Searching for the Parallel Growth of Cities in China

Abstract

Based on the parallel growth implications of the four urban growth theories (endogenous growth theory, random growth theory, hybrid growth theory and locational fundamentals theory), this paper uses Chinese city size data from 1984 to 2006 and time-series econometric techniques to test for parallel growth. The results from various types of stationarity tests show that city growth is generally random. Conditioning on growth trend and structural change, certain groups of cities with common location-specific characteristics, such as a similar natural resource endowment or policy regime, grow parallel in the long run, suggesting that locational fundamentals may have a persistent impact on city growth.

1. Introduction

City growth across countries exhibits two striking traits. First, cities keep growing in many countries, in terms of both city size (city population) and the number of cities. Secondly, the distribution of city sizes in different countries fits the power law (Pareto distribution) very well. Especially, in the upper tail of city size distribution, the power exponent is equal to or very close to one, which is called Zipf’s law or the rank–size rule, meaning that a city size is proportional to the inverse of its rank.¹

Although city size distribution has attracted many theoretical and empirical studies, the dynamics of the urban growth process have received less attention. Empirically, we are still not clear whether city growth is random or deterministic. If it is deterministic, does the growth converge (small cities grow faster), diverge (large cities grow faster) or become parallel (all cities grow at the same speed)? If it is random, does the growth of cities move together in the long run (also termed parallel growth in the long run) or move purely independently? Identifying these empirical regularities can shed light on city growth forecasting and urban public policy design related to regional inequality.

Of particular interest is to test the parallel growth hypothesis. Theoretically speaking, the four strands of urban growth theories that have been developed to explain the two stylised traits—namely, the endogenous growth theory, the random growth theory, the hybrid growth theory and the locational fundamentals theory—all have implications related to parallel growth. Empirically, to the best of our knowledge, we find only one study (Sharma, 2003) that has detected directly the parallel growth of city sizes in the long run in India. Whether parallel growth of cities exists in other countries remains unexplored.

The endogenous urban growth theory predicts deterministic, persistent, parallel growth of cities, meaning that, in the steady state, cities of different sizes grow at the same constant speed. The Black–Henderson model (Black and Henderson, 1999) assumes localised information spillovers and human capital accumulation as the engines of urban growth and concludes that sizes of different types of cities grow at the same rate proportional to the growth rate of human capital accumulation. Eaton and Eckstein (1997) also construct a similar model and predict that the growth of a system of cities is parallel, with relative city sizes depending upon the local knowledge spillover effects.

The random growth theory assumes that the city size growth process follows Gibrat’s law (the growth processes have the common expected growth rate and a common standard deviation), implying that the city size growth process is a random walk or a unit root process.² If at least for a certain range of size, the cities follow Gibrat’s law, then, in the steady state, the distribution of city sizes in that range will follow Zipf’s law (Cordoba, 2007; Gabaix, 1999). Although the random growth theory does not directly predict parallel growth, for cities of random growth to grow parallel, the growth pattern should be in the form of co-movement of unit root processes in the long run, or cointegration of city sizes.

The hybrid urban growth model (Rossi-Hansberg and Wright, 2007) combines both endogenous growth and Gibrat’s law and predicts the same two city growth traits. A random total factor productivity shock is introduced to the model so that the balanced growth of city sizes is also random. This theory predicts parallel growth in the long run (same expected growth rates in the steady state) but random deviation in the short run due to external productivity shocks, which is a special case of cointegration with the cointegrating coefficient equal to one.

The locational fundamentals theory states that natural advantage determines the existence and growth of cities (Fujita and Mori, 1996; Krugman, 1996). Locational characteristics may be considered randomly distributed over space; they are the initial conditions that play a crucial role in shaping the formation and evolution of the city size of that location. Even if the initial conditions become unimportant, their effects may still persist, which is called the path dependence effect or the lock-in effect of self-reinforcing agglomeration forces (Fujita and Mori, 1996). Such self-reinforcing agglomeration forces could be due to initial natural advantages creating secondary natural advantages through cumulative processes (Krugman, 1993). Furthermore, contrary to the random growth theory, strong location-specific advantages may even revert the strong, temporary shocks to city growth. Although this theory makes no clear prediction about the pattern of city growth, an inference can be drawn that, with all other things equal, if two cities have similar key locational fundamentals, their growth should be close to parallel due to the path dependence effect.

Table 1 summarises the implication of each urban growth theory for parallel growth. Only a few empirical studies have tested the parallel growth of cities. Junius (1999) uses cross-country data and finds that the population proportion of the largest cities in a country (urban primacy ratio) has a bell-shaped relationship with economic development, rejecting the parallel growth of cities.³ Although top cities might not exhibit parallel growth, whether other groups of cities tend to grow parallel is unknown. Using Indian decennial population census data from 1901 to 1991 and applying unit root and cointegration tests, Sharma (2003) concludes that city growth may be parallel in the long run, but the short-run growth may deviate from the long-run rate of growth due to exogenous shocks; and temporary shocks may take less than a decade to dissipate. However, Sharma specifies a strong assumption that the time trend of growth is zero or negligible to produce parallel growth. She also neglects a key feature of city size evolution: structural change.

