﻿ 房地产周期性波动的谱分析研究 Study on Spectral Analysis of Real Estate Cyclic Fluctuation

Statistics and Application
Vol.07 No.02(2018), Article ID:24671,13 pages
10.12677/SA.2018.72026

Study on Spectral Analysis of Real Estate Cyclic Fluctuation

Dongmei Xing

Department of Mathematics, Nanchang University, Nanchang Jiangxi

Received: Apr. 3rd, 2018; accepted: Apr. 21st, 2018; published: Apr. 28th, 2018

ABSTRACT

This paper first briefly describes the theoretical tools on studying economic data with periodic characteristics, then key indicators both of the supply and demand are identified by extracting the principal components from statistical data of real estate market in China from 1988 to 2016. Discussing the stability characteristics of the key indicators, the cycles both of the supply and demand is calculated by using spectral analysis techniques. The results show that both supply and demand fluctuate periodically. The supply has a main cycle about 3.5-year long and a secondary 5.6-year cycle. In the meantime, the demand has a main cycle of 4.67 and a sub-cycle of 2.8 years. From the long-term development, the supply and demand would have the trend of balanced development.

Keywords:Real Estate Market, Cycle, Eigenvectors, Principal Component, Spectral Analysis, Spectral Peak, Supply and Demand

1. 引言

2. 数据处理的理论基础

2.1. 主成分分析法

${\beta }_{1}\triangleq {\alpha }_{1}^{\text{T}}x={\alpha }_{11}{x}_{1}+{\alpha }_{12}{x}_{2}+\cdots +{\alpha }_{1n}{x}_{n}$

s.t. $\mathrm{var}\left({\beta }_{1}\right)$ 最大，且 ${\alpha }_{1}^{\text{T}}\cdot {\alpha }_{1}=1$

${\beta }_{2}\triangleq {\alpha }_{2}^{\text{T}}x={\alpha }_{21}{x}_{1}+{\alpha }_{22}{x}_{2}+\cdots +{\alpha }_{2n}{x}_{n}$

s.t. $\mathrm{var}\left({\beta }_{2}\right)$ 最大，且 ${\alpha }_{2}^{\text{T}}\cdot {\alpha }_{2}=1$${\beta }_{1}$${\beta }_{2}$ 线性无关

2.2. 谱分析方法

${X}_{t}={A}_{0}+\sum _{m=1}^{n}\left({A}_{m}\mathrm{cos}\left(2\text{π}mt/N\right)+{B}_{m}\mathrm{sin}\left(2\text{π}mt/N\right)\right)+{\epsilon }_{t},\text{\hspace{0.17em}}\text{\hspace{0.17em}}t=1,2,\cdots ,N$ , (1)

$\begin{array}{c}{X}_{t}={A}_{0}+\sum _{m=1}^{n}\left\{\left(\frac{2}{N}\sum _{t=1}^{N}{X}_{t}\mathrm{cos}\left(2\text{π}mt/N\right)\right)\mathrm{cos}\left(2\text{π}mt/N\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(\frac{2}{N}\sum _{t=1}^{N}{X}_{t}\mathrm{sin}\left(2\text{π}mt/N\right)\right)\mathrm{sin}\left(2\text{π}mt/N\right)\right\}+{\epsilon }_{t}\\ ={A}_{0}+2\sum _{m=1}^{n}\left\{\left(\frac{1}{N}\sum _{t=1}^{N}{X}_{t}\mathrm{cos}\left(2\text{π}mt/N\right)\right)\mathrm{cos}\left(2\text{π}mt/N\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(\frac{1}{N}\sum _{t=1}^{N}{X}_{t}\mathrm{sin}\left(2\text{π}mt/N\right)\right)\mathrm{sin}\left(2\text{π}mt/N\right)\right\}+{\epsilon }_{t}\end{array}$

${\overline{A}}_{m}=\frac{1}{N}\sum _{t=1}^{N}{X}_{t}\mathrm{cos}\left(2\text{π}mt/N\right),\text{}m=1,2,\cdots ,n$ (2)

${\overline{B}}_{m}=\frac{1}{N}\sum _{t=1}^{N}{X}_{t}\mathrm{sin}\left(2\text{π}mt/N\right),\text{}m=1,2,\cdots ,n$ (3)

$I\left({f}_{m}\right)=N\left({\overline{A}}_{m}^{2}+{\overline{B}}_{m}^{2}\right),\text{}m=1,2,\cdots ,n$ (4)

$周期长度=\mathrm{max}\left\{N\left({\overline{A}}_{1}^{2}+{\overline{B}}_{1}^{2}\right)\to N,N\left({\overline{A}}_{2}^{2}+{\overline{B}}_{2}^{2}\right)\to \frac{N}{2},\cdots ,N\left({\overline{A}}_{n}^{2}+{\overline{B}}_{n}^{2}\right)\to \frac{N}{n}\right\}$ .

3. 实证分析研究

3.1. 供给类与需求类指标选择

3.2. 供需双方数据及其规范化

Table 1. Variables of both suppliers and demanders

Table 2. Data of both supply and demand in real estate industry (1987-2016)

Table 3. Standardized data of both supply and demand in real estate industry (1988-2016)

3.3. 主成分提取

Table 4. From the Supply data: Eigenvalues and the corresponding principal components

Table 5. From the Demand data: Eigenvalues and the corresponding principal components

Figure 1. Time-series graphs of the supplies’ principal components

Figure 2. Time-series graphs of the demands’ principal components

3.4. 频率谱分析

Table 6. Synthetical variable information of supply and demand

Figure 3. Time-series graph of the synthetical viables of supply and demand

Figure 4. First-order difference time-series graph of the synthetical viables of supply and demand

Figure 5. Second-order difference time-series graph of the synthetical viables of supply and demand

$yy=0.8481\ast xx,\text{}且\text{bint}=\left[0.\text{7997},0.\text{8965}\right],$

Figure 6. Relation of the synthetical viables of supply and demand

Figure 7. Residual case order plot on the relation of the synthetical viables of supply and demand

Figure 8. Second-order difference Spectral density Plot of synthetical viables on supply and demand

3.5. 周期性说明

4. 结论

1988~2016年全国房地产市场的供给类与需求类均具有明显的短周期的周期特征。我们给出的周期与张红 [11] 、李玉梅 [13] 等得出的主周期或次周期不完全一致，但相差不大，说明全国性房地产行业的供需周期与地方性房地产行业供需情况或价格波动联系比较紧密，我国1988~2016年房地产行业周期与1998~2010年我国经济主周期2.23也基本吻合。

Study on Spectral Analysis of Real Estate Cyclic Fluctuation[J]. 统计学与应用, 2018, 07(02): 221-233. https://doi.org/10.12677/SA.2018.72026

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17. NOTES

1这里的x已经标准化，即已中心化、均值化。

2尽管主成分分析没有忽略协方差和相关性，但是更注重方差。