The Spatial Statistics Analysis of Housing Market Bubbles

2.1 State-Space Model

The benefit of using state pattern to represent the dynamic system can facilitate the incorporation of unobservable variables into observable model. Combining with the powerful iterative algorithm of Kalman filter, we can easily obtain the estimation results over this process. State-space model includes two equations: the measurement equation and state equation; among which, the first can reflect the relations of observable variables and unobservable variables, and the latter can depict the dynamic process of state variables. Two equations can be illustrated as:

where, y_t is k × 1 observable vector, x_t is exogenous vector or variables determined by observable variables, θ_t is unobservable m × 1 state vector, H is k × m vector, D is r × k vector, F is m × m vector, and ε_t and ζ_t are independent white noise for separate equation.

Based on Kalman filter, the parameters in state-space model can be estimated by predicting error decomposition and calculating maximum likelihood function. Furthermore, the values of state vector can be consistently corrected after the acquirement of new observables. The process is demonstrated as:

It’s assumed that the system matrix of state-space model is known when conducting iterative algorithm on state vector estimation from Kalman filter. However, parameter matrix of F,H,V and W are unknown here; therefore, we must adopt maximum likelihood method in the estimation process:

2.2 Spatial Autocorrelation Method

The spatial pattern analysis of economic variables is essentially to explore the structure patterns of various attributes among bodies, and the precondition is that economic variables are spatially interdependent to each other, i.e., there are consistent when it comes to similarities of attributes and the similarities of different positions among bodies. As one important form of spatial interdependence, spatial autocorrelation is embodied as the correlation of variables and their spatial positions, and is considered as the prerequisite of spatial interpolation analysis for economic variables. Only if the economic variables are spatially correlated can unbiased economic variable estimation to unknown space be conducted based on known samplings space.

Spatial autocorrelation mainly focuses on testing whether there are significant correlations among given spatial attributes and other adjacent spatial attributes, and it has two forms: global spatial correlation and local spatial correlation. Global spatial correlation is to test whether there is agglomeration effect for spatial variables in global region, and is always conducted by calculating Moran index I. Moran index I can be expressed as[23]:

where, n is the number of sampling, w_ij is spatial weight, x_i and x_j are the attributes of sample i and j ; x is the mean of attributes, and S² is variance.

Standardized statistics Z is used to test whether there have autocorrelations for an observable in a given region, and it is signified as:

where, E(I) and V AR(I) are expectation and variance of Moran index I respectively. When the index stays between −1 and 1, we can test the spatial pattern referring to Z statistics: When I > 0 and |Z| > 1.96, it signifies the bodies of similar attributes clustering together, and they have positive correlations; when I < 0 and |Z| > 1.96, it signifies the bodies of dissimilar attributes clustering together, and they have negative correlations; when I is approaching to 0, it signifies the attributes are randomly distributed, and they have no spatial correlations.

Local spatial autocorrelation is generally adopted to test the agglomeration degree of spatial variables in local regions. Local Moran’s (LISA), seeing (11)[24], is often taken to measure local spatial autocorrelation.

where, all parameters have same meanings as in formula (9). And we can find that every local Moran index I_i is proportionate to global Moran index I. Standardized statistics Z(I_i) is:

Similar to (10), E(I_i) and V AR(I_i) in (12) are expectation and variance of Local Moran index I_i respectively.

We can test the spatial pattern of observable body and its adjacent region using Moran index I_i and statistics Z(I_i): When I_i is positive and significant, it indicates that observable body has similar attributes with its adjacent region (high value is surrounded by high values and low value is surrounded by low values); when I_i is negative and significant, it shows the observable body is different from its adjacent region in the attributes (high value is surrounded by low values or low value is surrounded by high values).

3.1 The Specification of State-Space Model

We will use rental rate to estimate the fundamental of housing market. Assuming the risk premium of investing on housing market is constant, we have expected return on housing investment as:

where, P_t is housing price, D_t is the rent of house, E_tP_{t + 1} is the expected price in next stage, r_t is risk-free interest rate, and α is risk premium. We obtain the fundamental value by solving Equation (13), thus solution is expressed as:

Assuming risk-free interest rate has limited unconditional mean, i.e., E(r_t) ≡ r, we define long-term mean discount factor as β ≡ (1 + r + α)^{− 1}. Then, conducting first-order Taylor expansion of based on Ptf(Dt+i,rt+i,α), we obtain Ptf(Dt+i,r,α), among which:

Assuming rent has constant increase rate, then we express the rate as D_t = φ D_{t − 1} + ε_t, φ > 1. Conducting first-order autoregression on risk-free interest rate, we then have r_t = ρ₀ + ρ₁r_{t − 1} + η_t, in which, ε_t and η_t are zero-mean and uncorrelated disturbance terms. We use E_tD_{t + i} = φⁱD_t to signify predicted rental rate after i stages, r = ρ₀/(1 + ρ₁) represent the long-term predicted mean to risk-free interest rate. We then denote Etrt+i−r=ρ1irt−r as predicted risk-free interest rate after i stages. We can simplify (15) and (16) as:

The fundamental value of housing can be expressed as:

We define bubble price of housing as the deviation of actual price to fundamental, indicated as:

where, ξ_t is zero-mean, uncorrelated disturbance. According to Alessandri[25], bubble price of current stage is affected by interest rate and bubble price from prior stage; thus, we can use first-order autoregression model to estimate its dynamic process, and the equation is:

In which, η_{t + 1} is zero-mean and consistent uncorrelated disturbance term.