Table 1.

Implications of urban growth theories for parallel growth

Urban growth theory	Representative studies	Implication for growth rate of city size	Implication for parallel growth
Endogenous growth theory	Black and Henderson (1999); Eaton and Eckstein (1997)	Constant across cities in steady state	Deterministic, parallel growth
Random growth theory	Cordoba (2007); Gabaix (1999)	Same mean and variance for all cities across time; independent of city sizes	Stable city size distribution following Zipf’s law implies parallel growth in the long run
Hybrid growth theory	Rossi-Hansberg and Wright (2007)	Same expected rate across cities in steady state	Parallel growth in the long run but random deviation in the short run
Locational fundamentals theory	Fujita and Mori (1996); Krugman (1996)	Initial conditions affect future growth rate	Cities with similar locational fundamentals tend to grow parallel

This paper adapts Sharma’s method to test the parallel growth of cities in the long run by relaxing the time trend assumption and taking into account the endogenous structural change of city sizes. Specifically, we use Chinese city size time-series data from 1984 to 2006 to identify the dynamic patterns of city growth.

Rapid urbanisation has taken place in China since the 1980s. The urbanisation rate increased from 23.01 per cent in 1984 to 43.90 per cent by the end of 2006, and the total number of cities increased from 295 in 1984 to 661 in 2006. The dramatic change in China’s economic structure and policies may have had a strong impact on the evolution of city sizes and size distribution. Some cities benefit from the strong agglomeration economies from nearby super-large cities; some other cities, however, still suffer from locational disadvantages. Cities in special economic zones have been blessed by favourable government policies and grow much faster, but cities in the western interior are left behind. These features have attracted a small number of studies on Chinese city size distribution (Anderson and Ge, 2005; Au and Henderson, 2006; Song and Zhang, 2002), but few on the dynamics of Chinese city growth.

We do not intend to test which of these four urban growth theories better predicts parallel growth; instead, we focus only on identifying empirically the growth patterns of Chinese cities. The results from various types of stationarity and cointegration tests show that in general city growth is not parallel; however, once trend-stationarity with endogenous structural change is allowed, cities with certain common group characteristics, in terms of geographical region, natural resource endowment and policy regime, grow parallel. If any location-specific factors are considered locational fundamentals, then our findings lend some support to the locational fundamentals growth theory in the sense that cities with similar locational fundamentals tend to grow parallel in the long run.

We recognise that our annual data from between 1984 and 2006 are not perfect. A longer time-series is desired for testing long-run growth, as in the literature (Dobkins and Ioannides, 2001; Henderson and Wang, 2007; Ioannides and Overman, 2003; Sharma, 2003). We hope that urban growth models focusing on short-run dynamics will be developed to provide more insights on urban growth.

The next section discusses the methodology of testing for parallel growth. Section 3 describes the dataset; section 4 presents the results; and section 5 concludes.

2. Testing Parallel Growth of Cities

The endogenous urban growth theory predicts simple, deterministic, parallel growth in the steady state. Empirically, testing for deterministic, parallel growth is straightforward: simply test whether individual cities’ growth rates are constant across time-periods. However, obviously, since city growth is affected by many factors and stochastic shocks, such a strict version of parallel growth is rare if not impossible.

If city size is a random process, then, testing parallel growth can be indirect, meaning that a stable city size distribution over time implies parallel growth. Alternatively, the test can be direct by means of the cointegration method, but non-stationarity of or unit root in the city size data needs to be confirmed first. This section focuses on the methodologies for the indirect test for the stability of city size distribution and the direct test for co-movement of random growth.

2.1 Indirect Test for Parallel Growth

A few studies have identified the relative stability of city size distribution in some countries at different time-periods (Black and Henderson, 1997; Eaton and Eckstein, 1997), suggesting that cities of different sizes may grow relatively parallel. Yet the issue with indirect testing is that, if some cities change their ranks, then stable size distribution does not imply perfect parallel growth. In general, the rank order of top cities rarely changes, as evidenced by Polèse and Denis-Jacob (2010) who use city size data from 126 countries since 1900. Medium cities probably change rank order more often. Nevertheless, the stability of size distribution can still provide suggestive evidence for or against parallel growth. Eaton and Eckstein (1997) use city size data from France and Japan and estimate the Markov transition matrix of city size evolution. They conclude that a wide range of city size growth is persistent, with quite a stable distribution close to the rank–size rule. This is consistent with parallel growth. However, they use only the top 40 urban areas. As many studies have noted, the threshold of city size matters in estimating the power exponent (Eeckhout, 2004). Also, as Sharma (2003) points out, Eaton and Eckstein do not provide statistical inference so it is hard to know how large the diagonal transition probability should be to justify the persistent, parallel growth.

