{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 基础概念"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 均值回归局限\n",
"\n",
"只有满足古典假定，估计量才具有优良性质：BLUE\n",
"\n",
"<div align=center><img src=\"https://gitee.com/lei940324/picture/raw/master/img/2020/0619/205104.png\" width=\"683\" ></div>\n",
"\n",
"## 为什么需要分位数回归\n",
"\n",
"在迄今为止的同归模型中，我们着重考察解释变量 x 对被解释变量 y 的条件期望 E $(y | \\boldsymbol{x})$ 的 影响，实际上是均值回归。但我们真正关心的是 x 对整个条件分布 $y | x$ 的影响，而条件期望 $E(y| \\boldsymbol{x})$ 只是刻画条件分布 $y | \\boldsymbol{x}$ 集中趋势的一个指标而已。如果条件分布 $y | \\boldsymbol{x}$ 不是对称分布，则条件期望 E( $y | \\boldsymbol{x}$ )很难反映整个条件分布的全貌。如果能够估计出条件分布 $y | x$ 的若干重要的条件分位数，比如中位数、1/4 分位数 ，3/4 分位数，就能对条件分布 $y | \\boldsymbol{x}$ 有更全面的认识。另一方面， 使用 OLS 的古典“均值同归”，由于最小化的目标函数为残差平方和 $\\left(\\sum_{i=1}^{n} e_{i}^{2}\\right),$ 故容易受极端值的影响。 \n",
"\n",
"为此， Koenker and Bassett( 1978 ) 提出“分位数同归”(Quantile Regression，简记 QR ) ，使用残差绝对值的加权平均（比如 $\\sum_{i=1}^{n}\\left|e_{i}\\right|$ ) 作为最小化的目标函数，故不易受极端值影响，较为稳健。更重要的是，分位数回归还能提供关于条件分布 $y | \\boldsymbol{x}$ 的全面信息。\n",
"\n",
"<div align=center><img src=\"https://gitee.com/lei940324/picture/raw/master/img/2020/0619/205146.png\" width=\"750\" ></div>\n",
"\n",
"## 原理\n",
"\n",
"### 模型表示\n",
"\n",
"$Q_{t}(y | x)=x^{T} \\beta_{\\tau}$\n",
"\n",
"其中 $\\tau$ 为分位点, $\\beta_{\\tau}$ 为依赖于分位点的回归系数\n",
"\n",
"### 损失函数\n",
"\n",
"* 二次损失：生成均值 $\n",
" E(Y)=\\underset{y}{\\operatorname{argmin}} E(Y-y)^{2}$\n",
"* 绝对值损失：生成中位数 $A L(u)=|u|\n",
" $\n",
"* 非对称绝对值损失：生成分位数 $\\quad \\rho_{\\tau}(u)=u(\\tau-I(u<0)) \\quad Q_{\\tau}(Y)=\\underset{y}{\\operatorname{argmin}} E \\rho_{\\tau}(Y-y)$\n",
"\n",
"<div align=center><img src=\"https://gitee.com/lei940324/picture/raw/master/img/2020/0619/205807.png\" width=\"547\" ></div>\n",
"\n",
"### 估计方法\n",
"\n",
"由于分位数回归的目标函数带有绝对值，不可微分，故通常使用线性规划。\n",
"\n",
"详情可参考：[statsmodels 官方文档](https://www.statsmodels.org/stable/generated/statsmodels.regression.quantile_regression.QuantReg.html#statsmodels.regression.quantile_regression.QuantReg)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 使用 statsmodels 库实现\n",
"### 第一种方法"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>QuantReg Regression Results</caption>\n",
"<tr>\n",
" <th>Dep. Variable:</th> <td>data['hs300']</td> <th> Pseudo R-squared: </th> <td> 0.7878</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Model:</th> <td>QuantReg</td> <th> Bandwidth: </th> <td> 57.44</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Method:</th> <td>Least Squares</td> <th> Sparsity: </th> <td> 149.5</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Date:</th> <td>Sat, 11 Jul 2020</td> <th> No. Observations: </th> <td> 460</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Time:</th> <td>22:06:40</td> <th> Df Residuals: </th> <td> 458</td> \n",
"</tr>\n",
"<tr>\n",
" <th> </th> <td> </td> <th> Df Model: </th> <td> 1</td> \n",
"</tr>\n",
"</table>\n",
"<table class=\"simpletable\">\n",
"<tr>\n",
" <td></td> <th>coef</th> <th>std err</th> <th>t</th> <th>P>|t|</th> <th>[0.025</th> <th>0.975]</th> \n",
"</tr>\n",
"<tr>\n",
" <th>Intercept</th> <td> -173.7703</td> <td> 45.949</td> <td> -3.782</td> <td> 0.000</td> <td> -264.066</td> <td> -83.474</td>\n",
"</tr>\n",
"<tr>\n",
" <th>data['sz']</th> <td> 1.2845</td> <td> 0.016</td> <td> 82.112</td> <td> 0.000</td> <td> 1.254</td> <td> 1.315</td>\n",
"</tr>\n",
"</table><br/><br/>The condition number is large, 3.55e+04. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
],
"text/plain": [
"<class 'statsmodels.iolib.summary.Summary'>\n",
"\"\"\"\n",
" QuantReg Regression Results \n",
"==============================================================================\n",
"Dep. Variable: data['hs300'] Pseudo R-squared: 0.7878\n",
"Model: QuantReg Bandwidth: 57.44\n",
"Method: Least Squares Sparsity: 149.5\n",
"Date: Sat, 11 Jul 2020 No. Observations: 460\n",
