{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 收益率概念与平稳性"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 资产收益率"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"多数金融研究针对的是资产收益率而不是资产价格。\n",
"\n",
"使用收益率有两个主要理由：\n",
"1. 对普通的投资者来说，资产收益率完全体现了该资产的投资机会，且与其投资规模无关；\n",
"2. 收益率序列比价格序列更容易处理，因为前者有更好的统计性质\n",
"\n",
"然而，资产收益率有多种定义，设 $P_t$ 是资产在 $t$ 时刻的价格。下面给出常见的几个收益率定义.暂时假定资产不支付分红。"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 单期简单收益率"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"若从第 $t-1$ 天到第 $t$ 天(一个周期)持有某种资产，则**简单毛收益率**为：\n",
"\n",
"$$1+R_{t}=\\frac{P_{t}}{P_{t-1}} \\quad \\text { 或 } \\quad P_{t}=P_{t-1}\\left(1+R_{t}\\right)\\tag{1}$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"对应的**单期简单收益率**或称**简单收益率**为：\n",
"\n",
"$$R_{t}=\\frac{P_{t}}{P_{t-1}}-1=\\frac{P_{t}-P_{t-1}}{P_{t-1}}\\tag{2}$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 连续复合收益率"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"资产的简单毛收益率的自然对数称为**连续复合收益率** 或 **对数收益率**：\n",
"\n",
"$$r_{t}=\\ln \\left(1+R_{t}\\right)=\\ln \\frac{P_{t}}{P_{t-1}}=p_{t}-p_{t-1}\\tag{3}$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"其中 $p_{t}=\\ln P_{t}$，与简单净收益率 $R_t$ 相比，连续复合收益率 $r_t$ 有一些优点。首先，对多期收益率，我们有：\n",
"\n",
"$$\\begin{aligned}\n",
"r_{t}[k] &=\\ln \\left(1+R_{t}[k]\\right)=\\ln \\left[\\left(1+R_{t}\\right)\\left(1+R_{t-1}\\right) \\cdots\\left(1+R_{t-k+1}\\right)\\right] \\\\\n",
"&=\\ln \\left(1+R_{t}\\right)+\\ln \\left(1+R_{t-1}\\right)+\\cdots+\\ln \\left(1+R_{t-k+1}\\right) \\\\\n",
"&=r_{t}+r_{t-1}+\\cdots+r_{t-k+1}\n",
"\\end{aligned}\\tag{4}$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"这样，连续复合多期收益率就是它所包含的连续复合单期收益率之和。其次，对数收益率具有更容易处理的统计性质。"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>日期</th>\n",
" <th>hs300</th>\n",
" <th>sz</th>\n",
" <th>sz收益率</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>2018-01-03</td>\n",
" <td>4111.3925</td>\n",
" <td>3369.1084</td>\n",
" <td>0.618765</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>2018-01-04</td>\n",
" <td>4128.8119</td>\n",
" <td>3385.7102</td>\n",
" <td>0.491555</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>2018-01-05</td>\n",
" <td>4138.7505</td>\n",
" <td>3391.7501</td>\n",
" <td>0.178235</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>2018-01-08</td>\n",
" <td>4160.1595</td>\n",
" <td>3409.4795</td>\n",
" <td>0.521360</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>2018-01-09</td>\n",
" <td>4189.2977</td>\n",
" <td>3413.8996</td>\n",
" <td>0.129558</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" 日期 hs300 sz sz收益率\n",
"0 2018-01-03 4111.3925 3369.1084 0.618765\n",
"1 2018-01-04 4128.8119 3385.7102 0.491555\n",
"2 2018-01-05 4138.7505 3391.7501 0.178235\n",
"3 2018-01-08 4160.1595 3409.4795 0.521360\n",
"4 2018-01-09 4189.2977 3413.8996 0.129558"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import pandas as pd\n",
"import numpy as np\n",
"\n",
"stock = pd.read_excel('../数据/上证指数与沪深300.xlsx')\n",