The by-product of the indirect test is that it can also test indirectly whether city growth is random or not. The random growth theory states that, if city growth processes follow Gibrat’s law, then the steady state distribution will obey Zipf’s law. Therefore, if the city size distribution is not consistent with Zipf’s law, we would be able to cast doubt on random growth. We use Chinese city size data and estimate the standard rank–size model for each year to trace the change of the power exponent and the stability of the city size distribution

\ln R = α - β \ln P + ε,

(1)

where, R and P are the rank and population size of a city, with R = 1 for the largest city; ln denotes natural logarithm; and $β$ is Zipf’s exponent.

We estimate equation (1) using the ordinary least squares (OLS) method and the full sample of cities and discuss the results in section 4.1. Using OLS to estimate equation (1) will underestimate β and its standard error when the sample size is small. Gabaix and Ibragimov (2011) propose that replacing $\ln R$ by $\ln (R - 1 / 2)$ can reduce the bias significantly. We re-estimate model (1) by using $\ln (R - 1 / 2)$ as the dependent variable but obtain very similar results. Other methods, such as the Hill estimator, also underestimate the standard error or have other drawbacks (Gabaix and Ioannides, 2004). We also apply the Hill method to estimate Zipf’s exponent but the estimates are consistently smaller than the OLS estimates although the dynamic pattern across years is very similar. The smaller estimator of the Hill method is consistent with what Dobkins and Ioannides (2000) find using US city data.

2.2 Testing Stationarity of City Sizes

The sizes of a city at different time-periods are most likely to be correlated due to reasons such as the durability of urban infrastructure and housing (Glaeser and Gyourko, 2005). The intertemporal correlation of city sizes implies that a temporary random shock to city size may have persistent impact on future city growth. The random growth theory indicates that city size time-series is a random walk, implying that a temporary shock will have a permanent effect on city growth. If a random shock is identified as having only temporary effects on city size, then the random growth theory can be rejected. Davis and Weinstein (2002) find that the Allied bombing of Japanese cities during World War II had only temporary effects: most cities returned to their relative positions in the distribution of city sizes within about 15 years from the bombing. This strongly strikes at random growth theory. Brakman et al. (2004) and Bosker et al. (2008) apply similar methodology and also find that city growth in Eastern Germany follows a random walk but city growth in Western Germany does not, suggesting that different post-war economic systems may have played an important role in shaping urban growth dynamics.

Testing the persistence of a random shock effect on city size boils down to testing for the stationarity or unit roots of city sizes. If city size is detected to be non-stationary, then the cointegration approach can be applied to investigate parallel growth between cities.

Let $\ln P_{it}$ denote the logarithm of the population of city i at time t; then, the simplest specification for a stationarity test is to assume that city size is a first-order autoregressive (AR(1)) process

\ln P_{it} = ϕ_{i} \ln P_{i, t - 1} + ε_{it}

(2)

where $ϕ_{i}$ is the first-order autocorrelation coefficient; and $ε_{it}$ is the random shock at time t.

The Dickey–Fuller test for non-stationarity (Dickey and Fuller, 1979) of population levels takes the form

Δ \ln P_{it} = γ_{i} \ln P_{i, t - 1} + ε_{it}

(3)

where, $γ_{i} = ϕ_{i} - 1$ ; and $Δ$ indicates the first-order differencing.

The null hypothesis is non-stationarity: $ϕ_{i} = 1$ . If $ϕ_{i} < 1$ , city population will converge to a constant in the steady state. Since the conclusion of the unit root test is sensitive to the specification of the model (augmented Dickey-Fuller (ADF) test), we specify the model in the general form, including a drift $α_{i}$ and a linear, deterministic time trend $β_{i} t$

\begin{matrix} Δ \ln P_{it} = α_{i} + β_{i} t + γ_{i} \ln P_{i, t - 1} \\ + \sum_{j = 1}^{k_{i}} ρ_{ij} Δ \ln P_{i, t - j} + ε_{it} \end{matrix}

(4)

where, $k_{i}$ is the number of lagged difference terms for city i. We choose the optimal lags using Bayesian information criterion (BIC).⁴

To make sure that the stationarity tests are reliable and robust, we also apply the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test (Kwiatkowski et al., 1992). The null hypothesis of the KPSS test is that the underlying time-series is trend-stationary, which is complementary to the ADF test. The KPSS test selects the optimal lag length automatically with Newey–West methodology. If the results from both the ADF and KPSS tests agree with each other, then we will be confident of the stationarity of city sizes.

