In the recent years, due to the impact of a couple of factors such as the entry of urbanization into the middle and later periods and the rapid aging in cities, there has been slowing population mobility. According to the data released by the National Health Commission of the People's Republic of China, the total flowing population nationwide has shifted from the previous continual rise to slow decrease since 2015. In 2018, the total flowing population throughout the country registered 241 million, a decrease of 12 million as compared with that of 2014. Under the background of slowing population mobility, there has been big differentiation in population aggregation, and an obvious trend of population aggregation in megacities and megacity clusters^{[1, 2]}. The proportion of permanent residents in 31 PCs & MIPSs¹ of China among the country's total population increased from 13.68% in 2005 to 16.85% in 2018; and the growth of permanent residents in some of the third-tier and fourth-tier cities was lower than the current the natural population growth of the whole country, about 0.5%, indicating large population outflow from these cities. The inflow and aggregation of population would increase housing demand, and cause housing price rise in relevant regions. The high housing prices would, to some extent, constitute a barrier for population mobility. However, even though the housing prices were at a comparatively high level and the rise in the housing prices was comparatively fast in PCs & MIPSs, there were still inflows of a large number of people each year. With the differentiation in population aggregation, the inter-city differentiation had become increasingly serious^[3]. In 2018, the average selling price of commodity housing in China registered 8, 726 yuan/${\mathrm{m}}^2$, with a year-on-year growth of 10.57%, 3.88 times up as compared with that in 2002. In 2018, this figure reached 13, 005 yuan/${\mathrm{m}}^2$, with a year-on-year growth of 14.44%, 5.45 times up as compared with that in 2002. Among them, in 2018, the average selling price of commodity housing in Xi'an registered a year-on-year growth of 19.48%, that in Hangzhou 17.51%, and that in Chengdu 12.98%.

¹ Lhasa in Tibet Autonomous Region and Taibei, Taiwan province are excluded. The provincial capitals and municipalities with independent planning status mentioned in this paper do not include either of them.

To curb the precipitous housing price rise and ensure the sound and stable development of the real estate market, the Central Government of China adhered to the keynote of "Houses are for living in and not for speculative investment" in the regulation and control of the market while local governments adopted the differentiated regulation measures based on the specific conditions of their own cities. At the same time, the Central Finance and Economics Committee (CEFC) pointed out at the fifth plenary that central cities and city clusters in China are becoming the main spatial form for carrying the development elements; and under the new circumstance, it is necessary to strengthen the economic and population carrying capacity of the central cities and city clusters. The national development strategy has already shifted from urbanization to metropolitanization or from the previous encouragement of development of small towns to that of megacities and megacity clusters. Under such background, megacities' population carrying capacity has been constantly enhanced, but to what extent will it influence the housing prices of megacities? And will it accelerate the housing price difference between megacities and other cities?

This paper, with 31 PCs & MIPSs in China as research samples, explains and analyzes from the perspective of demand side how the continuous population inflows into and the population aggregation in PCs & MIPSs, which formed the metropolitanization effects, affect the housing price differentiation within the provincial-level administrative divisions in China as well as its roles and mechanism. The research in this paper found that the constant inflows of population into PCs & MIPSs will boost the housing price difference between PCs & MIPSs and other cities in the relevant provinces. Among them, for every 10, 000 people added to the net inflow into PCs & MIPSs, the average selling price of the commodity housing in PCs & MIPSs will rise by 0.08%$\sim$0.1%, and the deviation of housing prices between PCs & MIPSs and other cities in relevant provinces will increase by 0.0014$\sim$0.0016. Further research discovered that the income gap between PCs & MIPSs and other cities in relevant provinces has strengthened the role of the increasing net inflow of population in promoting the housing price differentiation there. This paper constructs a house price deviation index, quantitatively evaluates the severity of house price differentiation within the scope of provincial administrative divisions, and explores the impact of PCs & MIPSs population agglomeration on housing price differentiation. The marginal contribution of this paper is to analyze the reasons for the emergence and aggravation of housing price differentiation from the perspective of changes in the spatial structure of population; constructed a new indicator to measure the differentiation of housing prices, the phenomenon of housing price differentiation is not only quantified from the relative level of housing prices and the relative speed of growth; based on the actual development situation, that is, when China's urbanization entered the middle and late stages, the trend of population flow gradually changed from rural inflows to cities and towns to small cities to large cities, and analyzed the impact of population agglomeration to large cities on housing prices and housing price differentiation.

