac{u_{i}}{\sqrt{X_{2i}}}\right)^{2}=\frac{1}{X_{2i}} E\left(u_{i}^{2}\right)=\sigma^{2}
\end {align}$$`

---

## WLS矫正4（方差正比于因变量平方项）

**矫正情形4**：方差
`$\sigma^2_i$`正比于
`$\hat{Y_i}^2$`

如果主模型存在异方差问题，且假设方差正比于
`$\hat{Y_i}^2$`，也即：

`$$\begin{equation}
var(u_i)=E(u^2_i)=\sigma^2_i=\sigma^2\hat{Y_i}^2
\end{equation}$$`

此时，加权最小二乘法（WLS）矫正可以采用权重
`$w_i=\frac{1}{\hat{Y_i}}$`

也即可以对主模型的变量进行如下加权变换：

`$Y^{\ast}=w_iY_i=\frac{Y_i}{\hat{Y_i}}$`；
`$X^{\ast}=w_iX_i=\frac{X_i}{\hat{Y_i}}$`；
`$v_i=w_iu_i=\frac{u_i}{\hat{Y_i}}$`。

---

## WLS矫正4（方差正比于因变量平方项）

从而得到WLS变换后的新模型：

`$$\begin{align}
Y_{i} &=\beta_{1}+\beta_{2} X_{2 i}+\cdots+\beta_{k} X_{k i}+u_{i}  \\
\frac{Y_i}{\hat{Y_i}} &=\frac{{\beta}_1}{\hat{Y_i}}+{\beta}_2\frac{X_{2i}}{\hat{Y_i}}+\cdots+{\beta}_k\frac{X_{ki}}{\hat{Y_i}}+\frac{u_{i}}{\hat{Y_i}} \\
Y^{\ast}_i &=\beta^{\ast}_1+\beta^{\ast}_2X^{\ast}_{2i}+ \cdots +\beta^{\ast}_kX^{\ast}_{ki}+v_{i}
\end{align}$$`

容易证明新模型将消除异方差性：

`$$\begin {align} 
E\left(v_{i}^{2}\right)=E\left(\frac{u_{i}}{\hat{Y_i}}\right)^{2}=\frac{1}{\hat{Y_i}^{2}} E\left(u_{i}^{2}\right)=\sigma^{2}
\end {align}$$`

---

## WLS矫正5（方差是某种组合形式）

**矫正情形5**：方差是某种组合形式

如果主模型存在异方差问题，且假设方差是某种组合形式，也即：

`$$\begin{equation}
var(u_i|X_{2i},X_{3i}\cdots)=E(u^2_i)=\sigma^2_i=\sigma^2 \cdot {f}(X_{2i},X_{3i}\cdots)
\end{equation}$$`

此时，加权最小二乘法（WLS）矫正可以采用权重
`$w_i=\frac{1}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}}$`。

也即可以对主模型的变量进行如下加权变换
`$Y^{\ast}=w_iY_i=\frac{Y_i}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}}$`；
`$X^{\ast}=w_iX_i=\frac{X_i}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}}$`；
`$v_i=w_iu_i=\frac{u_i}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}}$`。

---

## WLS矫正5（方差是某种组合形式）

从而得到WLS变换后的新模型：

`$$\begin{align}
Y_{i} &=\beta_{1}+\beta_{2} X_{2 i}+\cdots+\beta_{k} X_{k i}+u_{i}  \\
\frac{Y_i}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}} &=\frac{{\beta}_1}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}}+{\beta}_2\frac{X_{2i}}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}}+\cdots\\
&\quad +{\beta}_k\frac{X_{ki}}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}}+\frac{u_{i}}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}} \\
Y^{\ast}_i &=\beta^{\ast}_1+\beta^{\ast}_2X^{\ast}_{2i}+ \cdots +\beta^{\ast}_kX^{\ast}_{ki}+v_{i}
\end{align}$$`

容易证明新模型将消除异方差性：

`$$\begin {align} 
E\left(v_{i}^{2}\right)=E\left(\frac{u_{i}}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}}\right)^{2}=\frac{1}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}^{2}} E\left(u_{i}^{2}\right)=\sigma^{2}
\end {align}$$`

