Examining Convergence Clubs in Chinese Provinces (1952-2017): New Findings from the Simplified Clustering Convergence Test

This paper empirically investigates the convergence clustering in 31 Chinese provinces regarding the popular and important economic indicator of GDP per capita over the period 1952-2017. Using the club convergence and clustering procedure of Phillips and Sul (2007) with necessary simplifications, a few provincial clusters are identified. It is clearly verified as expected that the Chinese provincial GDP per capita series contain significant nonlinear components. It is found that there are two or three convergence clubs depending on different starting years or initial conditions, and the clustering results are somewhat stable with respect to different starting years. The results can help local and central governments to select appropriate growth promotion strategies for different groups of provinces in general and, due to the evidence that GDP per capita in China heavily inclines to a few major provinces (such as Beijing, Shanghai, Tianjin, Jiangsu and Zhejiang), can also help provide useful information to relevant authorities to fight against the increasing income inequality across provinces in particular.

of source markets within a panel. The method has been effectively used in identifying convergence clubs among, e.g., many EU economies (Bartkowska & Riedl, 2012;Fritsche & Kuzin, 2011;Simionescu, 2015). In the context of China, this test allows for transitional dynamics of the Chinese provinces to be different in convergence testing within a panel no matter some particular source markets are initially in a state of transition or near steady state equilibrium. Since the growth rates for the individual provinces take a wide range of values over the period from 1952 to 2017, and possibly have different convergence paths, this methodological approach is very appropriate to test for convergence of the Chinese provinces. And the major research goal of this paper is just to identify the convergence clubs in the Chinese provinces using this empirical approach as described below.

Relative Transition Paths
Let X it denote the variable of interest for province i at time t (i.e., time enters the model in a non-linear fashion), where i = 1, 2, …, N and t = 1, 2, …, T. Following Phillips and Sul (2007), the variable can be decomposed into two components: the common component of cross-sectional dependence in a panel, g it , and transitory component, a it , as follows: Phillips and Sul (2007) reformulate eq. (1) as common and idiosyncratic components are separated, and the model takes a nonlinear form as follows: where  t is a common component and  it = (g it + a it )/ t is a time varying idiosyncratic element. That is,  it measures the economic distance between the common trend component  t and a source market's individual value of X it at time t in eq. (2). In the scenario of the Chinese GDP per capita for example,  t stands for a common GDP per capita trend in the entire 31 provinces, while  it depicts a relative share of a specific province's GDP per capita in common provincial trend in the entire economy at time t.
The main procedure in the convergence test of Phillips and Sul (2007) is to calculate the time-varying loadings  it such that they can be used to determine the club convergence, if the loadings converge.
Next, Phillips and Sul (2007) define the transition coefficient as h it and to extract the time varying factor loading  it as follows: where h it is the transition parameter that measures  it in relation to the panel average at time t, and therefore it describes the transition path for source market i relative to the panel average. It is suggested that the original data could be smoothed using, e.g., the Hodrick and Prescott (1997) filter, to remove the cyclical trend from the original data (Phillips & Sul, 2007). But to avoid this additional dimension of complexity or uncertainty (which may also not be so necessary for annual macroeconomic series, especially for relative ratios like h it ), this step will be omitted in the already quite complex procedure.
The most important part of the club convergence test is to construct the cross sectional variance ratio H 1 /H t where: Phillips and Sul (2007) prove that the transition distance H t has a limiting form as such: where A is a positive constant, L(t) is a slowly varying function of t with L(t)→∞ as t→∞, and  denotes the convergence speed. To test for the null hypothesis of convergence, Phillips and Sul (2007) perform the log t regression such that the null hypothesis of convergence is: H 0 : δ i = δ and α ≥ 0, against the alternative H 1 : δ i  δ for all i or α < 0, where δ it → p δ i as t→∞ (i.e., plim t→∞ δ it = δ i ) and δ it is defined in eqs. (2) and (3).

