This paper discusses the main factors that affect the real estate price, finds out the approximate linear relationship between the price and its main factors, establishes a mathematical model to express the real estate price-multiple linear regression model, and discusses the model in an all-round way, obtains the method to solve each parameter, and finally obtains the real estate price. In the process of modeling, firstly, the main factors are determined by scientific analysis and abstracted mathematically, and then various mathematical methods are comprehensively used to analyze and solve the factors. Firstly, the approximate linear relationship between price and various factors is found out by using probability theory and mathematical statistics, and the model is determined; Secondly, the parameters in the model are solved by least square method; Thirdly, the accuracy of the model is determined by regression analysis and tested, so as to obtain a complete mathematical model; Fourthly, through this model, the main factors affecting the real estate price are deeply analyzed, and some policy suggestions are put forward to reduce the high development cost and adjust the supply structure. Fifth, make a reasonable prediction according to the model and suggestions, and finally analyze the advantages and disadvantages of the model and put forward the improvement direction.

Restatement of the problem

The so-called real estate bubble is that the price of commercial housing far exceeds the actual value. In recent years, housing prices in major cities in China have generally continued to rise and remain high. Rising housing prices have greatly increased the cost of living, which makes it difficult for many low-income people to buy a house. At present, the per capita living space of urban residents in China is only about half that of developed countries, even lower than that of many developing countries. It's not that residents don't have housing demand, but that the existing monetary payment ability can't make them realize their desire to buy a house. Although it is now possible to buy a house by installments with loans, residents are also required to have a fairly good income level. They will not be able to pay off their houses until middle age or even later, and the best time in their lives will be given to them. Therefore, how to effectively curb the rise of housing prices, or even reduce housing prices, is a social issue that has attracted much attention. We analyze this problem, establish a mathematical model, and study how to effectively curb the rise in housing prices.

2. Basic assumptions

There are many factors that affect housing prices, such as housing construction cost, market supply and demand, urban economic development, urban scale and so on. Assume that the relationship between house price and various factors is linear, and:

(1) Housing construction cost is replaced by completed housing cost.

(2) Urban economic development is expressed by per capita GDP.

(3) The scale of the city is expressed by the built-up area.

(4) The relationship between supply and demand in the market is reflected by consumers' ability to pay, which is measured by the average wage of employees on the job.

(5) The real estate price is expressed by the housing equilibrium price.

(6) Ignore the influence of consumers' preferences on housing prices, such as whether there are schools, greening rate, parking spaces, hot water supply, communication, building forms, etc.

(7) Ignore the impact of consumption cost on housing prices, such as transportation costs, property fees and parking fees.

(8) Ignore the impact of some hype on housing prices.

Basic symbols, variables and terms

A: Represents per capita GDP series (yuan)

B: Represents the average salary series of employees on the job (yuan)

C: indicates the cost sequence of completed houses (RMB/㎡)

D: urban and rural per capita savings balance series/yuan

Y: housing equilibrium price index series, which refers to the market price when the consumer's demand for a commodity is equal to the supply of the commodity provided by the producer. The equilibrium price is determined by both demand and supply. It is related to the gold absorption rate and transaction price. [ 1]

: is a random variable;

Uy, UA, UB and UCUD are the average sequences of Y, A, B, C and D sequences respectively.

Δ y, Δ a, Δ b, Δ c and Δ d respectively represent Y-uy, A-UA, B-UB, C-UC and D-UD sequences, that is, centralized sequences.

: variance of sequence

,,,: Model parameters

S(a): it is the sum of squares of residuals.

N: Number of statistical cities (number of samples)

R: covariance of centralized sequence

Fourth, build a model and analyze it.

I. Model derivation process

Table 1 is the national 12 statistical table of housing equilibrium prices and related factors in major cities. According to this table, we can get the correlation coefficient between each factor and the housing equilibrium price, and then judge the influence degree of each factor on the housing price, as shown in Table 2.

Table 1 12 Statistical Table of Housing Equilibrium Price and Related Factors in Major Cities [1]

Balanced price index of urban housing

Per capita GDP/yuan/square meter

/yuan built-up area

Km2 Non-agricultural Population Change Rate Average Wage of Employees on the Job

Per capita living area per yuan

/m urban and rural per capita savings balance/yuan completed housing cost

/yuan/㎡

1 Beijing 3494.9719846 488 0.072 654438+0405413.97+04536

2 Tianjin1636.215976 378 0.041123 8.612417.381.

Shijiazhuang1424.85104251080.1477983 3.168105.87767.

4 Taiyuan 859.2110678177 0.076 737812.2313147.438+07760.

5 Hohhot 872.57 7489 790.16 7346 6.22+0.47866

6 Shenyang1655.6214989 202 0.028 8510.0313317.48 978.

7 Dalian1935.4318429 234 0.07910259 8.4513857.8 978

8 Changchun1222.4910261540.073 8618 6.5 6949.561087.

9 Harbin1502.949142165 0.345 7577 6.96 6957.27 897

10 Shanghai3119.62 30805 5550 0.052166465 438+06 5438+04.9619 778.30089.000000000006

1 1 Nanjing1934.31161940.1081kloc-0/93/kloc.