---

## WLS矫正6（方差完全未知）

**矫正情形6**：方差
`$\sigma^2_i$`未知

如果主模型存在异方差问题，对主模型两边同时取对数
`$ln()$`通常能够减低异方差性问题。

`$$\begin{align}
Y_{i} &=\beta_{1}+\beta_{2} X_{2 i}+\cdots+\beta_{k} X_{k i}+u_{i}  \\
ln{Y_t} &={\beta}_1+{\beta}_2ln{X_{2i}}+\cdots+{\beta}_kln{X_{ki}}+v_{i}
\end{align}$$`

---

## WLS矫正（小结）

以上所有讨论的变换都是一种权宜之计。我们基本上是在猜测
`$\sigma^2_i$`的情形。

> “幸福的家庭大抵类似，不幸的家庭各不相同!”——列夫托尔斯泰

> “同方差只有同一种结果（理想状态），异方差则可以异得千奇百怪（普遍现实）”

在所讨论的变换中哪一种能行之有效，要看问题的性质和异方差性的严重程度。这样做也会带来新的问题：

- 当我们超出双变量模型的范围时，我们也许不能预先知道应选择哪一个
`$X_{pi}$`变量进行数据变换。

- 对数变换或开根号变换，当某些Y和X值为零或负数时便不适用。

---

## 稳健标准误协方差矫正法（White法）：基本介绍

使用使用稳健标准误（robust standard errors）协方差矫正，可以一定程度上降低异方差问题对模型的影响（假设检验和预测）。

这里我们将主要讨论采用White (1980)提出的异方差一致性矫正法（heteroskedasticity consistent covariance）进行模型处理<sup>注释1</sup>。

因此，我们可以看到两份可供对比的回归分析报告：

- 分析报告A：使用OLS方法，不进行稳健标准误协方差矫正。

- 分析报告B：使用OLS方法，进行（white方法的）稳健标准误协方差矫正。

.footnote[
注释1.实际的稳健标准误协方差矫正方法有很多，目的、作用和算法各不相同。
]

---

### 稳健标准误协方差矫正法（White法）:EViews菜单操作

EViews软件中，方程估计时，可以在引导设置中做出White稳健标准误协方差设定，具体是：

> **Equation Estimation** 
`$\Rightarrow$` **Options**设置框
`$\Rightarrow$`
**Coefficient covariance**设置框（系数协方差）
`$\Rightarrow$`
**Covariance method**下拉框（协方差方法）
`$\Rightarrow$`下拉选择`Huber-White`

---

### 稳健标准误协方差矫正法（White法）:EViews菜单操作

---

### 进行稳健标准误协方差矫正（White法）：B分析报告

在EViews软件中，White稳健标准误协方差矫正的报告结果如下：

---

### 没有进行稳健标准误协方差矫正（White法）：A分析报告

作为对照，下面是**没有**进行White稳健标准误协方差矫正的EViews报告：

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

# 8.6 案例（异方差问题诊断和矫正）

---
layout: true
  
<div class="my-header-h2"></div>
<div class="watermark1"></div>
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<div class="my-footer"><span>huhuaping@   <a href="#chapter08">第08章 放宽基本假设：异方差性 </a> 
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#example">8.6 案例展示（异方差问题的诊断和矫正）</a> </span></div>

---

## 吸烟案例（数据表）

<div class="figure" style="text-align: center">
<div id="htmlwidget-2f74b3f9f1f2359255f4" style="width:100%;height:auto;" class="datatables html-widget"></div>
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class=\"display\">\n  <thead>\n    <tr>\n      <th>cigs<\/th>\n      <th>income<\/th>\n      <th>cigpric<\/th>\n      <th>educ<\/th>\n      <th>age<\/th>\n      <th>restaurn<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tip","columnDefs":[{"targets":2,"render":"function(data, type, row, meta) {\n    return type !== 'display' ? data : DTWidget.formatRound(data, 2, 3, \",\", \".\", null);\n  }"},{"className":"dt-center","targets":"_all"},{"visible":false,"targets":0}],"pageLength":8,"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[8,10,25,50,100]}},"evals":["options.columnDefs.0.render"],"jsHooks":[]}</script>
<p class="caption">吸烟数量及其影响因素案例数据n=(807)</p>
</div>

> `cigs`作为因变量Y，表示吸烟数量（每天）；
其余变量表示自变量X，其中：`income`表示年收入；`cigpric`表示每包香烟价格；`educ`表示读书年数；`age`表示年龄；`restaurn`表示是否禁止在餐馆公共场所吸烟。

