Housing Bubble in China's Real Estate Market
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Since the implementation of the reforms and opens to the outside world for 3 decades, China has been a country with an important influence in the world, whose unbelievably fast development is concerned by more and more other countries. Nevertheless, with the too rapid development of economy in China, some problems appear in the macro economy as a result of the system, management and other aspects. One of them is the problem of real estate market.
In 1998, Chinese government went in for innovations of real estate market. Since then, the Chinese real estate market has developed rapidly. And now it becomes a significant pillar industry and plays an increasingly important role in the development of economy. However, it appears abnormality simultaneously as well. Because of limitation and rareness of land resource, great public demand caused by quite large amount of population and relative stability compared with other financial assets, house prices continue to show an upward tendency. Especially during recent years, house prices in China increase extremely so fast that it seems not consistent with the development of Chinese economy——house prices in some cities are almost as high as those cities of developed countries, while people's average income is much lower. As we see that the reason of the financial crisis from 2007 until now is related to the collapse of real estate bubbles in US, this unusual phenomenon in China is more controversial because it has the possibility to cause another economic crisis. Thus, the issue about the housing bubbles in China becomes quite popular around the world. Therefore, I want to explore into this issue to test whether the housing bubbles really exist in Chinese real estate market by using some appropriate models and tests in the dissertation.
The plan of my dissertation is as follows. First, I review some prior related fields' studies that conducted before and conclude that index tests, present value model, unit root and cointegration tests should be chosen to use to test whether there are housing bubbles in the real estate market of China. Whereafter, I will state the reasonability and deficiency of those chosen models and methods and describe the specific models and approaches for the housing bubble tests. After that, I will collect the relative available statistic data about real estate market of China, monthly house prices index, monthly house rents index and mortgage rates of China, since 2003 till now. Then make some necessary process to those data sets and compare them to reveal the abnormal growth of the house prices. And the following step is to calculate and do the tests by using Eviews and report the results, which is that it does (not) exist housing bubbles in the real estate market in China. At last, I will analyze the results, discuss the main reasons behind and offer some suggestions to improve the danger situation of Chinese real estate market.
Literature Review 900
From the definition of Wikipedia, a real estate bubble or property bubble (or housing bubble for residential markets) is a type of economic bubble that occurs periodically in local or global real estate markets. It is characterized by rapid increases in valuations of real property such as housing until they reach unsustainable levels relative to incomes and other economic elements, followed by a reduction in price levels. 
Simply, housing bubbles are characterized by an exaggerated discrepancy between house prices and other fundamentals. As the definitions of fundamentals, financial series should be stationary. So by testing the explosion of house prices, a large amount of studies analyze the potential occurrence and existence of the real estate bubbles.
To test the real estate bubbles, one popular approach is the index tests. Gallin (2004) studied the relationship between house prices and rents. Through the model of house prices and rents, he assumes that in a frictionless market, prices should be high relative to rents when, among other things, interest rates are low and expected capital gains are high.  Thus in his paper, he addresses that the indicator of rent-price ratio should be stationary and discusses its predictive power to the changes on prices and rents. As a result, he concludes that even if the rent-price ratio could be considered as a measure of valuation to assess whether house prices are relatively too high and how house prices and rents would move in real estate market, it is not a precise indicator because price movements are hard to predict.
In the other hand, Himmelberg, Mayer, and Sinai (2005) thought that accelerating house price growth and outsized price increases in certain markets are not intrinsically signs of a bubble. They also apply the formula of house prices and rents, and introduce another similar index, price-to-income. Moreover, they highlight the influence of user costs on the price-to-rent ratio and price-to-income ratio. They believe that comparing rental costs or income levels with house prices is not so convincing without calculating the cost of owning a house. That is, conventional indices such as price-rent and price-income ratios are probably misleading in a test for bubble occurrence.
Therefore, though index test is commonly used, it is not so powerful or definite to test whether there is a bubble in housing market.
Another popular and more accepted tests for housing bubbles are unit root test and cointegration test, which are widely used on panel data. Malpezzi (1999) performs unit root tests based on the error correction model, using panel data of 133 major metropolitan areas in the US in order to research the ratio of housing prices to per capita income. Having done the tests, it is demonstrated that these series do have unit roots by failing to reject their presence and that the null hypothesis of cointegration is correct. Meanwhile, he also researches into the determinants of house prices and finds that government policies are important to the level of housing prices, which may be useful experiences to the market for housing in China.