"stock['sz收益率'] = 100*np.log(stock['sz']/stock['sz'].shift(1))\n",
"stock = stock.dropna() #删除缺失值\n",
"stock = stock.reset_index(drop=True)\n",
"stock.head()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 平稳性\n",
"### 概念"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"平稳性是时间序列分析的基础。\n",
"\n",
"时间序列 {$r_t$} 称为**严平稳**的(strictly stationary)，如果对所有的 $t$，任意正整数 $k$ 和任意 $k$ 个正整数($t_1,\\cdots,t_k$)，($r_{t_1},\\cdots,r_{t_k}$)的联合分布与($r_{t_1+t},\\cdots,r_{t_k+t}$)的联合分布是相同的.换言之，严平稳性要求($r_{t_1},\\cdots,r_{t_k}$)的联合分布在时间的平移变换下保持不变。这是一个很强的条件，难以用经验方法验证，经常假定的是平稳性的一一个较弱的形式。\n",
"\n",
"时间序列 {$r_t$} 称为**弱平稳**的（weakly stationary），如果 $r_t$ 的均值与 $r_t$ 和 $r_{t-l}$ 的协方差不随时间而改变，其中 $l$ 是任意整数.更具体地说，{$r_t$} 是弱平稳的，若：\n",
"1. $E_{r_t}=\\mu$，$\\mu$ 是一个常数；\n",
"2. $\\operatorname{Cov}\\left(r_{t}, r_{t-l}\\right)=\\gamma_{l}$，$\\gamma_{l}$ 只依赖于 $l$。\n",
"\n",
"在实际中，假定我们有 $T$ 个数据观测点$\\left\\{r_{t} | t=1, \\cdots, T\\right\\}$，弱平稳性意味着数据的时间图显示出 $T$ 个值在一个常数水平上下以相同幅度波动。在应用中，弱平稳性使我们可以对未来观测进行推断，即预测。\n",
"\n",
"在弱平稳性的条件中，我们隐含地假定了 $r_t$ 的前两阶矩是有限的.由定义可见，若 $r_t$ 是严平稳的且它的前两阶矩是有限的，则 $r_t$ 也是弱平稳的.反之，一般是不成立的.但如果时间序列 $r_t$ 是正态分布的，则弱平稳性与严平稳性是等价的.本内容主要考虑弱平稳序列。\n",
"\n",
"协方差 $\\gamma_{l}=\\operatorname{Cov}\\left(r_{t}, r_{t-l}\\right)$ 称为 $r_t$ 的间隔为 $l$ 的自协方差、它具有两个重要性质：\n",
"1. $\\gamma_{0}=\\operatorname{Var}\\left(r_{t}\\right)$\n",
"2. $\\gamma_{-l}=\\gamma_{l}$\n",
"\n",
"第二个性质成立是因为$\\operatorname{Cov}\\left(r_{t}, r_{t-(-l)}\\right)=\\operatorname{Cov}\\left(r_{t-(-l)}, r_{t}\\right)=\\operatorname{Cov}\\left(r_{t+l}, r_{t}\\right)=\\operatorname{Cov}\\left(r_{t_{1}}, r_{t_{1}-l}\\right)$，其中$t_{1}=t+l$\n",
"\n",
"在金融文献中，通常假定资产收益率序列是弱平稳的.只要有足够多的历史收益率数据，这个假定可以用实证方法验证，例如，我们可以把数据分成若干子样本，然后检验它们的一致性。"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 单位根所带来的问题\n",
"对于 AR(1)，一般从理论上认为，不太可能出现 | $\\beta_{1} |> 1$ 的情形，否则任何对经济的扰动都将被无限放大。因此，经济学家通常只担心存在单位根的情形， 即 $\\beta_{1}=1_{\\circ}$ 如果时间序列存在单位根，则为非平稳序列，可能带来以下问题:\n",
"1. 自回归系数的估计值向左偏向于 0\n",
"2. 传统的 t 检验失效\n",
"3. 两个相互独立的单位根变量可能出现伪回归(spurious regression)或伪相关"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Augmented Dickey-Fuller 单位根检验(ADF检验)"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-2.3069400422968207,\n",
" 0.16973943854299967,\n",
" 7,\n",
" 451,\n",
" {'1%': -3.444932949082776,\n",
" '5%': -2.867969899953726,\n",
" '10%': -2.57019489663276},\n",
" 4390.386487977853)"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import statsmodels.tsa.stattools as ts\n",
"ts.adfuller(stock['sz'])"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>Augmented Dickey-Fuller Results</caption>\n",
"<tr>\n",
" <td>Test Statistic</td> <td>-2.307</td>\n",
"</tr>\n",
"<tr>\n",
" <td>P-value</td> <td>0.170</td>\n",
"</tr>\n",
"<tr>\n",
" <td>Lags</td> <td>7</td>\n",
"</tr>\n",
"</table><br/><br/>Trend: Constant<br/>Critical Values: -3.44 (1%), -2.87 (5%), -2.57 (10%)<br/>Null Hypothesis: The process contains a unit root.<br/>Alternative Hypothesis: The process is weakly stationary."
