{ "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": [ "
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日期hs300szsz收益率
02018-01-034111.39253369.10840.618765
12018-01-044128.81193385.71020.491555
22018-01-054138.75053391.75010.178235
32018-01-084160.15953409.47950.521360
42018-01-094189.29773413.89960.129558
\n", "
" ], "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": [ "\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
Augmented Dickey-Fuller Results
Test Statistic -2.307
P-value 0.170
Lags 7


Trend: Constant
Critical Values: -3.44 (1%), -2.87 (5%), -2.57 (10%)
Null Hypothesis: The process contains a unit root.
Alternative Hypothesis: The process is weakly stationary." ], "text/plain": [ "\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": [ "\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
KPSS Stationarity Test Results
Test Statistic 0.874
P-value 0.005
Lags 12


Trend: Constant
Critical Values: 0.74 (1%), 0.46 (5%), 0.35 (10%)
Null Hypothesis: The process is weakly stationary.
Alternative Hypothesis: The process contains a unit root." ], "text/plain": [ "\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": [ "\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
Dickey-Fuller GLS Results
Test Statistic -0.563
P-value 0.492
Lags 7


Trend: Constant
Critical Values: -2.61 (1%), -1.99 (5%), -1.67 (10%)
Null Hypothesis: The process contains a unit root.
Alternative Hypothesis: The process is weakly stationary." ], "text/plain": [ "\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": [ "\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
Phillips-Perron Test (Z-tau)
Test Statistic -2.108
P-value 0.241
Lags 18


Trend: Constant
Critical Values: -3.44 (1%), -2.87 (5%), -2.57 (10%)
Null Hypothesis: The process contains a unit root.
Alternative Hypothesis: The process is weakly stationary." ], "text/plain": [ "\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 }