The structure of this paper are as follows: Part 1 is the introduction, which analyses the trend of population mobility and the housing price differentiation, describes the research purpose of this paper, i.e., to discuss to what extent the population aggregation in PCs & MIPSs has caused housing price differentiation between PCs & MIPSs and other cities in relevant provinces, and probes into the influential mechanism in them; Part 2 consists of literature review and theoretical hypothesis, in which the research achievements of both Chinese and foreign scholars about such aspects as urbanization and housing price differentiation are reviewed and formulate research hypotheses; Part 3 is models and data, which give an account of the selected data samples and the adopted empirical methods for the research issues; Part 4 contains empirical analysis, in which examines metropolitanization impact of PCs & MIPSs upon housing prices and the differentiation in them and the role of income gap in the metropolitanization impact upon housing price differentiation are analyzed; and Part 5 is the conclusion.

At present, there are not so many researches on housing price differentiation, and there is a lack of clear concept and definition on it; however, most scholars, based on the housing prices or the housing price rise of each city, have judged the housing price differentiation, and held that the occurrence of housing price differentiation is caused by the increasing housing price gap between different cities. In the international research on the theme of housing price differentiation, similar research has been conducted on the inter-city differences in housing prices. Through the analysis of the real estate market in some of the German cities, the anticipation of population change obviously affects the differences in inter-city housing prices^[4]. The rapid housing price rise in some "superstar" regions of USA, and the increasing housing price gap from other regions are due to a lack of elasticity of these "superstar" regions in land supply, as well as an increasing number of high-income households^[5]. Urban public service supply is an important cause of differences in housing prices between cities in Belgium^[6]. A small number of domestic researches on the housing price differentiation have analyzed the key causes of housing price differentiation from the perspective of the supply side. Among them, From the policy of spatial allocation of land, after 2003 the land supply leaned to central and western China, enabling the mismatching between the land supply and demand in space, which further resulted in the differentiation in inter-city housing prices^[7]. From the land supply strategy adopted by local governments, different natural endowments and conditions made local government adopt different strategies in land supply and the differences in the quantity and method of land supply and in the land distribution selected by local governments leads to housing price differentiation^[8]. Separately, there are a few literature which have emphasized the impact of urban characteristics on housing price differentiation; for example, the unbalanced supply of urban public service causes the housing price differentiation in different cities^[9], and the huge differences in the inter-city innovation capability forms the pattern of differentiation in the real estate market^{[10, 11]}. Till now, there is no literature which has analyzed the impact of spatial structure change upon housing price differentiation from the perspective of the demand side.

Currently, there are a large number of researches on the relationship between the spatial structure change in population and housing prices, with the domestic literature mainly focusing on the analysis on the relationship between urbanization and housing prices. Since the reform and opening up policy was introduced, China has witnessed rapid urbanization, with its urbanization rate increasing from 17.92% in 1978 to 59.58% in 2018. The rapid urbanization means the inflows of a large amount of population into cities, which will certainly produce rigid housing demand. Since the reform of the housing system was adopted in 1998, China has witnessed the constant rise of its overall housing prices, with the average selling price of commodity houses climbing from 2, 062.57 yuan/m² in 1998 to 8, 725.67/m² in 2018. The increasing housing demand of migrants invoked by the rapid urbanization and the positive expectation for urbanization in the days to come are important reasons for the rising housing prices^[12]. Based on panel data from Chinese provinces, found that the role of urbanization on housing prices produced the spatial spillover effect and threshold effect, and the lifting of urbanization level would drive up the housing prices of a region and its neighboring areas; this is more evident in the areas with higher economic growth and higher congregation of labor and capital^[13]. There exists an obvious positive correlation between urbanization level and the supply and demand of the real estate market^[14]. During the constant urbanization, real estate price will fall within the long-term rising zone. The research conducted by Wang and Zhou indicated that rapid urbanization is another important reason for the housing price rise, and there are direct and indirect effects on the housing price rise, which would not only drive up the housing prices of localities, but cause the rise of those in relevant neighboring areas; moreover, the indirect effect of urbanization is more evident in megacities^[15]. Analyzed the regional spillover effects of central city housing regulation policies on surrounding cities within the urban agglomeration by combining spatiotemporal models and event analysis methods, and there are obvious regional and cross-regional spillover effects of central city real estate market policies. The regional interaction of the housing market also reflects the flow and concentration of population and resources in regional economic development^[16]. On the basis of the research on how urbanization boosts the housing price rise, a couple of scholars have conducted detailed discussions on the mechanism of how urbanization influences housing prices. Lu, et al. found that after such factors as the urban economic development level and the size of urban population were put under control, the higher the proportion of migrants in a city, the higher its housing prices will be, which is due to the migration of urbanites and the migrants with a higher income level. Foreign literature has mostly focused on the research on the relationship between population mobility and relocation and housing prices^[12]. Foreign literature has mostly focused on the research on the relationship between population mobility and relocation and housing prices. The research conducted by Saiz found that immigrants has pushed up the housing demand at the destination cities in the United States, resulting in a rise in rents and housing prices of these regions. When the immigrants account for 1% of the original population of these cities, the rents and housing prices there would rise by 1%^[17]. Chen, et al. found that the change of inter-province urbanization and the status of population migration exerts an obvious impact on the housing prices in cities in China^[18]. The empirical research conducted Akbari and Aydede, found that immigrants produced an obvious impact on the housing prices in Canada, whose influential effect is comparatively small^[19]. However, Gonzalez and Ortega held that it is the immigrants who has driven the development of the real estate market in Spain, and the empirical results showed that from 2000 to 2010 the immigrants caused the annual growth of 1.2%$\sim$1.5% in newly-added houses, or the average annual growth of about 2% in housing prices in Spain^[20]. Wang, et al. discussed how the change in housing demand caused by population migration influences the housing prices in Chinese cities, and the research results showed that the increase of inter-regional population flow and the improvement of urbainization level would drive up the housing prices^[21]. Lin, et al. found, with 32 major Chinese cities as samples, that the cities with the national-level population inflow rate increase of 1% would lead to the rise of urban housing price by 0.31%; the cities with the regional-level population inflow rate increase of 1% would cause the rise of urban housing price in eastern China by 0.31%; however, the impact on the urban housing prices in central and west China is not so evident^[22].