Making transformation of (19), we can present housing price at time t as:

Due to unobservable part B_t in P_t, ordinary regression model to estimate the parameters is acceptable. Fortunately, it is permitted to contain more than one unobservable state variable in state spate-space model, and the dynamic process can be expressed as state equation. As a result, we can construct the state-space model depicting the relations of housing price and housing bubble price, which can be denoted as:

where, ξ_t and η_t are zero mean and consistent uncorrelated disturbances.

3.2 Measurement Results

In order to get insight into the market bubble differences among China’s different cities, we choose 63 large and medium-sized cities^[1] from China’s 293 cities, including prefectural cities, sub-provincial cities and municipalities directly under the central government. The sampling covers China’s 29 provincial regions. We have excluded the cities from Tibet and Sinkiang because these two provincial areas have no time series statistics on housing price and they have very limited micro samplings in housing market.

We choose the monthly data of housing price, housing rent and interest rate, and the period is from June, 2009 to June, 2014. As for the housing rental market, most of houses there are second-hand. In order to match the research objectives, we therefore solely choose the sales and rental of second-hand houses in selected cities, including collecting the second-hand house sales volume and rental rate. In previous study, risk-free interest rate can be collected from multiple sources: Benchmark one-year deposit rate, benchmark 5-year deposit rate or bond market benchmark interest rate. Considering we use monthly data, and deposit interest rate is comparatively with low fluctuation, we choose monthly average yield to maturity of treasure bond to represent risk-free interest rate.

All data concerning with second-hand house sales and rental rate are from CityRe database (http://www.cityre.cn/en/); yield to maturity of treasure bond is from Wind database (http://www.wind.com.cn/). The descriptive statistics is shown in Table 1.^[2]

The estimation of housing bubble state-space model is illustrated in Table 2, where all parameters are significant under the level of 5%. indicates interest rate is negatively correlated to housing price when rental price keeps stable; the estimation of risk premium coefficient is close to 0, signifying risk premium of housing investment is approximate to the compensation value of risk-free interest rate.

4.1 Global Spatial Autocorrelation Analysis

It is essential to determine the spatial weight before calculating Moran index I. We cannot adopt the adjacent standard to determine weights because cities cannot be seen as points. For this reason, we establish spatial weight matrix based on the inverse of spatial distance between any two cities. Table 4 is the Moran index I and related statistics for housing market from 2009 to 2014.

From Table 4 we know that global Moran I for every year is positive, Z is bigger than 1.96, and P value is less than 0.01. The result indicates that from 2009 to 2011, housing market bubble was positively correlated under the significance level of 1%, and it can be seen as clustering statistically; meanwhile, 2010 saw most obvious global spatial autocorrelation. Linking to the highest bubble in 2011 shown in descriptive statistics, we can conclude that China’s housing market bubble began to spill over spatially in 2010.

In order to better observe the trend of global spatial distribution, we further conduct trend-surface analysis of bubble from 2009 to 2014. The trends can be found in Figure 1.

Figure 1

The trend surface of housing market

From Figure 1, the sample data present an obvious trend of inverted U shape from the direction of Y axis, signifying housing market bubble showed a comparatively strong trend of decreasing from center to periphery in the direction of South-North during the period from 2009 to 2014. In the direction of X axis, different year had different trend. From 2009 to 2010, the trend was high in East and low in West. From 2011 to 2014, though whole trend was basically high in East and low in West, it had slightly trend of inverted U shape, indicating that bubble in Eastern region was higher than that in Central and Western region, but the East Central region had the trend of centrality. In addition, from the curves in Figure 1, the decreasing trend from center to periphery in the direction of North-South was not the simple form of linearity, but the complicated form of second-order curve.

4.2 The Analysis of Local Spatial Autocorrelation

Using Moran index, this sub-section will test whether there will be agglomerations in which similar or different observables will stay together in local regions. The result indicates that there were 30, 33, 32, 33, 31 and 31 cities^[3] that had passed 5% significance test from 2009 to 2014, and 23 cities^[4] passed local Moran test during this period. The cities that present the feature of local spatial agglomeration in latest 6 years were located in Shandong Peninsula, Yangtze River Delta and Pearl River Delta. Moreover, the selected cities from 3 Northeast provinces in China (Heilongjiang, Jilin, Liaoning) had passed significance test in the years of 2009, 2013 and 2014, and most cities selected from Fujian, Guangxi and Hainan had passed the test in the years of 2013 and 2014.

The detailed local spatial correlation pattern can be acquired using Moran scatter diagram. Moran scatter diagram graphs pairs of numerical number of wz and z, in which, z is the deviation of observable and mean, and wz is the product of spatial weight matrix and deviation. We map out the scatter diagrams reflecting housing market bubble from 2009 to 2014, seeing Figure 2.

