﻿ 基于残差自回归模型的北京市92#汽油价格的分析与预测 Analysis and Prediction of 92# Gasoline Price in Beijing Based on Residual Autoregressive Model

Vol. 08  No. 11 ( 2019 ), Article ID: 32996 , 9 pages
10.12677/ASS.2019.811257

Analysis and Prediction of 92# Gasoline Price in Beijing Based on Residual Autoregressive Model

Xiaogang Li

Yunnan University of Finance and Economics, Kunming Yunnan

Received: Oct. 28th, 2019; accepted: Nov. 11th, 2019; published: Nov. 18th, 2019

ABSTRACT

Oil price is related to the national economy and people’s livelihood. This paper selects the price data of 92# gasoline for price adjustment date from January 2013 to November 2018 in Beijing. The residual autoregression model is used to investigate its fluctuation trend, extract its internal change rule and make prediction, which can provide reference for the government to adjust and control price, enterprises to make production decisions and people to maintain normal life and production order.

Keywords:Gasoline Price, Residual Autoregressive Model, ARCH Model

1. 研究背景

2. 文献综述

3. 理论模型

3.1. 残差自回归模型

$\left\{\begin{array}{l}{x}_{t}={T}_{t}+{\epsilon }_{t}\\ {\epsilon }_{t}={\phi }_{1}{\epsilon }_{t-1}+\cdots +{\phi }_{p}{\epsilon }_{t-p}+{a}_{t}\end{array}$

$E\left({a}_{t}\right)=0,Var\left({a}_{t}\right)={\sigma }^{2},Cov\left({a}_{t},{a}_{t-i}\right)=0,\text{ }\forall i\ge 1$

3.1.1. 确定性模型拟合

${T}_{t}={\beta }_{0}+{\beta }_{1}\cdot t+\cdots +{\beta }_{k}\cdot {t}^{k}$，或 ${T}_{t}={\beta }_{0}+{\beta }_{1}\cdot {x}_{t-1}+\cdots +{\beta }_{k}\cdot {x}_{t-k}$

3.1.2. 残差自相关性检验及拟合

$E\left({\epsilon }_{t}{\epsilon }_{t-j}\right)\ne 0,\text{ }\exists j\ge 1$

3.2. 集群效应与ARCH模型

$\left\{\begin{array}{l}{x}_{t}=f\left(t,{x}_{t-1},{x}_{t-2},\cdots \right)+{\epsilon }_{t}\\ {\epsilon }_{t}=\sqrt{{h}_{t}}{e}_{t}\\ {h}_{t}=\omega +\underset{j=1}{\overset{q}{\sum }}{\lambda }_{j}{\epsilon }_{t-j}^{2}\end{array}$

$LM\left(q\right)=\frac{\left(SST-SSE\right)/q}{SSE/\left(T-2q-1\right)}$

4. 数据来源和处理

5. 实证分析

5.1. 残差自回归模型

$\left\{\begin{array}{l}{x}_{t}={T}_{t}+{\epsilon }_{t}，\text{ }t=1,2,3,\cdots \\ {\epsilon }_{t}={\phi }_{1}{\epsilon }_{t-1}+\cdots +{\phi }_{p}{\epsilon }_{t-p}+{a}_{t}\end{array}$

$E\left({a}_{t}\right)=0,Var\left({a}_{t}\right)={\sigma }^{2},Cov\left({a}_{t},{a}_{t-i}\right)=0,\text{ }\forall i\ge 1$

Figure 1. Sequence diagram of 92# gasoline price from 2013 to 2018 (August) in Beijing

5.1.1. 确定性模型拟合

(1) 时间t的幂函数为变量：

${T}_{t}={\beta }_{0}+{\beta }_{1}\cdot t+{\beta }_{2}\cdot {t}^{2}+{\beta }_{3}\cdot {t}^{3}，\text{ }t=1,2,3,\cdots$

${x}_{t}=7.958-0.0007611{t}^{2}+0.000005572{t}^{3}+{\epsilon }_{t}\begin{array}{cc},& {\epsilon }_{t}~N\left(0,{0.3895}^{2}\right)\end{array}$

Table 1. Fitting the cubic function regression model about t

(2) 1阶延迟 ${x}_{t-1}$ 为变量：

${T}_{t}={\beta }_{0}+{\beta }_{1}\cdot {x}_{t-1}$

${x}_{t}=0.13576+0.98037{x}_{t\text{-}1}+{\epsilon }_{t}\begin{array}{cc},& {\epsilon }_{t}~N\left(0,{0.1769}^{2}\right)\end{array}$

Table 2. Fitting autoregressive model for delay variable

5.1.2. 残差自相关性检验及拟合

Figure 2. The fitting effect of two trend-fitting models

Figure 3. Residual and its square after the first trend effect

Table 3. DW-test on correlation of residual sequence

${\epsilon }_{t}=-0.8969{\epsilon }_{t-1}+{e}_{t}\begin{array}{cc},& {e}_{t}~N\left(0,0.02886\right)\end{array}$

Figure 4. Autocorrelation and partial autocorrelation of residual sequence

Table 4. Fitting the AR (1) model to the residual sequence

Table 5. White noise test

$\left\{\begin{array}{l}{x}_{t}=7.958-0.0007611{t}^{2}+0.000005572{t}^{3}+{\epsilon }_{t}\\ {\epsilon }_{t}=-0.8969{\epsilon }_{t-1}+{e}_{t}\begin{array}{cc},& {e}_{t}~N\left(0,0.02867\right)\end{array}\end{array}$

Table 6. Table of predicted values, true values and relative errors

5.2. 集群效应与ARCH模型

$E\left({\epsilon }_{t}^{2}|{\epsilon }_{t-1}\right)=0.022557+0.939331{\epsilon }_{t-1}^{2}$

Table 7. ARCH (1) model for residual square sequence

Figure 5. Confidence interval comparison of conditional heteroscedasticity and homogeneity of variance

6. 结论

Analysis and Prediction of 92# Gasoline Price in Beijing Based on Residual Autoregressive Model[J]. 社会科学前沿, 2019, 08(11): 1887-1895. https://doi.org/10.12677/ASS.2019.811257

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