In summary, the impact of population flow and agglomeration on the real estate market is obvious, that is, increasing the demand for real estate. From a quantitative point of view, the flow and agglomeration of the population is reflected in the expansion of the population size of the net inflow of the city, and the reduction of the population of the city with the net outflow of the population. As the total population changes, so does the demand for real estate. Cities with an increasing population scale, the real estate market demand expands. From a structural point of view, the human capital of cities with net population inflow has been further improved, and a large number of high-end talents have flowed and gathered in PCs & MIPSs. These talents are potential buyers with purchasing ability, which strongly supports the effective demand of the real estate market. From a spatial point of view, due to China's special household registration system and the development characteristics of PCs & MIPSs with better economic development, more employment opportunities, and higher wages, a large number of low-end laborers gather in PCs & MIPSs to seek job opportunities and high wages. At the same time, this group of people often come from small cities around PCs & MIPSs, so this group of people often buy housing in the city where their household registration is located after earning higher wages in PCs & MIPSs. Therefore, metropolitanization promotes the flow and agglomeration of the population, which in turn brings about the differentiation and spillover effects of housing prices. Based on the above analysis, this paper proposes the following hypotheses:

Hypothesis 1  The metropolitanization effect of population promotes the rise of housing prices in large cities;

Hypothesis 2  The metropolitanization effect of population aggravates the differentiation of housing prices among cities;

Hypothesis 3  The income gap strengthens the effect of population metropolitanization on housing price differentiation.

The research in this paper focuses on how the metropolitanization effect of PCs & MIPSs influences the housing price difference between PCs & MIPSs and other cities in relevant provinces, as well as the mechanism which causes this influence; and taking this as the objective, it makes an analysis by setting the measurement model and measuring relevant variables

Firstly, how the metropolitanization of PCs & MIPSs influences their housing prices is examined and the following basic measurement model is established:

where, $i$ represents PCC & MIPS, $t$ represents the year and ${\epsilon }_{it}$ represents time fixed effects. The dependant variable $\ln \mathrm{p}\mathrm{r}\mathrm{i}$ is the logarithm of the average selling price of commodity housing which reflects the housing price level; the core explanatory variable pop refers to the net inflow of population, which reflects the metropolitanization; the control variable includes the industrial structure difference dinv, the degree of government function dgov, the fixed assets investment intensity dinv, the household consumption intensity dcon, the financial development level dfin and the land supply intensity dlan.

Secondly, it checks how the metropolitanization of PCs & MIPSs influences the housing price differentiation in PCs & MIPSs and other cities of relevant provinces; and on the basis of Formula (1), the core explanatory variable is replaced with the housing price deviation index which measures the housing price differentiation, and the following measurement model is established:

where, the dependant variable dpri is the housing price deviation, which reflects the housing price differentiation in PCs & MIPSs and other cities of relevant provinces.