---

## 吸烟案例（主模型）

首先构建如下**主回归模型**：

`$$\begin{equation} \begin{alignedat}{999} &cigs=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} log(income)&& + \hat{\beta}_{3} log(cigpric)&& + \hat{\beta}_{4} educ\\ &\text{(cont.)}&& + \hat{\beta}_{5} age&& + \hat{\beta}_{6} I(age^2)&& + \hat{\beta}_{7} restaurn&&+e_i\\ \end{alignedat} \end{equation}$$`

主回归模型回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{cigs}=&&-3.64&&+0.88log(income)&&-0.75log(cigpric)&&-0.50educ\\ &\text{(t)}&&(-0.1512)&&(1.2095)&&(-0.1301)&&(-3.0016)\\&\text{(se)}&&(24.0787)&&(0.7278)&&(5.7733)&&(0.1671)\\&\text{(cont.)}&&+0.77age&&-0.01I(age^2)&&-2.83restaurn\\&\text{(t)}&&(4.8132)&&(-5.1765)&&(-2.5410)\\&\text{(se)}&&(0.1601)&&(0.0017)&&(1.1118)\\&\text{(fitness)}&& R^2=0.0527;&& \bar{R^2}=0.0456\\& && F^{\ast}=7.42;&& p=0.0000\\ \end{alignedat} \end{equation}$$`

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## 吸烟案例（主模型EViews报告）

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## 吸烟案例（残差数据）

得到主回归模型的残差序列
`$e_i$`、残差平方和序列
`$e_i^2$`（及其他变换数据）：

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> **提示**：后面BPJ检验中需要用到的数据
`$P_{i}=\frac{\sum{e_i^2}} {\tilde{\sigma}^2};\quad \tilde{\sigma}^2=\frac{\sum{e_i^2}} {n}$`

---

## 残差分布模式（按观测样本）

**对数化**残差平方序列
`$log(e_i^2)$`的**描点图**（dot plot）为：

---

## 残差分布模式（对收入）

**对数化**残差平方序列
`$log(e_i^2)$`与**对数化**收入序列
`$log(income)$`的**散点图**（scatter plot）为：

---

## 残差分布模式（对香烟价格）

**对数化**残差平方序列
`$log(e_i^2)$`与**对数化**香烟价格序列
`$log(cigpric)$`的**散点图**（scatter plot）为：

---

## 残差分布模式（对受教育年数）

**对数化**残差平方序列
`$log(e_i^2)$`与受教育年数序列
`$educ$`的**散点图**（scatter plot）为：

---

## 残差分布模式（对年龄）

**对数化**残差平方序列
`$log(e_i^2)$`与年龄序列
`$age$`的**散点图**（scatter plot）为：

---

## 帕克检验（关注收入变量）

为验证**主模型**是否存在异方差问题。我们可以构建如下的**帕克检验辅助方程1**

`$$\begin{equation} \begin{alignedat}{999} &log(ei^2)=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} log(income)&&+e_i\\ \end{alignedat} \end{equation}$$`

**帕克检验辅助方程**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{log(ei^2)}=&&-0.41&&+0.48log(income)\\ &\text{(t)}&&(-0.5366)&&(6.0171)\\&\text{(se)}&&(0.7700)&&(0.0793)\\&\text{(fitness)}&& R^2=0.0430;&& \bar{R^2}=0.0419\\& && F^{\ast}=36.21;&& p=0.0000 \end{alignedat} \end{equation}$$`

从而可以得到**帕克检验**初步结论：F检验极显著，拒绝原假设
`$H_0$`，认为**主模型**为**异方差**。

---

## BPG检验（LM检验）

为验证**主模型**是否存在异方差问题。我们可以构建如下的**BPG检验辅助方程**

`$$\begin{equation} \begin{alignedat}{999} &Pi=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} log(income)&& + \hat{\beta}_{3} log(cigpric)&& + \hat{\beta}_{4} educ\\ &\text{(cont.)}&& + \hat{\beta}_{5} age&& + \hat{\beta}_{6} I(age^2)&& + \hat{\beta}_{7} restaurn&&+e_i\\ \end{alignedat} \end{equation}$$`