],
"text/plain": [
"<class 'arch.unitroot.unitroot.ADF'>\n",
"\"\"\"\n",
" Augmented Dickey-Fuller Results \n",
"=====================================\n",
"Test Statistic -2.307\n",
"P-value 0.170\n",
"Lags 7\n",
"-------------------------------------\n",
"\n",
"Trend: Constant\n",
"Critical Values: -3.44 (1%), -2.87 (5%), -2.57 (10%)\n",
"Null Hypothesis: The process contains a unit root.\n",
"Alternative Hypothesis: The process is weakly stationary.\n",
"\"\"\""
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from arch.unitroot import ADF\n",
"ADF(stock['sz'])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### KPSS 平稳性检验"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"G:\\anaconda\\lib\\site-packages\\statsmodels\\tsa\\stattools.py:1685: InterpolationWarning: p-value is smaller than the indicated p-value\n",
" warn(\"p-value is smaller than the indicated p-value\", InterpolationWarning)\n"
]
},
{
"data": {
"text/plain": [
"(0.8735590412113318,\n",
" 0.01,\n",
" 12,\n",
" {'10%': 0.347, '5%': 0.463, '2.5%': 0.574, '1%': 0.739})"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ts.kpss(stock['sz'], nlags='auto')"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>KPSS Stationarity Test Results</caption>\n",
"<tr>\n",
" <td>Test Statistic</td> <td>0.874</td>\n",
"</tr>\n",
"<tr>\n",
" <td>P-value</td> <td>0.005</td>\n",
"</tr>\n",
"<tr>\n",
" <td>Lags</td> <td>12</td>\n",
"</tr>\n",
"</table><br/><br/>Trend: Constant<br/>Critical Values: 0.74 (1%), 0.46 (5%), 0.35 (10%)<br/>Null Hypothesis: The process is weakly stationary.<br/>Alternative Hypothesis: The process contains a unit root."
],
"text/plain": [
"<class 'arch.unitroot.unitroot.KPSS'>\n",
"\"\"\"\n",
" KPSS Stationarity Test Results \n",
"=====================================\n",
"Test Statistic 0.874\n",
"P-value 0.005\n",
"Lags 12\n",
"-------------------------------------\n",
"\n",
"Trend: Constant\n",
"Critical Values: 0.74 (1%), 0.46 (5%), 0.35 (10%)\n",
"Null Hypothesis: The process is weakly stationary.\n",
"Alternative Hypothesis: The process contains a unit root.\n",
"\"\"\""
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from arch.unitroot import KPSS\n",
"KPSS(stock['sz'])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### DFGLS 检验"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>Dickey-Fuller GLS Results</caption>\n",
"<tr>\n",
" <td>Test Statistic</td> <td>-0.563</td>\n",
"</tr>\n",
"<tr>\n",
" <td>P-value</td> <td>0.492</td>\n",
"</tr>\n",
"<tr>\n",
" <td>Lags</td> <td>7</td>\n",
"</tr>\n",
"</table><br/><br/>Trend: Constant<br/>Critical Values: -2.61 (1%), -1.99 (5%), -1.67 (10%)<br/>Null Hypothesis: The process contains a unit root.<br/>Alternative Hypothesis: The process is weakly stationary."
],
"text/plain": [
"<class 'arch.unitroot.unitroot.DFGLS'>\n",
"\"\"\"\n",
" Dickey-Fuller GLS Results \n",
"=====================================\n",
"Test Statistic -0.563\n",
"P-value 0.492\n",
"Lags 7\n",
"-------------------------------------\n",
"\n",
"Trend: Constant\n",
"Critical Values: -2.61 (1%), -1.99 (5%), -1.67 (10%)\n",
"Null Hypothesis: The process contains a unit root.\n",
"Alternative Hypothesis: The process is weakly stationary.\n",
"\"\"\""
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from arch.unitroot import DFGLS\n",
"DFGLS(stock['sz'])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### PhillipsPerron（PP）检验"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>Phillips-Perron Test (Z-tau)</caption>\n",
"<tr>\n",
" <td>Test Statistic</td> <td>-2.108</td>\n",
"</tr>\n",
"<tr>\n",
" <td>P-value</td> <td>0.241</td>\n",
"</tr>\n",
"<tr>\n",
" <td>Lags</td> <td>18</td>\n",
"</tr>\n",
"</table><br/><br/>Trend: Constant<br/>Critical Values: -3.44 (1%), -2.87 (5%), -2.57 (10%)<br/>Null Hypothesis: The process contains a unit root.<br/>Alternative Hypothesis: The process is weakly stationary."
],
"text/plain": [
"<class 'arch.unitroot.unitroot.PhillipsPerron'>\n",
"\"\"\"\n",
" Phillips-Perron Test (Z-tau) \n",
"=====================================\n",
"Test Statistic -2.108\n",
"P-value 0.241\n",
"Lags 18\n",
"-------------------------------------\n",
"\n",
"Trend: Constant\n",
"Critical Values: -3.44 (1%), -2.87 (5%), -2.57 (10%)\n",
"Null Hypothesis: The process contains a unit root.\n",
"Alternative Hypothesis: The process is weakly stationary.\n",
"\"\"\""
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from arch.unitroot import PhillipsPerron\n",
"PhillipsPerron(stock['sz'])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"更多单位根检验请参考：[Unit Root Testing](https://arch.readthedocs.io/en/latest/unitroot/unitroot.html)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.6"
}
},
"nbformat": 4,
"nbformat_minor": 4
}