{
"cells": [
{
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"metadata": {},
"source": [
"# ARMA模型"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 原理讲解\n",
"\n",
"为了使模型更好地拟合数据，可以将 $AR(p)$ 与 $MA(q)$ 结合起来，得到 $ARMA(p,q)$ ：\n",
"\n",
"$$y_{t}=\\beta_{0}+\\beta_{1} y_{t-1}+\\cdots+\\beta_{p} y_{t-p}+\\varepsilon_{t}+\\theta_{1} \\varepsilon_{t-1}+\\cdots+\\theta_{q} \\varepsilon_{t-q}\\tag{1}$$\n",
"\n",
"其中，{$\\varepsilon _t$} 为白噪声。在给定 $\\left\\{y_{1}, y_{2}, \\cdots, y_{p}\\right\\}$ 与“$\\varepsilon_{0}=\\varepsilon_{-1}=\\cdots=\\varepsilon_{-q+1}=0$”的条件下，可以使用条件 MLE 来估计 $ARMA(p,q)$。为了估计 $ARMA(p,q)$，首先必须确定 $(p,q)$，而经济理论通常不能提供这些信息，故只能根据数据来估计 $(p,q)$。\n",
"在实践中，常常先考察数据的自相关函数（ACF）与偏自相关函数（PACF），以判断是否存在 p=0 或 q=0 的情形。"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## statsmodels 库实现"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>ARMA Model Results</caption>\n",
"<tr>\n",
" <th>Dep. Variable:</th> <td>sz</td> <th> No. Observations: </th> <td>460</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model:</th> <td>ARMA(1, 1)</td> <th> Log Likelihood </th> <td>-2294.749</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Method:</th> <td>css-mle</td> <th> S.D. of innovations</th> <td>35.352</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Date:</th> <td>Fri, 29 May 2020</td> <th> AIC </th> <td>4597.498</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Time:</th> <td>21:03:01</td> <th> BIC </th> <td>4614.023</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Sample:</th> <td>0</td> <th> HQIC </th> <td>4604.006</td> \n",
"</tr>\n",
"<tr>\n",
" <th></th> <td> </td> <th> </th> <td> </td> \n",
"</tr>\n",
"</table>\n",
"<table class=\"simpletable\">\n",
"<tr>\n",
" <td></td> <th>coef</th> <th>std err</th> <th>z</th> <th>P>|z|</th> <th>[0.025</th> <th>0.975]</th> \n",
"</tr>\n",
"<tr>\n",
" <th>const</th> <td> 2987.0655</td> <td> 142.111</td> <td> 21.019</td> <td> 0.000</td> <td> 2708.533</td> <td> 3265.597</td>\n",
"</tr>\n",
"<tr>\n",
" <th>ar.L1.sz</th> <td> 0.9902</td> <td> 0.006</td> <td> 161.182</td> <td> 0.000</td> <td> 0.978</td> <td> 1.002</td>\n",
"</tr>\n",
"<tr>\n",
" <th>ma.L1.sz</th> <td> 0.0012</td> <td> 0.047</td> <td> 0.025</td> <td> 0.980</td> <td> -0.090</td> <td> 0.093</td>\n",
"</tr>\n",
"</table>\n",
"<table class=\"simpletable\">\n",
"<caption>Roots</caption>\n",
"<tr>\n",
" <td></td> <th> Real</th> <th> Imaginary</th> <th> Modulus</th> <th> Frequency</th>\n",
"</tr>\n",
"<tr>\n",
" <th>AR.1</th> <td> 1.0099</td> <td> +0.0000j</td> <td> 1.0099</td> <td> 0.0000</td>\n",
"</tr>\n",
"<tr>\n",
" <th>MA.1</th> <td> -842.3029</td> <td> +0.0000j</td> <td> 842.3029</td> <td> 0.5000</td>\n",
"</tr>\n",
"</table>"
],
"text/plain": [
"<class 'statsmodels.iolib.summary.Summary'>\n",
"\"\"\"\n",
" ARMA Model Results \n",
"==============================================================================\n",
"Dep. Variable: sz No. Observations: 460\n",
"Model: ARMA(1, 1) Log Likelihood -2294.749\n",
"Method: css-mle S.D. of innovations 35.352\n",
"Date: Fri, 29 May 2020 AIC 4597.498\n",
"Time: 21:03:01 BIC 4614.023\n",
"Sample: 0 HQIC 4604.006\n",
" \n",
"==============================================================================\n",
" coef std err z P>|z| [0.025 0.975]\n",
"------------------------------------------------------------------------------\n",
"const 2987.0655 142.111 21.019 0.000 2708.533 3265.597\n",
"ar.L1.sz 0.9902 0.006 161.182 0.000 0.978 1.002\n",
"ma.L1.sz 0.0012 0.047 0.025 0.980 -0.090 0.093\n",
" Roots \n",
"=============================================================================\n",
" Real Imaginary Modulus Frequency\n",
"-----------------------------------------------------------------------------\n",
"AR.1 1.0099 +0.0000j 1.0099 0.0000\n",
"MA.1 -842.3029 +0.0000j 842.3029 0.5000\n",
"-----------------------------------------------------------------------------\n",
"\"\"\""
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from statsmodels.tsa.arima_model import ARMA\n",
"import pandas as pd\n",
"\n",
"data = pd.read_excel('../数据/上证指数与沪深300.xlsx')\n",
"res = ARMA(data['sz'], order=(1,1)).fit()\n",
"res.summary()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## matlab 实现\n",
"可以参考 [MA 模型](https://nbviewer.jupyter.org/github/lei940324/econometrics/blob/master/金融时间序列/移动平均(MA)模型.ipynb)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Eviews 估计\n",
"估计方程 `y c ma(p) ar(q)`\n",
"<div align=center><img src=\"https://gitee.com/lei940324/picture/raw/master/img/2020/0529/172512.png\" width=\"430\" ></div>"
]
}
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