"Time: 22:06:40 Df Residuals: 458\n",
" Df Model: 1\n",
"==============================================================================\n",
" coef std err t P>|t| [0.025 0.975]\n",
"------------------------------------------------------------------------------\n",
"Intercept -173.7703 45.949 -3.782 0.000 -264.066 -83.474\n",
"data['sz'] 1.2845 0.016 82.112 0.000 1.254 1.315\n",
"==============================================================================\n",
"\n",
"The condition number is large, 3.55e+04. This might indicate that there are\n",
"strong multicollinearity or other numerical problems.\n",
"\"\"\""
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"import statsmodels.formula.api as smf\n",
"\n",
"data = pd.read_excel('../数据/上证指数与沪深300.xlsx')\n",
"\n",
"mod = smf.quantreg(\"data['hs300'] ~ data['sz']\", data)\n",
"res = mod.fit(q=0.3)\n",
"res.summary()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 第二种方法"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>QuantReg Regression Results</caption>\n",
"<tr>\n",
" <th>Dep. Variable:</th> <td>hs300</td> <th> Pseudo R-squared: </th> <td> 0.7878</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Model:</th> <td>QuantReg</td> <th> Bandwidth: </th> <td> 57.44</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Method:</th> <td>Least Squares</td> <th> Sparsity: </th> <td> 149.5</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Date:</th> <td>Sat, 11 Jul 2020</td> <th> No. Observations: </th> <td> 460</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Time:</th> <td>22:06:40</td> <th> Df Residuals: </th> <td> 458</td> \n",
"</tr>\n",
"<tr>\n",
" <th> </th> <td> </td> <th> Df Model: </th> <td> 1</td> \n",
"</tr>\n",
"</table>\n",
"<table class=\"simpletable\">\n",
"<tr>\n",
" <td></td> <th>coef</th> <th>std err</th> <th>t</th> <th>P>|t|</th> <th>[0.025</th> <th>0.975]</th> \n",
"</tr>\n",
"<tr>\n",
" <th>const</th> <td> -173.7703</td> <td> 45.949</td> <td> -3.782</td> <td> 0.000</td> <td> -264.066</td> <td> -83.474</td>\n",
"</tr>\n",
"<tr>\n",
" <th>sz</th> <td> 1.2845</td> <td> 0.016</td> <td> 82.112</td> <td> 0.000</td> <td> 1.254</td> <td> 1.315</td>\n",
"</tr>\n",
"</table><br/><br/>The condition number is large, 3.55e+04. This might indicate that there are<br/>strong multicollinearity or other numerical problems."
],
"text/plain": [
"<class 'statsmodels.iolib.summary.Summary'>\n",
"\"\"\"\n",
" QuantReg Regression Results \n",
"==============================================================================\n",
"Dep. Variable: hs300 Pseudo R-squared: 0.7878\n",
"Model: QuantReg Bandwidth: 57.44\n",
"Method: Least Squares Sparsity: 149.5\n",
"Date: Sat, 11 Jul 2020 No. Observations: 460\n",
"Time: 22:06:40 Df Residuals: 458\n",
" Df Model: 1\n",
"==============================================================================\n",
" coef std err t P>|t| [0.025 0.975]\n",
"------------------------------------------------------------------------------\n",
"const -173.7703 45.949 -3.782 0.000 -264.066 -83.474\n",
"sz 1.2845 0.016 82.112 0.000 1.254 1.315\n",
"==============================================================================\n",
"\n",
"The condition number is large, 3.55e+04. This might indicate that there are\n",
"strong multicollinearity or other numerical problems.\n",
"\"\"\""
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import statsmodels.regression.quantile_regression as qr\n",
"import statsmodels.api as sm\n",
" \n",
"X = sm.add_constant(data['sz'])\n",
"mod = qr.QuantReg(data['hs300'], X)\n",
"res = mod.fit(q=0.3)\n",
"res.summary()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## matlab实现\n",
"可以参考 matlab 代码：[分位数回归](https://github.com/lei940324/econometrics/tree/master/matlab代码/分位数回归/分布滞后模型回归)"
]
}
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