Im, Pesaran, and Shin (2003) provide a unit root test in dynamic heterogeneous panels. They discuss both the situations of no serial correlations panels and serially correlated series panels and Monte Carlo simulations and augmented Dickey-Fuller regressions are used for the tests. At last, they also think that the power of the unit root tests might be substantially augmented when applied to a single time series. That is exactly what I do in this dissertation and it should be convincing.
As for the development of testing methods for cointegration, Pedroni (1999) describes a test of no cointegration in dynamic panels with multiple regressors and provide approximate critical values for the test, which could be thought of as extensions of the two-step residual-based Engle and Granger (1987) method to test the cointegration of heterogeneous panels. Pedroni (2004), as a further work, studies the properties of Engle and Granger (1987)'s residual-based tests for the cointegration hypothesis on nonstationary panel data as well as the properties of small sample statistics by Monte Carlo experiments. As a simplification of panels, the cointegration tests could be also performed on a simple time-series.
Campbell and Shiller (1987) develop a test on the present value model between financial assets prices and their cash flows which are stationary in first difference. Under the present value model, asset price and its cash flows should be of the same order of integration and cointegrated if they are non-stationary but stationary in the first differences. In this paper, the above tests only examined the stocks and bonds markets, but houses could be considered as assets as well and rents as their cash flows. Therefore, using likelihood logic, analogous present value model and cointegration tests will be used in the housing market in this dissertation.
Obviously, contrary views exist definitely. Gallin (2006) investigates on the long-run relationship between house prices and per capita income in the local housing market. He did more powerful panel data tests than standard tests for cointegration on 95 metro areas, combining the unit root tests and cointegration tests from Pedroni (1999) and Maddala and Wu (1999), both of which use Engle and Granger's (1987) augmented Dickey–Fuller (ADF) tests, but he could not reject the null hypothesis of no cointegration. In his opinion, tests for cointegration have low power and he finds little evidence for cointegration of house prices and fundamentals.
As we described in the survey of prior research on a variety of markets above, especially in the real estate market, a number of methods have been applied into those related literature, in which indices studies and cointegration hypothesis tests are the main and essential approaches that being used. In this dissertation, I aim to develop a set of homogenous tests with these reference materials mentioned above as a basis for application to examine the existence of bubbles in housing market in China with reasonable data collection and also analyze both the strengths and limitations for each technique.
Index test is a kind of widely used method to test the housing bubbles in housing market in the world. It is common because it is simple, convenient and intuitional. We could assess the performance of the real estate markets by observing the variation of some indices directly. If such indices are too high or too low, then we could intuitively say that the housing markets may have some problems or may be not effective. However, this approach is not so strict. It is really difficult to determine the exactly critical value for a certain index since it would be changed for different markets. For instance, population, mortgage rates and nominal rates will influent the environment of the housing market and in equilibrium the index of house price-to-income ratios are higher in some markets with more stringent regulatory environments.  So far, some global academic institutions have derived a series of approximate but accepted critical values of indices for different regions through large amount of studies. Seriously, the critical values of indices are subject to market fluctuation and characteristics.
As we known, a large number of ratios are provided as targets to identify the occurrence of the housing bubbles by comparing the values of the ratios to the accepted ones. Here, I will introduce some most common indices ratios and due to the weak power of this method, I only choose some indices for the initial identification. One of the most frequently used indices is the house rent-to-price ratio. The relationship between house prices and housing rents has been discussed in lots of academic articles and a model of house prices, rents and user costs is provided by Hendershott and Slemrod(1983) and Poterba(1984):
: the housing rents at time t
: the housing prices at time t
: the real interest rate at time t
: the property tax rate at time t
: the marginal income tax rate at time t
: the maintenance and depreciation rate at time t
: the risk premium at time t
: expected capital gains at time t+1
And the user cost:
If the housing market is in equilibrium, the annual user cost of owning a house should be equal to the annual rental cost.
Substitute (2) into (1) and we get:
which presents that the house rents-to-price ratio is equal to the user cost. Therefore, the research on housing rent-to-price ratio is to research the user cost and without housing bubbles, the changes in user costs lead to predictable changes in the housing rent-to-price ratio. 
Besides house rent-to-price ratio, many other indices could be applied to judge the performance of the housing market as well, such as house prices-income ratio, housing vacancy rate and the ratio of growth rate of real estate investment to growth rate of investment in fixed assets. In addition, we also have the data of the housing market wide range of the indices in equilibrium. For example, as Andrew Hamer suggested in 1992 by studying urban housing in China, without bubbles in housing market, house prices-income ratio should be between 2 and 6.  As for the housing vacancy rate and other construction indices, personally I think they are not suitable to China on account of the progress of large–scale urban housing construction as a developing country.