**BPG检验辅助方程**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Pi}=&&-3.57&&+0.14log(income)&&+0.34log(cigpric)&&-0.01educ\\ &\text{(t)}&&(-0.9752)&&(1.2493)&&(0.3898)&&(-0.5266)\\&\text{(se)}&&(3.6630)&&(0.1107)&&(0.8783)&&(0.0254)\\&\text{(cont.)}&&+0.11age&&-0.00I(age^2)&&-0.40restaurn\\&\text{(t)}&&(4.4750)&&(-4.5474)&&(-2.3626)\\&\text{(se)}&&(0.0244)&&(0.0003)&&(0.1691)\\&\text{(fitness)}&& R^2=0.0400;&& \bar{R^2}=0.0328\\& && F^{\ast}=5.55;&& p=0.0000\\ \end{alignedat} \end{equation}$$`

---

## BPG检验（LM检验）

**BPG检验辅助方程**的回归平方和为
`$ESS=$` 113.5691；**LM统计量**（绝对值）为
`$LM \equiv {\chi^2}^{\ast}=ESS/2=$` 56.7845。给定显著性水平
`$\alpha=0.05$`时卡方分布的理论查表值为
`$\chi^2_{1-\alpha}(m-1)=\chi^2_{0.95}(6)=$` 12.59159。从而可以得到**BPG检验**初步结论：卡方检验极显著，拒绝原假设
`$H_0$`，认为**主模型**存在**异方差性**。

>**提示**：
`$P_{i}=\frac{\sum{e_i^2}} {\tilde{\sigma}^2};\quad \tilde{\sigma}^2=\frac{\sum{e_i^2}} {n}$`

---

## 怀特检验

通过构建**怀特检验辅助方程**：

`$$\begin{equation} \begin{alignedat}{999} &I(ei^2)=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} log(income)&& + \hat{\beta}_{3} I((log(income))^2)\\ &\text{(cont.)}&& + \hat{\beta}_{4} log(cigpric)&& + \hat{\beta}_{5} I((log(cigpric))^2)&& + \hat{\beta}_{6} educ\\ &\text{(cont.)}&& + \hat{\beta}_{7} age&& + \hat{\beta}_{8} I(age^2)&& + \hat{\beta}_{9} restaurn&&+e_i\\ \end{alignedat} \end{equation}$$`

---

## 怀特检验

进而可以得到如下辅助回归的诊断结果：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{I(ei^2)}=&&+19699.17&&+43.49log(income)&&-1.04I((log(income))^2)\\ &\text{(t)}&&(1.0527)&&(0.1572)&&(-0.0683)\\&\text{(se)}&&(18712.4188)&&(276.6317)&&(15.2518)\\&\text{(cont.)}&&-10054.87log(cigpric)&&+1251.96I((log(cigpric))^2)&&-2.34educ\\&\text{(t)}&&(-1.0865)&&(1.0932)&&(-0.5116)\\&\text{(se)}&&(9254.3280)&&(1145.2348)&&(4.5649)\\&\text{(cont.)}&&+19.51age&&-0.22I(age^2)&&-67.03restaurn\\&\text{(t)}&&(4.4901)&&(-4.5759)&&(-2.2029)\\&\text{(se)}&&(4.3442)&&(0.0473)&&(30.4296)\\&\text{(fitness)}&& R^2=0.0414;&& \bar{R^2}=0.0318\\& && F^{\ast}=4.31;&& p=0.0000\\ \end{alignedat} \end{equation}$$`

---

## 怀特检验

**怀特检验辅助方程**的判定系数为
`$R^2=$` 0.0414；样本卡方统计量（绝对值）为
`${\chi^2}^{\ast}=nR^2=$` 33.4198。给定显著性水平
`$\alpha=0.05$`时卡方分布的理论查表值为
`$\chi^2_{1-\alpha}(m-1)=\chi^2_{0.95}(8)=$` 15.50731。从而可以得到**怀特检验**初步结论：卡方检验显著，应拒绝原假设
`$H_0$`，认为**主模型**存在**异方差问题**。