Hangzhou 2312+01.061961.1.14712/kloc-.

At the same time, the average value of each factor sequence can be obtained, as shown in table 1.

Average value of each factor sequence in the attached table 1

Housing equilibrium price index

Per capita GDP/yuan/square meter

/yuan built-up area

Km2 Non-agricultural Population Change Rate Average Wage of Employees on the Job

Per capita living area per yuan

/m urban and rural per capita savings balance/yuan completed housing cost

/yuan/㎡

The average value is1830.815401.67 0.110300 8.85121/kloc-.

Table 2 Correlation coefficient table of various factors and housing equilibrium price

Per capita GDP, built-up area, change rate of non-agricultural population, average salary of on-the-job workers, per capita living area, and per capita savings balance in urban and rural areas.

Cost of completed house

Correlation coefficient r 0.848 0.824-0.236 0.910 0.766 0.836 0.894

As can be seen from Table 2, the correlation coefficient between the housing equilibrium price and the non-agricultural population, per capita housing area and the change rate of built-up area is small, so here we ignore their influence and only consider the influence of other main factors, including: per capita GDP, average wage of employees on the job, completed housing cost, urban per capita savings balance and so on.

Through the table 1, we make a diagram of the relationship between the main factors and the balanced housing price in turn:

Figure 1

Figure 2

Figure 3

Figure 4

From the charts of balanced house price and per capita GDP, balanced house price and per capita salary, balanced house price and completion cost, and balanced house price and per capita savings, we can see that there is a dependence between balanced house price and per capita GDP, per capita salary, completion cost and per capita savings, so it is easy to think of using multiple linear regression model.

Y= A+ B+ C+ D+……。 +

Represents the dependence of dependent variable Y on independent variables A, B, C, D ..., where,,, ... are parameters.

The model features are as follows:

1, a, b, c, d ... are general variables and random variables;

2.Y is a linear combination of general variables and random variables, and the value of Y sequence depends not only on A, B and C sequences, but also on.

As shown in Table 3, each sequence

Generally, it is assumed to be a white noise sequence, and it is assumed to obey a normal distribution with a mean value of 0 and a variance of 0.

Table 3

Serial number city Y A B D C

1 Beijing 3494.97198461405421447.032037

2 Tianjin1636.215976112312417.381061.

3 Shijiazhuang1424.8510425 7983 8105.87 767

4 Taiyuan 859.2110678 737813147.17 760

5 Hohhot 872.57 7489 7346 6721.47 866

6 Shenyang1655.621498985113317.48 978.

7 Dalian1935.43184291025913857.8 978

8 Changchun1222.49102618618 6949.561087.

9 Harbin1502.949142 7577 957.27 897

10 Shanghai 3119.62 308051664119778.24 2232

1 1 Nanjing1934.31193110569.5910.

12 Hangzhou 2311.061.9961.1218712054./kloc-0

After concentration, you will get

y-Uy = *(A-Ua)+*(B-Ub)+*(C-Uc)+*(D-Ud)+

The above formula is

δY = *δA+*δb+ *δC+*δD+

Now the parameters of the model are estimated by least square method.

The values of δ y, δ a, δ b, δ c and δ d are shown in table 4.

Table 4

Serial number city Δ y Δ a Δ b Δ d Δ c

1 Beijing1664.538+09744366

Second, Tianjin-194.573.59989.99999999895

3 Shijiazhuang-405.923-4976.42-2317.67-4004.37-382.67

4 Taiyuan-971.563-4723.42-2922.671036.93-389.67

5 Hohhot-958.203-7912.42-2954.67-5388.77-283.67

6 Shenyang-175.153-412.42-1789.671207.24-171.67.

7 Dalian104.657 3027.58-41.671747.56-171.67.

8 Changchun-608.283-5140.42-1682.67-5160.68-62.67.

9 Harbin-327.833-6259.42-2723.67-55438+052.97-252.67

10 Shanghai1288.84715403.58 6340.36681082.33

1 1 Nanjing103.5371414.581630.33-1540.74-239.67

12 Hangzhou 480.287 4559.581886.33-56.08 67.33

Let a= (,,,), then the least square estimation of A should minimize the sum of squares of residuals, where

S (a) = = (δ y t-* δ at-* δ b t-* δ c-* δ dt), take S(a) =0 to get:

S (a) = 2 * (δ y t-* δ at-* δ b t-* δ CT-* δ d) * (-δ at) = 0-formula 6544.

Rya represents the covariance of sequences Δ y and Δ a, Raa represents the variance of sequences Δ a, Rba represents the covariance of sequences Δ b and Δ a, and Rca represents the covariance of sequences Δ c and Δ a: Equation 1 can be written as follows:

-Rya+* Raa+* RBA+* RCA+* RDA = 0-Formula 2.

Similarly, s (a) = o deduces:

-ryb+* rab+* rbb+* rcb+* RDB = 0-Formula 3.