Unit root and cointegration test
The approach of unit root and cointegration tests is an important way to measure stationarity of serials to be tested and long-run stable relationship between two series through linear equations. A large amount of studies argue that house prices and fundamentals are cointegrated and examine the existence of housing bubbles by identifying the cointegration of house prices and fundamentals. Also the technique for testing shows high reliability through finished empirical tests results. Consequently, I assume that house prices and fundamentals are cointegrated if the real estate market is efficient and apply the unit root and cointegration tests to detect the housing market in China.
It is known that, for econometrics theory, if the real estate market is in equilibrium of supply and demand, or effective in other words, house prices should be a stationary time series and randomly fluctuate around a baseline.
Denote by Pt the actual house price at time t. If the real estate market is efficient, the time series of the house prices should be a random and stationary process around a base value:
Pt = P*t + εt (4)
where P*t is the base value or the theoretical housing price at time t and εt is the stationary error series. From bubbles angle to analyze, εt represents the housing bubbles.
Speaking simply, in order to detect the relationship between Pt and P*t we need to compute P*t first and perform unit root tests on Pt and P*t respectively to test whether they are stationary. If Pt is stationary, we could say that there is no bubble in housing market; if neither Pt or P*t is stationary, we cannot derive inefficiency for the housing market. Additional cointegration tests need to be done instead to test the null hypothesis of cointegration between them in accordance with no bubbles.
Calculation for P*t
Real estate is a kind of fixed asset with quite long service life--70 years in China. The same as other assets, owning a house is of benefit for all the service life. Here, I consider a house as an infinite-period asset and rents as its cash flows. Then the price of such an asset is the present value of all its cash flows. From the present-value formula we could compute how much the house prices should be:
P*t: house price under present-value model at time t
Rt: house rent at time t
rt: mortgage rate at time t
For simplification, I ignore the effect of tax, depreciation and other risk factors on the property, and so is an approximation of house prices depending on mortgage rate.
Having computed the theoretical house price, we could carry out the tests to detect the existence of the housing bubbles by analyzing the relationship between Pt and P*t. That is, we need to perform unit root tests and cointegration tests on Pt and P*t.
Unit root tests
Testing stationarity of series by unit root tests is the first step to examine housing bubbles. Make regressions for variables at time t on that at time t-1 and we have:
Pt = βPt-1 + μt (6)
and P*t = ϒP*t-1 + νt (7)
The null hypothesises of the tests above are β=1 and ϒ=1 respectively, or equivalently Pt as well as P*t has a unit root, which illustrate the two series are non-stationary. In contrary the alternative hypothesises are β<1 and ϒ <1 which lead to stationary time series.
A widely used method for unit root test is the Augmented Dickey-Fuller (ADF) test, whose general form is:
Testing for the existence of a unit root is calculated as a t-statistic on with H0: =0, using the same critical values as those for the DF tests. If <0 for both Pt and P*t, then the two series are stationary and we conclude that no bubble exists. If we fail to reject the null of a unit root for Pt and P*t, derive whether they are of the same order of integration and a further test, cointegration test, need to be performed.
When massive speculations appear in the property market, high yield on houses and rapid growth on house prices make consumers and speculators generate high expectation on housing market, which causes more investment on purchasing houses and higher house prices. Wide fluctuation of house prices might be a sign of the existence of housing bubbles. To prove this point, we need to observe that whether the movement of the actual housing prices deviates from the justifiable prices. In other words, we have to detect whether the long-run linear and steady relationship is still existent between Pt and P*t by cointegration tests.
I perform the cointegration test between Pt and P*t by two-step procedure of Engle and Granger(1987) described as follows:
First, estimate the long-run equilibrium equation:
Pt = β0 +β1 P*t + εt (9)
and take the OLS residuals above
εt =Pt - β0 -β1 P*t (10)
Second, test whether εt is stationary by ADF tests. On condition that εt presents stationarity, Pt and P*t are cointegrated.
Generally speaking, if both Pt and P*t are non-stationary and of the same order of integration, through cointegration tests we come to the conclusion that εt is stationary and Pt and P*t are cointegrated, we say the market house prices still move on the basis of theoretical prices so that the property market is efficient and there is no bubbles, and vice versa.