Since China’s economic reform and transition to a market economy may have had very different impacts on city size dynamics during different time-periods, it is worth considering structural change in the trajectory of city sizes. A methodology testing both unit root and endogenous structural break point simultaneously is the Zivot–Andrews (ZA) test (Zivot and Andrews, 1992). The null hypothesis of the ZA test is that the underlying time-series $\ln P_{it}$ has a unit root with drift $μ$

\ln P_{it} = μ + \ln P_{i, t - 1} + ε_{it} .

(5)

The alternative hypothesis is that $\ln P_{it}$ is trend-stationary with a one-time break in the trend occurring at an unknown point of time. Specifically, suppose that $\ln P_{it}$ has one structural change in the level, then, the regression equation for testing unit root is

\begin{matrix} Δ \ln P_{it} = μ + θ D U_{t} (λ) + β t + γ \ln P_{i, t - 1} \\ + \sum_{j = 1}^{k} ρ_{j} Δ \ln P_{i, t - j} + ε_{it} \end{matrix}

(6)

where, $λ = t / T$ is the location of the break point and is chosen to give the least favourable result for the null hypothesis (5) and $D U_{t} (λ) = 1$ if $t > T λ$ . Again, BIC is applied to select the optimal lag in the ZA test.

The three types of unit root testing results are reported in section 4.2.

2.3 Testing Parallel Growth of Cities

Even if city sizes evolve in a non-stationary way, they still could move together as city growth is affected by many common factors, such as national or regional macroeconomic factors. A special case is when cities grow parallel in the long run but deviate in the short run, as the hybrid urban growth theory predicts. Parallel growth of cities also implies that their population levels may move together with the national urban population. Sharma (2003) uses the Indian census data from 1901 to1991 and conducts unit root and cointegration tests. She concludes that city growth in the long run may be parallel to the national urban growth, but in the short run growth may deviate from the expected long-run rate of growth due to exogenous shocks.

However, two issues remain in Sharma’s study. First, she does not consider the impact of structural changes on city sizes. The period of rapid industrialisation and urbanisation in each country is also accompanied by dramatic changes in economic structure and policy regime, which may have had persistent effects on later urban growth and different impacts on different cities. Therefore, structural changes should also be taken into account.

Secondly, Sharma specifies a strong assumption that the time trend of growth is zero or negligible to generate parallel growth. Specifically, she details the following model to test parallel growth

\ln P_{it} = α_{i} + δ t + β_{i} \ln P_{jt} + ε_{it}

where, $P_{it}$ denotes the population of city i at time t.

She argues that $β_{i} = 1$ implies parallel growth. This argument is true only if the time trend $δ$ is zero or very small. To see this, let us assume that $\ln P_{it}$ is a general AR(1) process with a unit root, a drift and a time trend

\ln P_{it} = α_{i} + β_{i} t + \ln P_{i, t - 1} + ε_{it} .

The expected growth rate $E (g_{it})$ at time t is

E (g_{it}) = E (\ln P_{it} - \ln P_{i, t - 1}) = α_{i} + β_{i} t .

By the same token, the expected growth rate of city j at time t is

E (g_{jt}) = E (\ln P_{jt} - \ln P_{j, t - 1}) = α_{j} + β_{j} t .

The random parallel growth requires that, for any t,

α_{i} + β_{i} t = α_{j} + β_{j} t .

Therefore, parallel growth implies the same time trend across all cities, regardless of the magnitude of the time trend.

We propose using a city size ratio test that does not require the small time trend assumption and is even simpler than the cointegration method. To demonstrate our approach, let us explain the cointegration method first. The long-run equilibrium relationship between two city sizes with parallel growth is

\ln P_{it} = α_{i} + \ln P_{jt} .

If both $\ln P_{it}$ and $\ln P_{jt}$ are unit root processes, the regression model for the cointegration test can be specified as

\ln P_{it} = α_{i} + γ \ln P_{jt} + ε_{it}

(7)

where, $P_{jt}$ is the population level of a chosen reference city.

If $\ln P_{it}$ and $\ln P_{jt}$ are cointegrated and $γ \neq 1$ , then city i grows at a rate different from the reference city j. If $\ln P_{it}$ and $\ln P_{jt}$ are cointegrated and $γ = 1$ , then the two cities grow parallel at the same expected long-run rate. While standard methods of cointegration testing are applicable here, a more convenient way is to test whether the logarithm of the ratio of two city sizes is stationary or not. To see this, suppose that two city sizes grow parallel; then, equation (7) can be rewritten as

\ln (P_{it} / P_{jt}) = α + ε_{it}

(8)

meaning that parallel growth of two cities requires that the ratio of the two city sizes be stationary around a constant. Furthermore, if city i grows faster than city j by a constant rate, the relationship will be characterised by a linear time trend

\ln (P_{it} / P_{jt}) = \ln P_{it} - \ln P_{jt} = α + β t + ε_{it} .