S (a) = 0 deduces:

-ryc+* RAC+* RBC+* RCC+* RDC = 0-Formula 4.

S (a) = 0 deduces:

-ryd+* rad+* rbd+* rcd+* rdd = 0-Formula 5.

Write Formula 2, Formula 3, Formula 4 and Formula 5 as matrix multiplication:

* =

The formula for calculating parameters is:

= *-Equation 6

Specific to this problem, we use the statistical data of previous years to solve the parameters in the model.

The calculated value of each covariance is: (using matlab software)

Raa=38730662

Rba=Rab= 18250255

Rca=Rac=2543343

Rda = rad =25327000

Rbb=8 106483

Rcb=Rbc= 1257098

Rdb=Rbd= 1 1269000

RCC = 2 1 1 174. 1

Rdc=Rcd= 1882000

Rdd=22936000

Rya=44757 18

Ryb=2 197259

Ryc=343656.3

Ryd=325 1000

Through matrix operation, the value of is: (using matlab software)

,=0.0583

=-0.0487

= 1. 162 1

=0.0059

Substitute coefficients,, into the original model:

y- 1830.77 = 0.0583 *(A- 1540 1.4)-0.0487 *(B- 10300)+ 1. 162 1 *(C- 1 149)+0.0059 *(D-666

Using the balanced house price, per capita GDP, average wage of employees on the job, completed house cost and per capita savings balance in urban and rural areas in Table 3, the values are reversed, namely:

= Y- 1830.77-〔0.0583 *(A- 1540 1.4)-0.0487 *(B- 10300)+ 1. 162 1 *(C- 1 149)+0.0059

The obtained value of 12 is:

Table 5 Remaining data

City serial number surplus

1 50 1.5639

2 -86.822 1

3 239.83 16

4 -39 1.56 1

5 -279.054

6 -45.85 12

7 1 15. 1803

8 -287.093

9 228.703 1

10 -604.037

1 1 387.9655

12 228. 186 1

The average value is 0.584425

Figure 5

Because the average value of is 0.584, which is very small relative to the y value, it can be approximately regarded as 0, so it is ignored.

Therefore, the model is further simplified as:

y- 1830.77 = 0.0583 *(A- 1540 1.4)-0.0487 *(B- 10300)+ 1. 162 1 *(C- 1 149)+0.0059 *(D-666

that is

y = 0.0583 *(A- 1540 1.4)-0.0487 *(B- 10300)+ 1. 162 1 *(C- 1 149)+0.0059 *(D- 12 1438

That is, the optimization of six models.

Although we use statistical laws to establish a multivariate linear regression model to represent the real estate price, the calculation results are generally consistent with the reality. But there are still many problems in this model.

First of all, there are many factors that affect the real estate price, and we neglected many factors that are considered unimportant when building the model.

In addition to the factors affecting housing prices considered in the above model, there are a series of other factors:

(1) The structure, quality, function and recency of a house are important factors that affect the house price. Due to different building materials, different construction institutions, different construction methods and different construction technical forces, the price difference is formed.

(2) The number, level and orientation of buildings. Houses are divided into high-rise, multi-storey and low-rise buildings. Due to the differences in equipment, construction technology and construction mechanization, houses with various floors and orientations have certain price differences.

(3) Environmental factors. Whether the location of the house is in the urban area or the suburbs, the bustling area with convenient transportation or the back streets and alleys, transportation, cultural education and community service facilities have a great influence on the house price.

(4) National policies. House prices are greatly influenced by policy factors, and in some cases, policy factors often become the decisive factors of house prices. For example, during the planned economy period, China implemented the welfare housing allocation policy that houses were not used as commodities, and the price was far below the value, which seriously affected the reproduction of houses.

(5) There are also some people who are speculative and want to take advantage of rising house prices to make profits, buy more houses and make profits through housing proliferation.

The above factors have a certain impact on housing prices, but due to the rush of time and limited capacity, we can't consider many factors one by one, only considering the factors with greater influence. Therefore, we adopt the method of "grasping the main contradiction and ignoring the secondary contradiction", so the model is universal and representative, representing a basic idea and algorithm. On this basis, considering other factors, this method is still applicable.

Secondly, the sample sequence we use to determine the model parameters is only 12 sets of data, which can be said to be a great precept for applying statistical laws, because statistical laws are originally only applicable to some large samples or even infinite sequences. If they are applied to small samples, the results will be very wrong, and sometimes even wrong. But we still use such a small sample to calculate here, in fact, just to illustrate a calculation method, and when we put forward the model, we did refer to a lot of data to make sure that the factors between them are linear. In order to save time and explain the problems in calculation, we only selected several sets of data.

In addition, some factors in the model have * * * linear problems, which need to be further improved.

In view of the problems existing in the model, we put forward the following suggestions for improvement.

(1), I think if more statistical data (samples) of cities are used for model operation, the accuracy will definitely be higher.

(2) Comprehensively consider all aspects of the city, such as the built-up area, floating population, traffic environment and other factors.

(3) Considering the linear problem, we try to use independent factors or some other more classical models.