Data  500
Since our goal in this dissertation is to explore into the relationship between actual house prices (Pt) and the theoretical prices (P*t) derived by rents (Rt) and mortgage rates (rt), we could neglect the influence of inflation rates, consumer price index (CPI) and other fundamental economic variables associated with prices for these factors lead to the same effect on Pt and Rt as well as P*t, which could be eliminated without changing the final results. Therefore, as those testing approaches stated above, I need to collect the data sets of monthly house prices index, monthly house rents index and mortgage rates of China.
In detail, I obtain the mortgage rates from the statistics data of People's Bank of China,  which shows the mortgage rates with different terms of the loans and its adjustment dates. On account of the long service life of a house, we choose those mortgage rates with term of longer than 5 years for calculations. And the data sets of housing price index and rental index, or the growth rates (%) of prices and rents, are taken from National Bureau of Statistics of China (NBSA).  Monthly housing sales price index is only available from July 2005 to June 2010 while quarterly index are from 2003. To ensure the efficiency of the tests, I have to choose samples as large as possible so I take both of the quarterly and monthly house price index and then change quarterly data into monthly data by assuming the growth rates for each month are the same in each quarter, which is:
monthly index = (quarterly index/100) 1/3 *100 (11)
Having done these, I have got all the monthly housing sales price index from January 2003 to June 2010. However, what data we use in tests are prices rather than their growth rates, we still need to compute the monthly prices with growth rates by taking the price on December 2002 as 100. With regard to the rental index, only quarterly data are available for rents are relatively stable comparing to prices. We process the data by same method: transform quarterly rental index into monthly ones according to the formula (11) and derive rents by rental index with the assumption that the rent on December 2002 is 1.
Lastly, with housing prices, housing rents and mortgage rates, we could derive the value of P*t. With regard to the present value formula (5), Rt and rt are the values at time t and would remain the same, thus we have:
As n goes to infinity,
which we applied to compute .
Here, it is worthwhile to note that I set 1 for rent and 100 for price in December 2002 arbitrarily. It does not matter what exact values they are because as I emphasized repeatedly I only focus on the linear relationship between Pt and P*t and the proportional movement of each time series. The absolute values of Pt, P*t and εt will neither change their property of stationarity nor cointegration. Consequently, tests under these random values have the same convincingness as the actual exact values.
Results for tests 980
Although I have introduced index tests before, I will only take the unit root and cointegration tests on account of the low power of index method and the property of the data sets. As I described above, the data sets are growth rates rather than actual values in the market so it is impossible to derive correct indices of ratios. Index tests make no sense with my data.
However, with the processed data, we have obtained available and necessary data sets of time series Pt and P*t from 01/2003 to 06/2010, implying we are well prepared for the unit root and cointegration tests for existence of housing bubbles. Next, I will present the empirical tests on Pt and P*t by using software Eviews.
Unit Root Tests
Figure 1 and Figure 2 plot the variation of both the time series Pt and P*t. As we seen in Figure 1, the series of house sales prices (Pt) has an obvious increasing trend through all the periods and the market price now is almost three times as much as that of 7 years ago. It does grow so fast. Since its graph looks like a random walk process with a drift, we use regression with constant and time trend as expressed in formula (8) for the ADF unit root test:
Differently, the series of P*t reflects fluctuation during the whole time duration. From the beginning of 2003 to the end of 2008, the justifiable prices reveal a downward trend with small increases, however, a sharp growth happens suddenly in the year of 2009 till now. Generally Figure 2 looks like a simple random walk process so that regression with constant will be used on P*t to examine the unit root:
The results details of ADF unit root test are shown in Table 1 and Table 2. The P value in Table 3 is 0.5297, implying we fail to reject the null hypothesis that Pt has a unit root. Equivalently, house prices is not a stationary series. In the other hand, in Table 4, the probability that the null hypothesis of unit root is right is 73.44%, even higher than Pt, which proves the correctness of non-stationarity of P*t.
Before carrying out the cointegration tests, we still have to check the order of integration of Pt and P*t since cointegration tests are only operative when series have the same order of integration. First, we will examine whether the first order difference of Pt and P*t are stationary or not. Here I omit the constant term because I do not expect the change of prices to have a nonzero mean or time trend. Table 3 reflects that the t-statistic of ADF of P*t is larger than the critical values of any level of significance, so we could reject the null and P*t is I(1). As for Pt, strictly speaking that the first difference of Pt is not stationary either for its P value is 0.14 as shown in Table 4. And Table 5 appears that the second difference of Pt is stationary. In order to implement the next cointegration tests, Pt and P*t should to be with the same order of integration then we have to choose the higher level of significance of 15% at which the first difference of Pt is stationary.