(9)

We can apply the same unit root test techniques in section 2.2 to test whether the transformed time-series data $\ln (P_{it} / P_{jt})$ is (trend) stationary. Another advantage of using equation (9) to test parallel growth is that we can take into account structural change in either or both cities under testing.

We test parallel growth for all city pairs whose sizes are non-stationary but focus on cities with the same location-specific characteristics, such as same region or same major natural resource endowment. Results from estimating equation (9) for some city pairs with the same locational fundamentals are presented in section 4.3.

3. Data

Before introducing the data, let us first summarise our testing methodologies based on the four urban growth theories. We will test whether Chinese city sizes obey Zipf’s law and whether the distribution is stable over time; if not, we will reject the random urban growth theory and parallel growth. We will also test whether Chinese city sizes follow Gibrat’s law or unit root processes; if yes, we will proceed to test for parallel growth. For the parallel growth test, we use a modified cointegration test strategy shown in equation (9).

We use the Chinese city size data from 1984 to 2006 drawn from each year’s China Urban Statistical Yearbook. Here, a city is defined as ‘city proper’, including both inner city area and suburban areas but excluding independent suburban counties.⁵ The city population is defined as the number of non-agricultural population in an urban area of a city (by permanent residence) at the year’s end. Non-agricultural population is defined as those who engage in non-agricultural vocations and their dependents.

Small sample size, especially in a short time-period, is a notorious problem in standard time-series analysis, especially for the stationarity test and cointegration analysis.⁶ An obvious remedy is to expand the time-span of the dataset. However, we are aware that before 1984 China employed different definitions of the urban population and annual data are not available for the majority of cities. Many studies on long-run urban growth use decade data (such as Sharma, 2003; Henderson and Wang, 2007), but such data by region or by city are not available in China. To deal with the small sample size problem and to date the structural break point precisely, we interpolate annual city size data and disaggregate them into quarterly data.⁷ Of the two commonly used interpolation methods—the quadratic method and the cubic spline method—we use the second, as recommended by Baxter (1998). The overall results from the transformed quarterly data are very similar to those from the original annual data except that in some cases the break point dates are different; therefore, in this paper we report only the results from annual data but the results from quarterly data are available on-line.⁸

Chinese cities are usually classified into five size categories according to their population: small, medium, large, extra large and super large-sized cities with populations less than 200 000, between 200 000 and 500 000, between 500 000 and 1 000 000, between 1 000 000 and 2 000 000, and above 2 000 000 persons respectively. By region, Chinese cities traditionally are assigned to one of the following three regional categories: Eastern, Middle and Western. A more disaggregated regional classification consists of seven sub-regions: North-eastern, North-western, South-western, Northern, Eastern, Southern and Middle China.

During the transition to a market economy and opening to the world, the Chinese government has favoured a small number of cities to implement reform and open policies, including five cities in special economic zones and 16 open coastal cities (starting from 1980) which we term ‘policy cities’. For reasons that will soon be clear, we are also interested in another type of city—tourism cities. In 1998, the National Tourism Administration of China nominated the first list of ‘Best Tourism City’, including 54 cities.

4. Results

4.1 Time Variations in the Zipf’s exponent

The random growth theory states that, if city growth processes follow Gibrat’s law, then the steady state distribution will obey Zipf’s law. Therefore, if the Chinese city size distribution is not consistent with Zipf’s law, we would be able to cast doubt on the random growth theory. Moreover, if the distribution is not stable over time, parallel growth can be rejected.

Table 2 reports the results from estimating equation (1) using the OLS method. The coefficient $β$ in the third column is estimated using the full sample. The power exponent has been increasing from less than one in 1984 to much greater than one in 2006, implying that the overall city size distribution becomes close to and, finally, more even than what Zipf’s law predicts. In 19 out of 23 years, an F test rejects the null hypothesis that $β$ is equal to 1. Column 8 presents Zipf’s exponent for the balanced panel, which includes 259 cities existing since 1984. The pattern is very similar to the full sample, suggesting that the balanced panel sample is representative.

Table 2.