Finally, it examines the mechanism of how the metropolitanization of PCs & MIPSs influences the housing price differentiation in PCs & MIPSs and other cities of relevant provinces; and on the basis of Formula (2), it adds two explanatory variables, the interaction terms which increase the income gap and the net inflow of population. The concrete measurement model is as follows:

where, the explanatory variable inc refers to income gap, which reflects the income level difference in PCs & MIPSs and other cities of relevant provinces. The explanatory variable pop$\times$inc refers to the interaction terms of the net inflow of population and the income gap, indicating that the role of net inflow of population on housing price differentiation is influenced by the income gap.

1) Housing price differentiation. Housing price is expressed by the average selling price of commodity housing, while the later is obtained by the amount of sales of the commodity housing divided by the area of such commodity housing for sales. In respect of the measurement of housing price differentiation, there is currently no definite and unified index and method; and a majority of literature directly use housing prices or housing price growth as the agency index for measuring the housing price differentiation^[23], while Ni (2019) classified 285 prefecture-level cities in China into first-tier, secondtier, third-tier and fourth-tier cities; also he measured the housing price differentiation in different cities by using the exponential method of spatial difference in specific directions, which is used for measuring the degree and status of spatial differentiation in two groups of regions and cities which feature the mode of "high-low"^[3]. This paper analyzes the housing price differentiation within the provincial-level administrative divisions in China. The housing price deviation of a provincial capital city or a municipality with independent planning status $=$ (the average selling price of commodity housing in the provincial capital city or the municipality with independent planning status–the average selling price of commodity housing in the relevant province)/the average selling price of commodity housing in the relevant province.

2) Metropolitanization. The metroplitanization involved in this paper refers to the metropolitanization in terms of population; namely, the population keeps gathering in megacities, thus constantly expanding the size of population in them. This paper expresses the net inflow of population and measures the metropolitanization by calculating the difference between the permanent residents and registered population in PCs & MIPSs. Among them, the number of permanent residents in the provincial capital or the municipality with independent planning status is obtained² by calculating its GDP and per capita GDP. The reason for employing the above index structure and the data processing method is 1) there is a lack of permanent residents in some cities in some years; and 2) the registered population data cannot directly reflect the population mobility even though they come from the public security authority and are comparatively accurate.

² The National Bureau of Statistics requested that from Jan.1, 2005, all regions should measure in a unified manner their per capita GDP based on their permanent populations. The regions which measured their per capita GDP based on registered population in the past can take it as a transitional measure, and can measure their per capita GDP using these two methods at the same time within two years to come (The method used shall be indicated behind the data); and the measurement of per capital GDP as per registered population shall be cancelled in two years.

3) Income gap. It is measured as per the difference between the average salary of the employees in PCs & MIPSs and that of the employees in relevant provinces.

4) Relevant control variables. The fundamentals of the macro economy and finance is the major determinant of housing price change^{[24, 25, 26, 27, 28, 29]}; hence, it considers such factors as industrial structure, fixed assets investment, financial development and household consumption in the selection of control variables and determines the variables in these four aspects of the difference in industrial structure, fixed assets investment intensity, financial development level and household consumption intensity. It measures the industrial structure as per the proportion of the output of the secondary industry and that of the tertiary industry, and calculates the difference between the industrial structure of PCs & MIPSs and that of the relevant provinces; measures the fixed assets investment intensity of PCs & MIPSs as per the proportion of the fixed assets investment of PCs & MIPSs in the relevant provinces; measures the financial development degree of PCs & MIPSs as per the proportion of the RMB loan balance of PCs & MIPSs in the relevant provinces; and measures the household consumption intensity of PCs & MIPSs as per the total retail sales of consumer goods of PCs & MIPSs in the relevant provinces. Lu, et al. held that land supply is the cause of urban housing price differentiation. The government measures the land supply intensity of PCs & MIPSs as per the proportion of the purchased land area of PCs & MIPSs in the relevant provinces, and by which the impact of land supply upon the housing prices and housing price differentiation is controlled^[30]. Besides, many scholars held that the public service supply level and the completeness of infrastructure produced an obvious impact upon housing prices^{[31, 32, 33]}. The public service and infrastructure essentially represent the government function, so this paper measures the degree of the government function of PCs & MIPSs as per the fiscal expenditure of PCs & MIPSs in the relevant provinces.

This paper selects 31 PCs & MIPSs in China as research samples, and considering the availability of data, it selects the samples for the period from 2006 to 2017. All kinds of data come from the 2005$\sim$2017 statistical yearbooks of all provinces, the 2005$\sim$2017 statistical yearbooks of all PCs & MIPSs, and wind database. For the descriptions of all variables and data, please see Table 1.