---

## WLS矫正1（收入变量）

根据散点图，可以认为主模型方差正比于收入（income）的平方，因此加权最小二乘法（WLS）矫正模型如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{I(cigs/income)}=&&+0.00&&+0.00I(log(cigpric/income))&&+0.07I(educ/income)\\ &\text{(t)}&&(5.4182)&&(6.7093)&&(2.1392)\\&\text{(se)}&&(0.0006)&&(0.0001)&&(0.0331)\\&\text{(cont.)}&&+0.00age&&-0.00I(age^2)&&-0.00restaurn\\&\text{(t)}&&(3.6951)&&(-4.0142)&&(-2.3069)\\&\text{(se)}&&(0.0000)&&(0.0000)&&(0.0001)\\&\text{(fitness)}&& R^2=0.1486;&& \bar{R^2}=0.1432\\& && F^{\ast}=27.95;&& p=0.0000\\ \end{alignedat} \end{equation}$$`

---

### 对WLS矫正（收入）模型进行BPG检验

我们可以构建如下的**BPG检验辅助方程**

`$$\begin{equation} \begin{alignedat}{999} &Pi=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} I(log(cigpric/income))&& + \hat{\beta}_{3} I(educ/income)\\ &\text{(cont.)}&& + \hat{\beta}_{4} age&& + \hat{\beta}_{5} I(age^2)&& + \hat{\beta}_{6} restaurn&&+e_i\\ \end{alignedat} \end{equation}$$`

**BPG检验辅助方程**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Pi}=&&+3.77&&+0.95I(log(cigpric/income))&&+1090.12I(educ/income)\\ &\text{(t)}&&(1.4638)&&(2.2419)&&(8.0546)\\&\text{(se)}&&(2.5730)&&(0.4237)&&(135.3412)\\&\text{(cont.)}&&+0.08age&&-0.00I(age^2)&&-0.60restaurn\\&\text{(t)}&&(1.1710)&&(-1.2426)&&(-1.2390)\\&\text{(se)}&&(0.0704)&&(0.0008)&&(0.4858)\\&\text{(fitness)}&& R^2=0.1851;&& \bar{R^2}=0.1800\\& && F^{\ast}=36.38;&& p=0.0000\\ \end{alignedat} \end{equation}$$`

---

### 对WLS矫正（收入）模型进行BPG检验

**BPG检验辅助方程**的回归平方和为
`$ESS=$` 115.8392；**LM统计量**（绝对值）为
`$LM \equiv {\chi^2}^{\ast}=ESS/2=$` 57.9196。

给定显著性水平
`$\alpha=0.05$`时卡方分布的理论查表值为
`$\chi^2_{1-\alpha}(m-1)=\chi^2_{0.95}(4)=$` 9.487729。

从而可以得到**BPG检验**初步结论：卡方检验极显著，拒绝原假设
`$H_0$`，认为**WLS矫正模型**仍旧存在**异方差性**。

---

## WLS矫正2（香烟价格变量）

根据散点图，也可以认为主模型方差正比于香烟价格（cigpric）的平方，因此加权最小二乘法（WLS）矫正模型如下：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{I(cigs/cigpric)}=&&-0.06&&+0.01I(log(income/cigpric))&&-0.39I(educ/cigpric)\\ &\text{(t)}&&(-0.7733)&&(1.1926)&&(-2.4633)\\&\text{(se)}&&(0.0742)&&(0.0123)&&(0.1598)\\&\text{(cont.)}&&+0.01age&&-0.00I(age^2)&&-0.06restaurn\\&\text{(t)}&&(4.5097)&&(-4.8636)&&(-3.0092)\\&\text{(se)}&&(0.0027)&&(0.0000)&&(0.0185)\\&\text{(fitness)}&& R^2=0.0492;&& \bar{R^2}=0.0432\\& && F^{\ast}=8.29;&& p=0.0000\\ \end{alignedat} \end{equation}$$`

---

### 对WLS矫正（香烟价格）模型进行BPG检验

我们可以构建如下的**BPG检验辅助方程**

`$$\begin{equation} \begin{alignedat}{999} &Pi=&& + \hat{\beta}_{1} && + \hat{\beta}_{2} I(log(income/cigpric))&& + \hat{\beta}_{3} I(educ/cigpric)\\ &\text{(cont.)}&& + \hat{\beta}_{4} age&& + \hat{\beta}_{5} I(age^2)&& + \hat{\beta}_{6} restaurn&&+e_i\\ \end{alignedat} \end{equation}$$`