Time variations in Zipf’s exponent

	All cities		Cities with population >200 000		Cities with population >500 000		Balanced panel
Year	Number of cities	$β$	Number of cities	$β$	Number of cities	$β$	$β$
1984	295	0.891	131	1.266	50	1.463	0.879
1985	324	0.856	146	1.275	52	1.459	0.895
1986	342	0.857	149	1.260	54	1.447	0.904
1987	382	0.884	158	1.260	55	1.445	0.904
1988	434	0.927	168	1.273	58	1.440	0.919
1989	450	0.932	175	1.282	58	1.429	0.925
1990	468	0.902	177	1.253	60	1.381	0.929
1991	479	0.923	182	1.289	61	1.423	0.934
1992	514	0.950	201	1.302	62	1.433	0.941
1993	569	0.981	228	1.343	69	1.433	0.942
1994	622	1.007	248	1.377	73	1.453	0.961
1995	640	1.023	267	1.383	76	1.451	0.965
1996	665	1.038	273	1.388	78	1.454	0.968
1997	666	1.032	285	1.389	81	1.452	0.978
1998	665	1.047	291	1.383	86	1.457	0.988
1999	665	1.075	302	1.380	86	1.423	0.990
2000	661	1.025	293	1.364	92	1.412	0.989
2001	662	1.026	304	1.357	105	1.430	0.993
2002	656	1.001	311	1.339	108	1.448	0.972
2003	653	0.986	324	1.269	122	1.416	0.897
2004	655	1.024	348	1.290	130	1.412	0.945
2005	659	1.117	581	1.473	363	1.939	1.073
2006	659	1.129	586	1.480	368	1.936	1.072

Notes: The balanced panel includes 259 cities. All regressions have a good fit; in the full sample, 19 out of the 23 p-values for testing the null hypothesis $H_{0} : β = 1$ are less than 0.1.

Since many empirical studies confirm that the exponent is sensitive to the sample choice and is close to one for the upper tail of size distribution (Rosen and Resnick, 1980; Eeckhout, 2004), we also estimate the model by selecting only cities of a size greater than a certain threshold. The fifth and seventh columns report the results for cities of a size greater than 200 000 and 500 000 respectively. The estimated power exponents now are significantly greater than one. The values of $β$ increase when we move the cut-off to the upper tail, suggesting that larger cities distribute more evenly than predicted by Zipf’s law. One possible explanation is that the Chinese government has implemented policies that increasingly restrict the size of large cities since the 1980s. In 1980, the central government began the urban planning policy that controls the sizes of large cities, appropriately develops medium cities and actively develops small cities; however, in 1990 the Urban Planning Law enacted a policy that strictly controls the sizes of large cities and appropriately develops medium and small cities. Another interpretation is that the current size distribution may not be in a steady state. In fact, the overall size distribution of Chinese cities has been evolving. In 2007, the urbanisation rate in China was only 42.2 per cent, while the urbanisation rate of developed countries was above 70 per cent.⁹ China is still in a period of rapid urbanisation and the distribution of city sizes will keep evolving. Historically, the power exponent shows a U-shaped pattern in many countries (Parr, 1985). Cross-country studies also show that Zipf’s exponent is significantly different from one (Nitsch, 2005; Soo, 2005). In addition, using the Chinese city size data up to 1999, Anderson and Ge (2005) find that Chinese city sizes are better described by a log-normal distribution. Given all these pieces of mixed evidence, we tend not to make a conclusion about the random growth and parallel growth by looking at only the evolution of the power exponent.

4.2 Non-stationarity in City Sizes

To allow for the longest city size time-series data possible, we select cities in the balanced panel from 1984 to 2006, but exclude cities at the county level for the unit root tests.¹⁰ Overall, both the ADF and KPSS tests show that 153 out of 210 cities have unit roots, suggesting that the sizes of most cities are not stationary. As a demonstration, Table 3 reports the unit root test results for some selected cities: panel 1 presents the cases where the ADF and KPSS tests are not consistent; panel 2, the cases that are consistently stationary; and panel 3, the cases that are consistently non-stationary. It is worth noting that, in panels 2 and 3, when both the ADF and KPSS tests agree with each other, the ZA test tends to be consistent too.

Table 3.

Testing stationarity of city sizes

City	ADF test	KPSS test	ZA test	Break point date
Cities with inconsistent testing results
Anyang	−2.61	0.12	−6.75^*	2003
Baotou	−3.34	0.10	−3.53	2002
Binzhou	−1.90	0.08	−2.39	1998
Cities that are trend stationary
Daqing	−4.51^*	0.11	−6.12^*	1988
Deyang	−4.43^*	0.07	−5.03^*	1994
Xiangfan	−4.03^*	0.10	−6.64^*	2002
Cities that are non-stationary
Shanghai	−1.31	0.18^*	−2.55	1990
Beijing	−3.32	0.16^*	−3.90	1990
Chongqin	−0.95	0.17^*	−2.47	2004

Notes: For the ADF test, the null hypothesis is non-stationarity; the critical value for the 5 per cent level is -3.46; ^* indicates rejection of non-stationarity at least at the 5 per cent level. For the KPSS test, the null hypothesis is trend-stationarity; the critical value for the 5 per cent level is 0.15; ^* indicates rejection of stationarity at least at the 5 per cent level. For the ZA test, the null hypothesis is non-stationarity without a break point and the alternative hypothesis is trend-stationarity with an endogenous break point; the critical value for the 5 per cent level is −4.80; ^* indicates rejection of non-stationarity without a break point at least at the 5 per cent level.