In order to ensure the stationarity of the data and avoid the pseudo-regression phenomenon of the model, the stationarity test of the variables is carried out. In this paper, the HT test for homogeneous unit roots and the IPS test for heterogeneous unit roots are used to test the variables in the model and the first-order differences of variables respectively. The specific test results are shown in Table 2. Except for industrial structure and fixed assets investment intensity, which failed the HT test and IPS test, all other variables passed the HT test and IPS test. First-order difference processing is performed for each variable, and each variable after the first-order difference has passed the HT test and the IPS test. The above results show that the stationarity of each variable is good. In order to further examine the long-term equilibrium relationship between variables, the co-integration test is carried out on the variables. The specific results are shown in Table 3. Each statistic rejects the null hypothesis at the 1% significance level, indicating that each variable has passed the co-integration test and there is a long-term stable co-integration relationship between the variables. According to the above panel data unit root test results and cointegration test results, it shows that the data of each variable is stationary and that there is a long-term stable equilibrium relationship between variables, and the next step model calculation can be performed.

The empirical analysis of this paper consists of three parts: Firstly, the metropolitanization impact of PCs & MIPSs upon housing prices is checked, and whether the increasing net inflow of population into PCs & MIPSs will drive up their housing prices. Secondly, the metropolitanization impact of PCs & MIPSs upon the housing price differentiation is examined, and whether the increasing net inflow of population into PCs & MIPSs will accelerate the housing price difference in PCs & MIPSs and the cities in relevant provinces is analyzed. Thirdly, the influencing mechanism of PCs & MIPSs for the housing price differentiation, with the focus being placed on reviewing the mechanism of how income gap influences the housing price differentiation through the net inflow of population; namely, how the income level difference between PCs & MIPSs and the relevant provinces causes the population to flow into PCs & MIPSs is examined. If the income level of the population which flows into PCs & MIPSs is relatively high, it will enhance the housing demand, and then boost the rise of the housing prices in PCs & MIPSs; and finally, this will cause the housing price differentiation in PCs & MIPSs and the cities in the relevant provinces.

Table 4 reports the measurement results of how metropolitanization influences housing prices. Models (1) and (2) in Table 4 employ respectively the mixed regression method and the fixed effects method for variable checking, the results of which show that the net inflow of population exerts an obvious positive influence on the housing prices, indicating that the increasing net inflow of population into PCs & MIPSs will, to a certain degree, cause the housing price rise in PCs & MIPSs. However, theoretically speaking, there exists the endophytism in the net inflow of population which is taken as the housing demand factor, and the results anticipated by means of mixed regression and the fixed effects method may be unstable. This paper alleviates the estimation bias by using the Sys-GMM method for re-estimation and by trying to find the instrumental variables of the size of the net population inflow. Model (3) in Table 4 uses the one-step system generalized method of moments (one-step system GMM) for revaluation, the results of which show that the AR (1) $p$-value is smaller than 0.1, the AR (2) $p$-value is bigger than 0.1, and the $p$-value in the Hansen Test is bigger than 0.1. There exists no second order autocorrelation in the difference of the model disturbing terms, and the selected instrumental variables are both valid; the influential effect of net inflow of population upon housing prices should be obviously positive at the statistical level of 1%; the co-efficient is 0.0009, which lies between the estimated value of fixed effects and that of mixed regression; and the results of the model estimation is valid. The estimated results of Model (3) in Table 4 show that the net inflow of population exerts a positive influence on housing prices; in case that other variables are put under control, the average selling price of commodity housing in PCs & MIPSs will increase by 0.09% for each net inflow of 10, 000 people into PCs & MIPSs. According to the basic idea and logic of constructing instrumental variables, we find and find that the electricity consumption of urban residents can be used as a suitable instrumental variable for the scale of net population inflow. On the one hand, the increase in the urban population will inevitably be accompanied by the increase in the electricity consumption of urban residents. On the other hand, the impact of urban residents' domestic electricity consumption on urban housing prices is minimal. The data on the domestic electricity consumption of urban residents are mainly from China Urban Statistical Yearbook. Model (5) in Table 4 is the result of using the electricity consumption of urban residents as an instrumental variable of the scale of net population inflow and using the 2SLS estimation method. The instrumental variables set in this paper are appropriate, there is no weak instrumental variable problem, and the impact of net population inflow scale on housing prices is significantly positive at the 1% statistical level, with a coefficient of 0.0017. It further verifies the robustness of the conclusion that the scale of net population inflow has a significant positive impact on housing prices. By summarizing the estimated results of Models (1), (2), (3), (5) in Table 4, we can find that for PCs & MIPSs, the increasing net inflow of population will cause the housing price rise, and the metropolitanization effect of PCs & MIPSs should be an important factor for the rise of their housing prices.