Figure 2

Local Moran scatter diagrams of housing market bubble

From Figure 2, most cities lied in first quadrant and third quadrant from 2009 to 2014. Precisely, the year of 2009 saw 28 cities in first quadrant and 17 cities in third quadrant; 2011 saw 30 cities and 17 cities respectively; in 2012, related numbers had changed to 26 and 19; however, in 2014, there were 26 cities in first quadrant and 20 cities in third quadrant. Totally, more than 70% of cities were distributed among HH and LL areas.

Putting all cities and their quadrants in Table 2, we can clearly dig out the dynamic evolution of China’s housing market bubble. In Table 2, there were 24 cities (Anqing, Baoding, Beijing, Fuzhou, Hangzhou, Hefei, Huzhou, Jinan, Jiaxing, Nanjing, Nantong, Ningbo, Qingdao, Rizhao, Shanghai, Shijiazhuang, Suzhou, Tianjin, Wenzhou, Xuzhou, Yantai, Yancheng, Yangzhou, Zibo) lying in the first quadrant from 2009 to 2014, and most of them were located around Bohai Bay, Shandong Peninsula and Yangtze River Delta. The first quadrant is the high-high correlation pattern, indicating these 24 cities and their adjacent regions were with higher bubble level than China’s average. There were 12 cities (Changde, Chengdu, Dongguan, Foshan, Guiyang, Harbin, Haikou, Jilin, Jiangmen, Changchun, Changsha, Chongqing) in third quadrant (which is low-low correlation pattern) in the whole period, and majority of them were located in Northeast 3 provinces, Sichuan, Hunan and Guangdong, reflecting bubble there was less serious than China’s average. 5 cities (Dalian, Hohhot, Linyi, Quanzhou, Taiyuan) stayed at the second quadrant (which is characterized as low-high spatial correlation pattern), proving that these 5 cities had significantly lower bubble level than their neighboring areas have. 4 cities (Pingdingshan, Xiamen, Shenzhen, Zhuhai) lied in the fourth quadrant (this quadrant is featured as high-low correlation pattern), which demonstrates bubble level in these 4 cities was higher than that of neighbors. In conclusion, high bubble level appeared in the regions of Bohai Bay, Yangtze River Delta in last 6 years, and low bubble level appeared basically in the regions of northeast 3 provinces and western China. This finding confirms to spatial patters in global trend that the bubble level from direction of South-North had an inverted U shape; but the bubble level experienced a trend from high in East to low in West.

Some cities such as Zhengzhou, Wuhan, Nanchang, Shantou had changed greatly in their spatial patterns from 2009 to 2014. For example, Zhengzhou in Henan province changed its pattern from high-high to low-high, and Pingdingshan in Henan province experienced the pattern change from low-high to low-low; however, Hefei in Anhui province witnessed its pattern change from high-high to low-high, then to high-high at last, and Anqing in Anhui province stayed at its high-high pattern. From this result, we can prove that Henan province was more adjacent to Central and Western China from the perspective of agglomeration, but Anhui was more affected by Jiangsu and Zhejiang provinces. In 2011, Guangzhou changed its pattern from low-low to high-low, indicating that it began to be affected by Shenzhen and Zhuhai and it had shown spatial variance since then.

We can also see the changes happened in Beihai, Lanzhou, Kunming and Nanning. Both the high-low pattern of Beihai, Lanzhou and Kunming and the low-high pattern of Nanning tended to stay at low-low, verifying that these 4 cities were impacted by radiation from low value agglomeration. Under this trend, local agglomeration became more significant. The patterns in Wuhan, Nanchang and Hohhot had experienced the evolution from instable to stable, and ended at low-low, high-low and low-high respectively. Therefore, we can prove that Wuhan and Nanchang were clustered at western China. Unlike the pattern of Wuhan, Nanchang had high bubble level though it was located in low bubble region, signifying the appearance of local variance. Hohhot was affected by the region of Beijing, Tianjin and Hebei and became the city of low bubble level in the high bubble level region.

From the results of local and global spatial statistics, the bubble level in China can be divided into 3 plates: The plate of Northeast 3 provinces in China, the plate of Central and East China and the plate of Central and West China. As for plate of Northeast 3 provinces in China, it was the low bubble level region and it included provinces of Jilin, Heilongjiang and Liaoning (Dalian of Liaoning province is excluded). As for the plate of Central and East China, it comprised Beijing, Tianjin, Hebei, Jiangsu, Zhejiang, Shanghai, Shandong, Anhui, Shanxi, Shaanxi and Inner Mongolia, and it can be categorized as high bubble region. In this plate, some cities such as Dalian, Hohhot, Linyi, Taiyuan, Xi’an had local spatial variances. The plate of Central and West China, which included provinces of Yunnan, Guizhou, Sichuan, Guangdong, Guangxi, Hunan, Hubei, Gansu, Fujian, Jiangxi and Hainan, were low bubble regions. In this plate, cities such as Nanchang, Xiamen, Shenzhen, Zhuhai, Guangzhou, Sanya and Yuxi were with high bubble levels thought located in low agglomeration region.