It is well known that the power of a unit root test based on a single equation is poor, especially when the time-series is short. Panel unit root tests with a large N can improve the power. The time length of our dataset is not very long, but the number of cities is large enough. For a robustness check, for the 153 cities detected with unit root, we also conduct the Im–Pesaran–Shin panel unit root test (Im et al., 2003) and the Levin–Lin–Chu panel unit root test (Levin et al., 2002). Both test results cannot reject the null hypothesis that all cities in the panel have unit roots (the critical values of these two test statistics are 16.3 and 17.0 respectively).¹¹

We also conduct unit root tests for the population growth rate of the same set of cities. The null hypothesis of unit root of growth rate is rejected for all cities. Therefore, we conclude that the sizes of most of the cities in our sample are stochastic processes integrated with order one (I(1) processes).

Two inferences can be drawn from the unit root tests. First, there exists no steady state size for the majority of cities. This rejects the conditional convergence hypothesis. Secondly, since for the majority of cities, city size is a non-stationary process and the rate of growth is a stationary process, we cannot reject the random growth theory as confidently as do Davis and Weinstein (2002).

4.3 Results for Parallel Growth

After confirming that most of the city size time-series has a unit root, we continue to test the stationarity of equation (9). We first run the test for all pairs (11 628 pairs in total for the 153 cities).¹² Because the ADF and KPSS tests are not always consistent, the majority (about 80 per cent) of cities do not appear to grow parallel; however, cities with the same location-specific characteristics, such as the same region, same major natural resource endowment and same policy intervention, are more likely to show parallel growth. Therefore, as a representative demonstration, we present the results for seven groups of cities with similar locational fundamentals: tourist cities, capital cities, coastal cities, cities in the Yangtze River delta region and the Pearl River delta region, and cities in the South-western region and the North-eastern region.

Our classification is rather suggestive and intuitive as it is difficut to delineate and quantify locational fundamentals. In the literature, factors that represent locational fundamentals include both the natural characteristics of a city, such as latitude, rainfall and distance to coastline, and produced characteristics such as education infrastructure.

Table 4 provides the ADF, KPSS and ZA test results for testing parallel growth for the seven groups of cities. To save space, for each group, we present only three cities that grow parallel with the reference city of each group.

Table 4.

Testing for parallel growth of city sizes

Cities	ADF test	KPSS test	ZA test	Break point date
Group 1: tourism cities (Guilin as reference city)
Hangzhou	−1.33	0.16^*	−10.79^*	2001
Suzhou	−1.54	0.16^*	−10.33^*	2001
Xi’an	−3.31	0.10	−4.52	1995
Group 2: capital cities (Guangzhou as reference city)
Beijing	−2.32	0.12	−7.46^*	2000
Shanghai	−2.11	0.11	−6.45^*	2000
Shenyang	−2.52	0.15^*	−8.81^*	2000
Group 3: coastal cities(Shantou as reference city)
Sanya	−1.54	0.16^*	−14.64^*	2003
Shenzhen	−1.47	0.17^*	−7.32^*	2003
Xiamen	−1.91	0.15^*	−20.52^*	2003
Group 4: Yangtze River delta cities (Shanghai as reference city)
Hangzhou	−1.20	0.16^*	−11.56^*	2001
Nantong	−2.21	0.12	−7.40^*	2003
Suzhou	−1.40	0.16^*	−11.87^*	2001
Group 5: Pearl River delta cities (Jiangmen as reference city)
Guangzhou	−2.50	0.14	−8.45^*	2000
Shenzhen	−1.39	0.16^*	−7.44^*	2004
Zhongshan	−0.59	0.13	−13.68^*	2002
Group 6: south-western cities (Panzhihua as reference city)
Deyang	−2.14	0.08	−10.21^*	2001
Neijiang	−1.21	0.17^*	−7.08^*	1990
Luzhou	−2.36	0.13	−5.89^*	1996
Group 7: north eastern cities (Liaoyang as reference city)
Anshan	−1.38	0.12	−5.59^*	1994
Benxi	−1.61	0.10	−5.23^*	1994
Fuxin	−2.97	0.11	−6.54^*	1994

Notes: See notes of Table 3.