To further clarify the relationship between the net inflow of population and housing prices, this paper takes the housing price growth as the dependant variable for robustness test, and employs the one-step system GMM for estimation, the concrete results of which are shown in Model (4) in Table 4 as follows. The influential effect of the net inflow of population on housing price growth is obviously positive at the statistical level of 1%. The AR (1) $p$-value in Model (4) in Table 4 is smaller than 0.1; the AR(2) $p$-value is bigger than 0.1; and the $p$-value of Hansen Test is bigger than 0.1. There exists no second order autocorrelation in the difference of the model disturbing terms; the selected instrumental variables are both valid; and the results estimated by the one-step system GMM are accurate and reliable. This further demonstrates that the constant inflows of population into PCs & MIPSs will cause the rise of the housing prices in PCs & MIPSs, while the increasing net inflow of population will produce obviously positive influential effect upon their housing prices.

China has a vast territory, the economic and social development of various regions is extremely unbalanced, and the economic and social development gap between the eastern, central and western cities is large. In this paper, 31 PCs & MIPSs in China are divided into three sub-samples of east, middle and west according to the region where the province is located, reasearch the impact of population agglomeration in PCs & MIPSs on housing prices in eastern, central and western provincial, and examines the regional characteristic effect of the expansion of the net inflow of population in PCs & MIPSs on housing prices.

Table 5 shows the quantitative test results of the impact of population agglomeration in PCs & MIPSs on housing prices in the eastern, central and western regions. Among them, the effect of net inflow of population in PCs & MIPSs in the eastern region on housing prices is significantly positive at the statistical level of 1%, with a coefficient of 0.0004; the effect of net inflow of population in PCs & MIPSs in the central region on housing prices is significantly positive at the statistical level of 1%, with a coefficient of 0.0005; the effect of net inflow of population in PCs & MIPSs in the western region on housing prices is significantly positive at the statistical level of 1%, with a coefficient of 0.0010. According to the results in Table 5, it can be shown that whether it is the eastern region, the central or the western region, the continuous expansion of the net inflow of population in PCs & MIPSs will promote the rise of housing prices, and the metropolitanization effect of population in PCs & MIPSs can significantly promote housing prices. Judging from the coefficient of the net population inflow scale, the net inflow of population in PCs & MIPSs in the western region has a greater effect on housing prices than in the central and eastern regions. The economic and social development level of the western region is relatively low, the urbanization process is relatively slow, the overall housing price level is low, and the expansion of the population size has a relatively large impact on the real estate market. The economic and social development level of the eastern region is relatively high, the urbanization process is relatively fast, the population base is large, and the overall housing price level is relatively high. The marginal effect of housing price increases brought about by the expansion of the net inflow of population to PCs & MIPSs in the eastern region is relatively small.

In 2014, the State Council issued the Notice on Adjusting the Criteria for Urban Size Division formally adjusting the division criteria for urban size in China. Among them, cities with an urban resident population of more than 10 million are megacities, cities with an urban resident population of more than 5 million but less than 10 million are supercities, cities with an urban resident population of more than 1 million but less than 5 million are largecities, and urban resident populations of more than 500, 000 cities with a population of less than 1 million are medium-sized cities, and cities with an urban population of less than 500, 000 are considered small cities. In order to reasearch the impact of metroplitanization of PCs & MIPSs on housing prices of different city sizes, this paper divides 31 PCs & MIPSs into two categories, one includes supercities and megacities³, and the other is smaller than supercities.

³ Nine cities including Guangzhou, Shenzhen, Wuhan, Chengdu, Nanjing, Xi'an, Shenyang, Hangzhou and Harbin.

Table 6 shows the quantitative test results of the impact of population agglomeration in PCs & MIPSs of different scales on housing prices. Among them, the effect of net inflow of population in PCs & MIPSs of supercity and megacity on housing prices is significantly positive at the statistical level of 1%, with a coefficient of 0.0007; the effect of net inflow of population in PCs & MIPSs smaller than supercity on housing prices is significantly positive at the 1% statistical level, with a coefficient of 0.0009. According to the results in Table 6, it can be shown that the continuous expansion of the net inflow of population in PCs & MIPSs can significantly promote the rise of housing prices. From the perspective of the coefficient of net population inflow scale, the smaller the city scale is, the greater the effect of net population inflow scale on housing prices is in PCs & MIPSs.