The log t Regression
Next, the following regression model where  = 2α: is estimated using the ordinary least squares (OLS) and sample data as: where L(t) = log(t +1) is used as suggested by Phillips and Sul (2007) and the fitted coefficient of log t where a is the estimate of α in the null hypothesis. To take into account the impact of initial conditions on the test, the empirical regression is run after a fraction of the sample is removed. The data for this regression starts at some time point t = [rT] with r > 0. It is recommend by Phillips and Sul (2007) that r = 1/3. For inference purpose, a one-sided t test of the null hypothesis α ≥ 0 is performed based on the estimate of b and the standard error of the estimate which are calculated using a heteroskedasticity and autocorrelation consistent (HAC) estimator for the long-run variance of the residuals (so the test statistic t b is normally distributed). The decision rule is that the null hypothesis of convergence is rejected at the 5% significance level if t b < -1.645.

Club Convergence Algorithm
Phillips and Sul (2007) also argue that a strict rejection of the null of global convergence may not necessarily rule out the existence of sub-group convergence within the panel, and a club convergence algorithm is thus developed to detect possible convergence clusters. Following this procedure, the current paper will bring new information into the convergence process within the 31 provinces in China by revealing whether or not convergence clusters are present in the sample period from as early as 1952 to as recent as 2017. This examination of club convergence is important because the causality relationship within the clusters could be further investigated based on members' economic or structural characteristics within each club. The clustering algorithm is based on repeated log t regressions, and it contains the following four main steps.
Step 1 (Ordering): Order the members (i.e., the X it series) in the panel according to their last observations (i.e., the X iT values).
Step 2 (Core Group Formation): For some k = 2, 3, …, N, select k panel members that have the first k highest individual X iT values in the panel to form the subgroup G k ; run the log t regression using data of members in G k and calculate the convergence test statistic t b (k) = t(G k ) for this subgroup. Choose the core group size k * by maximizing t b (k) over k = 2, 3, …, N according to the criterion: Step 3 (Club Membership): Select provinces for membership in the core group G k* (Step 2) by adding ONE remaining province (not in G k* ) separately to the core group, and then the log t regression is run and the log t test is done. The new province (member) will be included, if the associated t-statistic is greater than zero (a conservative rule). Convergence criterion will be checked for the club as usual.
Step 4 (Recursion and Stopping): The log t test is run on the group of provinces not selected in Step 3.
If this set of provinces converges, then the second club is formed. Otherwise, repeat Steps 1-3 to reveal some sub-convergent clusters. If no subgroups are found ( Step 2), then these provinces display a divergent behavior.
It should be noticed that in the first round of calculations of the above algorithm, Step 2 needs to run the not-simple log t regression many (up to N-1) times for finding the first core group G k* , then Step 3 also needs to run the log t regression many (up to N-k*) times for determining the first convergence club. If the provinces not in the just-determined club 1 do not converge (based on a log t regression for them), then Steps 1-3 are repeated on them to determine the second convergence club, and so on.
Clearly, many log t regressions need to run on different groups of provinces and for each of such regressions many new h it and H t values as defined in eqs. (3) and (4) need to be calculated on the corresponding new group of provinces, which is really too complex and cumbersome, especially for large N as in this paper. Necessary simplifications are thus worth of considerations in practically applying the above algorithm, especially in the seemingly not so crucial Step 3, for clustering many regions into a number of convergence clubs, which will be carefully proposed when needed in the following real application of the convergence clustering algorithm.

Data
This paper investigates China's potential provincial convergence clusters for the popular and important provinces are ranked based on their actual GDP per capita in the last sample year of 2017, and the ranking is quite similar to that based on the last five years' average as shown in Table 1, showing certain stability in the provinces' average income positions. The relative share of each province's per capita GDP in the last report year of 2017 over the 31 provinces' total is also listed in the last column of comparing the average annual growth rates of the 31 provinces' GDP per capita in the last five, ten and twenty years (not reported here for the sake of saving space), it is interesting to notice that provinces with higher final-year GDP per capita generally grow more slowly, showing certain trend in growth convergence. This stylized fact is similar to that found in, e.g., Lau (2010).

