_{1}

^{*}

Economic system mathematical model often contains multiple variance change points about structure model. In the same mean, we combine the Bayesian method with the maximum likelihood method on the detection of the variance multiple change points. With Bayesian method, we can eliminate extra parameters first, and then use maximum likelihood method to find the change position. So we both can eliminate extra parameters and can avoid the change point on the prior distribution unknown problem. In addition, the benefit of the maximum likelihood method is just needed to find out likelihood density function of the maximal solution in the solution space, thus the variance of multiple change point detection problem is resolved. By making substantial analysis with an example from commodity house prices in Wuhan, three changes of variance are found in all. They correspond to the major structure changes of contemporary property market in Wuhan city. The method is practical and effective as the final results shown. It also has a certain practical significance.

With the rapid economic development of modern society, we need to deal with more and more economic and financial problems. Economic cycle of change point analysis has always been a problem concerned by economists and statisticians [

Generally, there are many ways to estimate change point, such as Schwarz information criterion method, Binary segmentation method, Bayesian method, maximum likelihood method, the likelihood ratio test, (weighted) Least square method, nonparametric method, cumulative sum method, etc. Schwarz information criterion is set up in 1978 by Schwarz [

A likelihood-ratio method to detect changes in gamma distribution parameter for general values of the scale parameter was adapted in the reference [_{0} is derived and the consistency of the test is also established. The reference [

The reference [

However, the parameters will change with time; in the reference [

Change point definition: according to the statistical definition, to a certain random variable sequences, if there is a point in time, the sequence before the point to a kind of probability distribution and after this point in time sequence to another kind of probability distribution (or the same kind of probability distribution but parameters of different), so there is a change point in the sequence. The problem of a change point contains two aspects as follows: 1) to confirm whether there is any change, 2) to estimate the number and position of unknown change-point.

The typical objective of a change point analysis is to identify how many change points a series has and where they occur.

The reference [

Pignatiello and Samuel (2001) used the EWMA and cusum charts and the maximum likelihood estimator (MLE) to estimate the change point of a process [

In view of the above research, for normal random variable variance change point problem, we combine the Bayesian method with maximum likelihood method. First using the Bayesian method eliminates extra parameters, and then using the maximum likelihood method find the place of change point. Thus, we can use the information to estimate the change point more accurately.

Consider

where

The main question now is how to estimate the location

We detect variance change point,

the variance is sample distribution density function, so the detection and analysis of the variance change point ultimately is considered to be the sample density function estimation, and then we use the maximum likelihood method to estimate the location of the change point.

where

In terms of the reference [

where

We want to solve the key parameter k and m, that is the change point location and the numbers of change points, and

To apply Bayesian method expunction extra parameters, we must construct the prior distribution of

With the above information,

Thus we can derive the distribution of

For

By type (1):

The likelihood function is as follows:

(, ,)

Therefore, we want to use the Bayesian method, also need to know the prior distribution of m. The prior information of m is unknown, which plays a crucial role in solving problem. We use the maximum likelihood method to calculate k and m according to the above joint density function that is the likelihood function. Thus we can avoid the problems of the prior distribution information of m unknown. We first give m with an initial value, and then we determine

Housing price problem is the core of the realty business market, housing price is related to the people to live and work in peace and contentment in China, relates to the healthy development of the real estate market [

The commodity house prices in Wuhan from 2000 to 2015 are as shown in

Year | Average sale price |
---|---|

2000 | 1983.52 |

2001 | 2065.32 |

2002 | 2327.21 |

2003 | 2353.4 |

2004 | 2667.64 |

2005 | 3345.75 |

2006 | 3622.20 |

2007 | 4685.32 |

2008 | 4883.01 |

2009 | 5265.91 |

2010 | 6184.49 |

2011 | 6414.72 |

2012 | 6349.74 |

2013 | 7213.70 |

2014 | 8179.25 |

2015 | 8861.00 |

rising trend of real estate industry has lasted for 15 years. With a nationwide real estate boom climate, statistics show that: 16 years in Wuhan commodity house average price rose from 1983.52 Yuan/square meters to 8861.00 Yuan/square meters, the average annual growth of 13.14%.

As is shown in

From the annual sales average increment data shows upward trend since 2001, the price fell slightly after 2012, down 1.01%, but the price increases for three consecutive years since 2013.