**BPG检验辅助方程**的回归结果为：

`$$\begin{equation} \begin{alignedat}{999} &\widehat{Pi}=&&-1.41&&+0.12I(log(income/cigpric))&&+0.01I(educ/cigpric)\\ &\text{(t)}&&(-2.1108)&&(1.0434)&&(0.0044)\\&\text{(se)}&&(0.6675)&&(0.1103)&&(1.4379)\\&\text{(cont.)}&&+0.10age&&-0.00I(age^2)&&-0.48restaurn\\&\text{(t)}&&(4.0588)&&(-4.1341)&&(-2.8722)\\&\text{(se)}&&(0.0242)&&(0.0003)&&(0.1663)\\&\text{(fitness)}&& R^2=0.0376;&& \bar{R^2}=0.0316\\& && F^{\ast}=6.27;&& p=0.0000\\ \end{alignedat} \end{equation}$$`

---

### 对WLS矫正（香烟价格）模型进行BPG检验

**BPG检验辅助方程**的回归平方和为
`$ESS=$` 117.8926；**LM统计量**（绝对值）为
`$LM \equiv {\chi^2}^{\ast}=ESS/2=$` 58.9463。给定显著性水平
`$\alpha=0.05$`时卡方分布的理论查表值为
`$\chi^2_{1-\alpha}(m-1)=\chi^2_{0.95}(5)=$` 11.0705。从而可以得到**BPG检验**初步结论：卡方检验极显著，拒绝原假设
`$H_0$`，认为**WLS矫正模型**仍旧存在**异方差性**。

---

## WLS矫正3（多个变量的组合）：思路

如果主模型存在异方差问题，且假设方差是某种组合形式，也即：

`$$\begin{equation}
var(u_i|X_{2i},X_{3i}\cdots)=E(u^2_i)=\sigma^2_i=\sigma^2 \cdot {f}(X_{2i},X_{3i}\cdots)
\end{equation}$$`

此时，加权最小二乘法（WLS）矫正可以采用权重
`$w_i=\frac{1}{\sqrt{{f}(X_{2i},X_{3i}\cdots)}}$`。

为了获得权重
`$w_i$`，可以考虑到如下的异方差模式方程：

`$$\begin{align}
ln(f(.))=ln{(\tilde{e}_i^2)} =\tilde{\lambda}_1 &+ \tilde{\lambda}_1ln{(income)}_i + \tilde{\lambda}_3ln{(cigpric)}_{i} + \tilde{\lambda}_4educ_i + \tilde{\lambda}_5age_i + \tilde{\lambda}_6restaurant_i\\
& + \tilde{\lambda}_7{(ln(income_i))}^2 + \tilde{\lambda}_8(ln(cigpric_i))^2 +  \tilde{\lambda}_9educ_i^2 +  \tilde{\lambda}_{10}age_i^2
\end{align}$$`

上述异方差模式方程的拟合值可以作为WLS的权重，也即
`$w_i=\frac{1}{\sqrt{f(.)}}=\frac{1}{\sqrt{exp(\tilde{e}_i^2)}}=\frac{1}{\tilde{e}_i}$`。

---

## WLS矫正3（多个变量的组合）：操作步骤

**步骤1**：构建OLS回归

`$$\begin{align}
ln{(\tilde{e}_i^2)} =\tilde{\lambda}_1 &+ \tilde{\lambda}_1ln{(income)}_i + \tilde{\lambda}_3ln{(cigpric)}_{i} + \tilde{\lambda}_4educ_i + \tilde{\lambda}_5age_i + \tilde{\lambda}_6restaurant_i\\
& + \tilde{\lambda}_7{(ln(income_i))}^2 + \tilde{\lambda}_8(ln(cigpric_i))^2 +  \tilde{\lambda}_9educ_i^2 +  \tilde{\lambda}_{10}age_i^2
\end{align}$$`

**步骤2**：得到拟合值
`$f(.)=exp(linear)=\tilde{e}_i^2$`，并计算权重
`$w_i=\frac{1}{\sqrt{f(.)}}=\frac{1}{\tilde{e}_i}$`

**步骤3**：WLS变量变换，
`$Y^{\ast}_i=w_i{Y_i}=\frac{Y_i}{\tilde{e}_i}$`，
`$X^{\ast}_i=w_i{X_i}=\frac{X_i}{\tilde{e}_i}$`。

**步骤4**：估计变换后的新模型
`$Y^{\ast}_i = \alpha_0 +\alpha_1X^{\ast}_i+v_i$`

---
exclude: true

### software code

EVIEWS code

```
series ei2 = ei^2
eq_fx.fit
series ei_tilde = ei2_tilde^0.5
series nincome = income/ei_tilde
series ncigs = cigs/ei_tilde
series neduc =educ/ei_tilde
series nage =age/ei_tilde
series ncigpric = cigpric /ei_tilde
```