Overall Convergence or Not
The practical algorithm for the econometric approach of Phillips and Sul (2007)  1) r = 0, i.e., all data will be used with 1952 as the starting year.
Since China's economic reform started in 1978 has very important effect in Chinese economy, one more test using data started from 1978 is also considered.
Firstly, equations for the overall log t regression with the four different starting years are estimated as follows using the professional EViews package: The full results are reported in Table 2 where, for example, the row corresponding to code 10 indicates that Chongqing is the province with the 10 th highest per capita GDP in 2017, and each t-statistic in that row is for the log t coefficient of the estimated regression equation for subgroup G 10 (of the 10 provinces with the 10 highest GDP per capita in 2017) with data started from a specific year. convergence clubs in the four cases (G 13 , G 19 , G 24 and G 16 respectively), the next row just shows that in each case, if the province with the next highest 2017 per capita GDP is included into the group, the group is no longer a convergence one because of the resulting HAC t-statistic < -1.645.  Tables 1-2 for which province is tagged with which code number.
If just the initial starting year of 1952 is used as the real base year without dropping any data, then G 13 of the 13 provinces with the highest 2017 per capita GDP forms the first convergence club. Then the same procedure is applied as above for the remaining 18 provinces (with codes 14-31) with lower 2017 per capita GDP to determine the next convergence club(s). A group with only the 14 th province (of Liaoning) is first formed initially and the province with the next higher 2017 per capita GDP is added sequentially one-by-one into it, then the log t regression is run for the group until the corresponding HAC t-statistic is still bigger than the critical value of -1.645. As expected and also as shown in rows 5-8 of the first panel (for 1952-2017) of Table 3, when more provinces are included into the group, the resulting HAC t-statistic generally decreases, and finally provinces 14-29 (with codes 14-29) form the second convergence club with a corresponding HAC t-statistic of - more strategically important areas such as investing more in education and providing favorable conditions to attract more and higher quality investments from other Chinese provinces and other countries, in order to establish some important competitive advantages so as to catch up with the wealthier ones in the long-run.

Conclusions and Future Research Directions
The current paper empirically investigates the convergence clustering in 31 Chinese provinces regarding the important and popular economic indicator of GDP per capita over the period from 1952 to the recent year of 2017. Using the club convergence and clustering procedure of Phillips and Sul (2007)  conditions. As such, absolute or overall convergence may not be expected for the Chinese provinces, at least at this stage, but as in some similar investigations in the Chinese context and in some studies for other countries and regions, the results of this paper show that Chinese provinces have revealed clear tendency to converge into a few clubs in terms of average income levels.
The current study produces a number of meaningful empirical findings in this direction. Firstly, as expected, it is observed that Chinese GDP per capita series contain significant nonlinear components as validated by Phillips and Sul's (2007) approach. Secondly, it is found that there are two or three significant convergence clusters based on the GDP per capita data in China's 31 inland provinces.
Although the clustering results vary somewhat depending on starting years (1952, 1962, 1973, or 1978) or initial conditions, in general the two or three convergence clubs are relatively similar in membership structures, indicating an important evidence for the conditional or club convergence of Chinese average income at provincial level. These results can help local and central governments implement different economic growth policies for different groups of provinces.
Economic convergence studies as this paper need data over longer time period to get more stable and convincing results, which can be expected to be conducted with the availability of more recent and future GDP per capita data at China's provincial level. To support policy analysis for different convergence clubs, future studies can be suggested to directly link GDP per capita to such more controllable variables as tax rates on labor and consumption using econometric models in a more general convergence clustering framework (Donath & Mura, 2019). In the future it is also meaningful www.scholink.org/ojs/index.php/jepf Journal of Economics and Public Finance Vol. 6, No. 3, 2020 Published by SCHOLINK INC.
to extend the current study to some other important economic variables such as income-consumption ratios, inflation rates, unemployment rates, industrial output, fixed-asset investments, monetary aggregates, and interest rates as once examined by, e.g., Brada et al. (2005), Kutan andYigit (2005, 2007), Chow (1985,2010,2011,2016), Chow and Wang (2010), Nagayasu (2011), Tillmann (2013), and Zhang (2011, 2013a, 2013b. This can be expected to produce more comprehensive and reliable results in grouping Chinese provinces into more specific convergence clusters to help promote China's economic growth and structural change more effectively. It is also expected that, applying the same club convergence and clustering procedure of Phillips and Sul (2007), internationally comparable studies can be done with other developing countries with available long-period data in average income (and inflation, consumption and unemployment rates) to further validate the ideas and results of (conditional) economic convergence from a broader perspective in the future.
Last but not the least, this paper suggests and actually uses some necessary simplifications of the popular convergence clustering algorithm of Phillips and Sul (2007) which usually involves many rounds of complex and cumbersome calculations on different groups of provinces. It is hoped that more simplifying efforts, including validations of these simplifications, can be done in the future to make the algorithm more welcome in economic growth convergence studies.