So far, I have got the results that neither Pt nor P*t is stationary but they have the same order of integration. As we tested before, at the significance level of 15%, the first difference of Pt and P*t are stationary so that cointegration tests are appropriate to be done at following parts under Engle and Granger(1987).
First, according to formula (9) we need to estimate the long-run equilibrium equation by regressing Pt on P*t. This OLS estimation reveals a long-run relationship between Pt and P*t and therefore will have serial correlation. Then the t statistics, as well as the significance of coefficients from the regression in Table 6 are not interpretable. Whereas this would not influent the outcomes of the tests since my purpose for estimating this equation is to collect the series of residuals εt other than anything else from Table 6.
Second, also the last step of the cointegration test, test whether the series of εt is stationary by ADF unit root test. On condition that εt presents stationarity, Pt and P*t are cointegrated. In Figure 3, the series of residuals presents an increasing trend but we expect that it does not have either time trend or a nonzero constant mean. So here we perform the ADF unit root test without time trend or constant term. Moreover we must notice that the critical values of the ADF test supplied by EViews are not correct for the ADF cointegration test. McKinnon (1991) provides an approach to adjust the critical values for cointegration tests by Engle and Granger to obtain the correct critical values. This adjustment formula for the critical values is:
where the values of , and are shown in Table 8. In this test, we have 2 variables (p=2), Pt and P*t, 90 observations (T=90) and no trend, which is in accordance with -3.3377, -5.967 and -8.98 for , and respectively at the level of significance of 5%. Then we could derive that the critical value is -3.40511 with formula (14). Similarly we could also compute that the critical values of 1% significance level and 10% significance level are -4.02085 and -3.09212 respectively. And in Table 8 the t statistic is -0.372869, which is larger than all the three critical values. Therefore, we say that we fail to reject the null hypothesis that the series of the residuals has a unit root. Without stationarity of residuals, Pt and P*t are not cointegrated.
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Reporting, analysis and discussion of Results 500
Having obtained the results of the tests above, we could conclude that whether the housing bubbles present in Beijing and analyze the reasons of such results:
no housing bubbles but abnormal development of the real estate market;
the presence of the bubbles in real estate market in China.
At last I may also offer some advices to prevent the deterioration and even bursting (or presence) of the real estate bubbles.
Conclusion and Future Work 500
 ( http://en.wikipedia.org/wiki/Real_estate_bubble)
 Page 3, Gallin, J. (2004). The long-run relationship between house prices and rents. Finance and Economics Discussion Series 2004-50, Federal Reserve Board
 Page 68, Charles Himmelberg, Christopher Mayer and Todd Sinai (2005). Assessing High House Prices: Bubbles, Fundamentals, and Misperceptions. Journal of Economic Perspectives, 2005, v19(4,Fall), 67-92. (åŽŸæ–‡å¼•ç”¨)
 because of availability of data, I collect indices of house prices and rents from the statistics of the whole housing market in China instead of that in Beijing, which is different with the proposal submitted before.
 Malpezzi, S. (1999). "A Simple Error Correction Model of House Prices", Journal of Housing Economics 8, 27-62.
 Charles Himmelberg, Christopher Mayer and Todd Sinai (2005). Assessing High House Prices: Bubbles, Fundamentals, and Misperceptions. Journal of Economic Perspectives, 2005, v19(4,Fall), 67-92.
 being introduced to China by Hamer, Andrew Marshall(1992). China: Implementation Options for Urban Housing Reform. World Bank. Washington, DC: The Bank, 1992, 172p. (A World Bank Country Study).
Campbell, John Y., and Robert J. Shiller. (1987). “Cointegration and Tests of Present Value Models.” Journal of Political Economy 95, 1062-88.
Pedroni, P., 1999. Critical values for cointegration tests in heterogeneous panels with multiple regressors. Oxford Bulletin of Economics and Statistics 61, 653-70
Pedroni, P., 2004. Panel cointegration: Asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis. Econometric Theory 20, 597-625
Hendershott, Patric H. and Joel Slemrod. "Taxes and the User Cost of Capital for Owner-Occupied Housing." AREUEA Journal, Vol. 10, No. 4 (Winter 1983) , pp. 375-393.
J. Poterba, "Tax Subsidies to Owner-Occupied Housing: An Asset Market Approach," Quarterly Journal of Economics 99 (1984), 729-752.
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