In the process of trying to find a place to change point, we first give m with an initial value, for m = 1, and then maximizing

We use the above analysis method to detect the 16 sequence data by the data of

Year | Average price increment |
---|---|

2001 | 81.8 |

2002 | 261.89 |

2003 | 26.19 |

2004 | 314.24 |

2005 | 678.11 |

2006 | 276.45 |

2007 | 1063.12 |

2008 | 197.69 |

2009 | 382.9 |

2010 | 918.58 |

2011 | 230.23 |

2012 | −64.98 |

2013 | 864 |

2014 | 965.55 |

2015 | 681.75 |

Change-point number m | Change-point position k | Logarithm likelihood density | Logarithm likelihood density increment |
---|---|---|---|

0 | - | −106.162 | - |

1 | (3) | −74.309 | 31.853 |

2 | (3, 6) | −55.012 | 19.297 |

3 | (3, 6, 9) | −53.906 | 1.106 |

From the analysis result shows that this set of data has two change points. Because the logarithmic likelihood density increment is 31.853 when

Raising interest rates in 2004 as a prelude to comprehensive adjust and control the real estate regulation policies to follow up. The province government in Hubei adopted a series of policies and measures such as subsidies and reducing the deed tax the purchase of second-hand housing, which have obvious positive role and effect. Accumulation fund, housing loan portfolio and the secondary housing market fully open, which the housing reform policies carry forward all-around. Residents purchase ability and the demand increase promote the rapid development of the real estate industry. As a result, prices rose in 2004 can locate as “rising” at a high speed.

Three years later in 2007, the macroeconomic regulation and control prices still rising, disappointed with the reality of policy generally bullish but lead to consumer expectations. In 2007, the density increases mainly by means of financial regulation. The central bank raises interest rates four times in 8 months time. Such a density increases in interest rates. Touched by month high CPI growth, mainly used to suppress the surge of excess liquidity, prevent the possible inflation later. Difference from the developers for housing prices to judge “to do” in 2004, 2007 is the prices of basic steady, firm, real estate prices rising more concentrated. As a result, prices rose in 2007 can locate for “high rise”.

Have to say is the hottest in 2009 that is an inevitable phenomenon. Because of the sudden regulation in 2008, the main demand is depressed for a year, concentrated burst out in a short time, the market suddenly rushed up, the developers are very confident, the price is carried higher. Panic buying appeared at the end of 2009. Prices uptrend continued in 2010. So the housing price in 2010 has gone up a lot, we also detected the corresponding results from the model.

Test results and the actual housing conditions in line with the prices in Wuhan. It shows that the variance change point analysis technology in our country are explained from the perspective of quantitative control measures on the housing prices have a certain effect. The real estate market regulation has achieved initial results, but the complexity of the market operation and unstable factors still exist, the real estate market regulation is still in a critical period. In 2015 to continue the current regulation policies do not relax, expected prices to keep smooth, not ups and downs.

Change point in the field of economics and finance, its application prospect is very broad, which can solve many practical problems. Find out the change point, and deduce which factors in the model or parameter changed, use them for policy evaluation, analysis and forecasting.

On the housing forecast model, the real estate market is a complex system, many factors affect the housing prices, and the influence degree is different [

In this study, we proposed the Bayesian approach combination with the maximum likelihood method for estimation of the variance change point. With Bayesian method, we can eliminate extra parameters, and then use maximum likelihood method to find the change position. So we both can eliminate extra parameters and can avoid the change point on the prior distribution unknown problem. In addition, the benefit of the maximum likelihood method is just needed to find out likelihood density function of the maximal solution in the solution space, thus the variance of multiple change point detection problem is resolved. Finally, by making empirical analysis with an example from commodity house prices in Wuhan from 2000 to 2015, we get the test results and the actual housing conditions in line with the prices in Wuhan. It suggests that the combination method to detect the variance change point is effective. There is also a certain practical significance and reference value.

I thank my tutor for helpful discussion and careful guidance, and this work is supported by the Program for Excellent Youths in Hubei Provincial Department of Education under grant Q20121902.

Shen, H.H. (2016) The Detection and Empirical Study of Variance Change Points on Housing Prices. Journal of Mathematical Finance, 6, 699- 710. http://dx.doi.org/10.4236/jmf.2016.65049