R code here

---

### WLS矫正3（多个变量的组合）：异方差探查方程的回归

**步骤1**：构建OLS回归，探明异方差的变量组合形式。

`$$\begin{equation}
\begin{alignedat}{999}
&\widehat{lnei2}=&&+13.3430&&+0.7081lnincome_i&&-8.4510lncigpric_i&&-0.0221educ_i\\ 
&(s)&&(73.6034)&&(1.0878)&&(36.3946)&&(0.1089)\\ 
&(t)&&(+0.18)&&(+0.65)&&(-0.23)&&(-0.20)\\ 
&(cont.)&&+0.2051age_i&&-0.0024age^2_i&&-0.6194restaurn_i&&-0.0232lnincome2_i\\ 
&(s)&&(0.0171)&&(0.0002)&&(0.1197)&&(0.0600)\\ 
&(t)&&(+11.96)&&(-12.89)&&(-5.18)&&(-0.39)\\ 
&(cont.)&&+1.0685lncigpric2_i&&-0.0023educ^2_i && &&\\ 
&(s)&&(4.5039)&&(0.0043) && &&\\ 
&(t)&&(+0.24)&&(-0.53) && &&\\ 
&(over)&&n=807&&\hat{\sigma}=1.4291 && &&\\ 
&(fit)&&R^2=0.2478&&\bar{R}^2=0.2393 && &&\\ 
&(Ftest)&&F^*=29.18&&p=0.0000 && &&
\end{alignedat}
\end{equation}$$`

---

### WLS矫正3（多个变量的组合）：权重和变量变换(数据表)

**步骤2**：得到变量组合形式的权重
`$w_i=\frac{1}{\tilde{e}_i}$`。

**步骤3**：利用权重
`$w_i=\frac{1}{\tilde{e}_i}$`，进行WLS变量变换并回归。

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---

### WLS矫正4（多个变量的组合）：WLS矫正后的回归

利用WLS权重变换后新变量进行OLS回归，结果如下：

`$$\begin{equation}
\begin{alignedat}{999}
&\widehat{ncigs}=&&+1.5511&&-0.0636nlnincome_i&&+0.1475nlncigpric_i&&-0.2390neduc_i\\ 
&(s)&&(0.8433)&&(0.0923)&&(0.2795)&&(0.1097)\\ 
&(t)&&(+1.84)&&(-0.69)&&(+0.53)&&(-2.18)\\ 
&(cont.)&&-0.0037nage_i&&+0.0003nage^2_i&&-0.0321restaurn_i &&\\ 
&(s)&&(0.0211)&&(0.0003)&&(0.1462) &&\\ 
&(t)&&(-0.18)&&(+0.99)&&(-0.22) &&\\ 
&(over)&&n=807&&\hat{\sigma}=1.5650 && &&\\ 
&(fit)&&R^2=0.0119&&\bar{R}^2=0.0045 && &&\\ 
&(Ftest)&&F^*=1.61&&p=0.1416 && &&
\end{alignedat}
\end{equation}$$`

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## 稳健标准误协方差矫正法（White法）：知识回顾

使用使用稳健标准误（robust standard errors）协方差矫正，可以一定程度上降低异方差问题对模型的影响（假设检验和预测）。这里我们将主要采用White (1980)提出的异方差一致性矫正法（heteroskedasticity consistent covariance）进行模型处理。

因此，我们可以看到两份可供对比的回归分析报告：

- 分析报告A：使用OLS方法，不进行稳健标准误协方差矫正。

- 分析报告B：使用OLS方法，进行（white方法的）稳健标准误协方差矫正。

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### 使用White稳健标准误法矫正异方差问题：B分析报告

使用White校正法矫正异方差的EViews报告如下：

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### 没有使用White稳健标准误法矫正异方差问题：A分析报告

作为对照，下面给出的是主模型的EViews报告：

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