Table 7 reports the measurement results of the metropolitanization impact upon housing price differentiation. Models (1) and (2) employ the mixed regression method and the fixed effects method respectively for variable checking; and Model (3) adopts the one-step system GMM for variable checking, which can effectively overcome the endogenous problems between the net inflow of population and the housing price deviation. Model (5) is still the result of using the electricity consumption of urban residents as an instrumental variable of the scale of net population inflow and using the 2SLS estimation method. The checking results of Models (1), (2), (3), (5) in Table 7 show that, regardless of which estimation methods to be adopted, the co-efficient of the net inflow of population will be significantly positive at the statistical level of 1%. Among them, the co-efficient of the net inflow of population estimated by means of the mixed regression method in Model (1) is 0.0014; the co-efficient of the net inflow of population estimated by means of the fixed effects method in Model (2) is 0.0014; and the AR (1) $p$-value in Model (3) is 0.009, the AR (2) $p$-value is 0.578 and the $p$-value of Hansen Test is 0.112, indicating that the model setting is reasonable and the estimated results are valid. Besides, the co-efficient of the net inflow of population estimated in this model is 0.0016; Model (5) has a coefficient of 0.0016 for the net inflow of population estimated by instrumental variables and using the 2SLS estimation method. According to the results estimated by the one-step system GMM, the housing price deviation will increase by 0.0016 in case of the net inflow of 10, 000 people into PCs & MIPSs or the deviation between the average selling price of commodity housing in PCs & MIPSs and that in the relevant provinces will add by 0.0016. The results in Table 7 show that the increasing net inflow of population will boost housing price differentiation; namely, in case of an increasing net inflow of population into PCs & MIPSs, the deviation of the housing prices between the housing prices in PCs & MIPSs and those in the relevant provinces will become bigger; and the metropolitanization effects of PCs & MIPSs will cause and accelerate the housing price differentiation between PCs & MIPSs and relevant provinces.

To ensure the reliability of the above conclusion, the core explanatory variable in this paper is replaced for the robustness check. The net inflow of population is replaced with the increment of registered population as the core explanatory variable, and the one-step system GMM is used to conduct the revaluation. The results of Model (4) in Table 7 show that the co-efficient of the increment of registered population is 0.0010, and is significantly positive at the statistical level of 5%, indicating that the housing price deviation will increase by 0.001 whenever the registered population of 10, 000 people in PCs & MIPSs is added. This further demonstrates that the metropolitanization effects of PCs & MIPSs will lead to and accelerate the housing price differentiation in PCs & MIPSs as well as other cities in relevant provinces Meanwhile, by comparing the net inflow of population in Model (4) and the co-efficient of the increment of registered population, we discovered that the co-efficient of the net inflow of population is bigger than that of the increment of registered population, which coincides with the actual situation that the permanent resident population in PCs & MIPSs is far larger than that of the registered population.

Table 8 reports the measurement results of the metropolitanization impact upon housing price differentiation of PCs & MIPSs in the eastern, central and western regions. Among them, the effect of net inflow of population in PCs & MIPSs in the eastern region on the differentiation of housing prices is significantly positive at the statistical level of 1%, with a coefficient of 0.0016; the coefficient of the net population inflow scale of PCs & MIPSs in the central region is 0.0001, which fails the test under the 10% statistical level; the effect of net inflow of population in PCs & MIPSs in the western region on housing prices is significantly positive at the statistical level of 1%, with a coefficient of 0.0005.

The results in Table 8 show that in the eastern and western regions, the continuous expansion of the net inflow of population in PCs & MIPSs will significantly exacerbate the phenomenon of housing price differentiation, and the increase in the net inflow of population in PCs & MIPSs of the eastern region has a greater impact on the differentiation of housing prices. Guangzhou, Shenzhen, Nanjing, Hangzhou and other cities are all located in the eastern region, with relatively high levels of economic and social development, large populations, high housing prices, and obvious polarization effects. At the same time, due to the serious imbalance of regional development, the level of economic and social development of cities in the western region is relatively low. In particular, the overall level of housing prices in Lanzhou, Xining, Yinchuan, Urumqi, Kunming and other cities is basically the same as that of other cities in the province However, as the provincial administrative regions concentrate various resources to develop the provincial capital cities, the population inflow of these cities has expanded rapidly, the real estate market demand has increased, and housing prices have risen rapidly, widening the gap between housing prices and other cities in the provincial administrative regions. The overall housing price level in the central region is relatively average, and the central region is geographically closer to the cities in the eastern region, a large number of people in the central region flow to PCs & MIPSs in the eastern region. The effect of metropolitanization in the central region did not significantly exacerbate the phenomenon of housing price differentiation.

Table 9 shows the quantitative test results of the impact of metropolitanization in PCs & MIPSs of different city scales on housing price differentiation. Among them, the effect of net inflow of population in PCs & MIPSs of supercity and megacity scales on housing price differentiation is significantly positive at the statistical level of 1%, with a coefficient of 0.0017, the effect of net inflow of population in PCs & MIPSs smaller than supercity on housing prices is not significant.

According to the results in Table 9, it can be seen that the impact of metropolitanization in PCs & MIPSs on housing price differentiation has the characteristic effect of city scale. Only when the city scale reaches the limit of supercity, the expansion of the net population inflow scale will significantly exacerbate the housing price differentiation phenomenon.

Table 10 reports the measurement results after incorporating the income gap and the interaction term of the net inflow of population and the income gap. Among them, Models (1) and (3) use the fixed effects method to estimate the incorporated income gap and the interaction term of the income gap, the net inflow of population and the income gap respectively; Models (2) and (4) use the one-step system GMM to respectively estimate the incorporated income gap and the interaction term of the income gap, the net inflow of population and the income gap. The results of Table 4 show that the co-efficient of the income gap, regardless of the estimation method, is significantly positive at the statistical level of 1%; and the interaction term of the net inflow of population and the income gap is significantly positive at the statistical level of 1% indicating that the income gap will strengthen the role of net inflow of population on housing price differentiation; namely, the bigger the income gap is, the bigger the promoting role of the expansion of net inflow of population on the housing price differentiation will be. That means, when the income in PCs & MIPSs is higher than the average income in relevant provinces, there will be bigger net inflow of population into PCs & MIPSs, which will further accelerate the housing price differentiation in PCs & MIPSs as well as other cities in relevant provinces. Income gap constitutes an important motive for population mobility, and the income level is also an important factor of attracting population to work and live in PCs & MIPSs, among others. When the income level in PCs & MIPSs is far higher than that in other cities of relevant provinces, the siphonic effect of PCs & MIPSs will become more obvious, the net inflow of population will get larger, and the income level of the population who flow into PCs & MIPSs will be higher. As a result, this will cause higher housing demand there, and lead to and accelerate the housing price difference between PCs & MIPSs and other cities in relevant provinces.

This paper uses the relevant data of 31 PCs & MIPSs in China from 2006 to 2017 to check the influence of population aggregation in megacities upon housing price differentiation. The research results show that 1) the increasing net inflow of population into PCs & MIPSs has promoted the rise in the housing prices of PCs & MIPSs, and the impact of net inflow of population in PCs & MIPSs on housing prices has regional heterogeneity and heterogeneity of city size, the net inflow of population from PCs & MIPSs in the western region has a greater effect on housing prices than in the central and eastern regions, the smaller the city size is, the greater the effect of net inflow of population in PCs & MIPSs on housing prices is. 2) The continuous expansion of the net inflow of population in PCs & MIPSs has exacerbated the housing price differentiation between the PCs & MIPSs and other cities in the province. From a regional perspective, the continuous expansion of the net inflow of population from PCs & MIPSs in the eastern and western regions will significantly exacerbate the phenomenon of housing price differentiation, and the increase in the net inflow of population from PCs & MIPSs in the eastern region will have a greater impact on housing price differentiation. From the perspective of city size, when the city scale reaches the limit of megacities, the expansion of the net inflow of population in PCs & MIPSs will significantly exacerbate the phenomenon of housing price differentiation. 3) the income gap between PCs & MIPSs and other cities in relevant provinces has strengthened the effect of population inflows upon housing price differentiation. The research found that the population aggregation in megacities has remarkably pushed up the housing prices in megacities and strengthened the housing price difference between megacities and other cities. The research of this paper analyzes the reason for housing price differentiation, and demonstrates that the differentiation in population aggregation is an important explanatory factor for housing price differentiation.

As the urbanization in China enters the middle and later periods and the population mobility and urbanization slows down, the population mobility trend has shifted to population aggregation in megacities, which causes an increase in housing demand in megacities and the decrease in housing demand of middle and small-sized cities, and further accelerates the housing price difference between megacities and other cities. Therefore, we should comply with the natural rules of population mobility and the general rules of urban development, further ease the policy restriction on the development of megacities, especially the household registration system which restricts population mobility and the land supply policies which affect the elasticity of housing so as to better develop megacities and make full use of the aggregation effects of them. Meanwhile, we should boost the construction of city clusters, strengthen the influencing role of them, and by virtue of the gradual extension in a couple of economic and social aspects such as industrial expansion, infrastructure and public service, promote the free flows of major elements between megacities and middle and small-sized cities nearby so as to makes resource complement to each other, and shorten the gap between them.