Tourism cities obviously have location-specific fundamentals—natural tourist attractions such as beautiful lakes, beaches and mountains. Using Guilin as the reference city, we conduct a parallel growth test for the 54 cities in the first list of the best tourism cities nominated by the National Tourism Administration of China in 1998. Panel 1 of Table 4 reports the test statistics for three cities that grow parallel with Guilin. It is interesting to see that Hangzhou and Suzhou satisfy the parallel growth condition, with one level shift occurring at the same date (2001 in the annual data and the fourth quarter of 2000 in the quarterly data). These two cities are geographical neighbours in Zhejiang province but Hangzhou is a capital city. Figure 1 plots the logarithm of population levels for Hangzhou, Suzhou and Guilin. Visually, we can see that the three cities grow parallel if we ignore the break point occurring in 2001 for Hangzhou and Suzhou. Figure 2 plots the logarithm of population ratio for Hangzhou and Suzhou compared with Guilin, and the patterns are remarkably similar. Taken together, we tentatively conclude that natural tourist attractions might play an important role in long-run growth for tourism cities.

Figure 1.

Population levels of three tourism cities.

Figure 2.

Logarithm of population ratio.

In the capital cities category, a provincial capital city or a municipality directly under the central government can be considered a ‘policy city’ in the sense that it receives many favourable economic policies and resource allocation from central and local governments, even more so during the planned economy period. If we treat location-specific policies as general locational fundamentals, then these capital cities have advantages comparable with the natural tourist attractions in tourism cities. There are 32 capital cities in China. Using Guangzhou as the reference city, panel 2 of Table 4 shows that Beijing, Shanghai and Shenyang grow parallel with Guangzhou.

Coastal cities are harbour cities and enjoy a natural location advantage in transport. They are also the cities that started their transition in the early stage of China’s economic reform and have received favourable government intervention. Choosing Shantou as the reference city, panel 3 shows that Sanya, Shenzhen and Xiamen grow parallel with Shantou. We should point out that Shantou, Shenzhen and Xiamen are three of the five cities in the special economic zones.

Yangtze River delta cities have both a transport advantage (near the Yangtze River and the Pacific Ocean) and a policy advantage (receiving favourable open and reform policies); furthermore, this region is a manufacturing industry cluster. Panel 4 shows that Hangzhou, Nantong and Suzhou grow parallel with Shanghai.

Pearl River delta cities enjoy advantages similar to Yangtze River delta cities. They also attract labour-intensive manufacturing firms. Panel 5 shows that Guangzhou, Shenzhen and Zhongshan grow parallel with Jiangmen.

South-western cities are located in less developed south-western China, with a less developed transport network and a large minority population. Taking cities in Sichuan province as examples, panel 6 shows that Deyang, Neijiang and Luzhou grow parallel with Panzhihua when a structural change is taken into account.

North-eastern cities are in a region with traditional heavy industry and mining. Panel 7 of Table 4 shows that Anshan, Benxi and Fuxin grow parallel with Liaoyang, conditional on a structural change. Anshan, Benxi and Liaoyang are well known for their steel industry; Fuxin, for its coal mining industry. This panel suggests that natural resource endowment might play an important role in the long-run growth of ‘resource cities’.

After examining these seven groups of cities, we find little evidence for simple, deterministic, parallel growth among Chinese cities. However, after taking into account growth trends and structural change in city sizes, we find that some pairs of cities with common group characteristics do grow parallel in the long run. Those common characteristics can be summarised as generalised locational fundamentals, including natural tourist attractions (tourist cities), transport advantages (coastal cities), natural resource endowments such as coal (resource cities) and similar government interventions (policy cities). If we extend the definition of locational fundamentals to any location-specific amenities, including natural resources, transport accessibility and government interventions, the identified parallel growth patterns of sub-groups lend some support to the locational fundamentals theory.

Our findings are consistent with a few studies that confirm the important role that locational fundamentals play in urban growth. For example, Beeson et al. (2001) use US census data from 1840 to 1990 and find that locational fundamentals, including both natural characteristics such as access to water transport and produced characteristics such as educational infrastructure, can explain cross-county variations in population in 1990 as well as in 1840; furthermore, produced characteristics in 1840 have a significant, persistent effect on population growth over the 150 years. Ayuda et al. (2010) find that many European countries show a similar spatial pattern, with population concentration persisting throughout the industrialisation and post-industrialisation periods in locations where population concentration in the pre-industrialisation period was already high. They also confirm this pattern by constructing data for Spain for the period 1787–2000 and regressing provincial population density on a set of natural advantage variables such as altitude, rainfall and coastal location. They find that, not only do location fundamentals affect population concentration contemporaneously, but also initial population concentration reinforces later population growth, suggesting that not only are initial natural advantages crucial to urban growth, but also that such an advantage may help to generate forces of increasing returns to scale in a later period. González-Val and Olmo (2010) and Bleakley and Lin (2010) also conclude that both locational fundamentals and increasing-returns-to-scale forces can explain the growth of income and city population in American cities and in portage sites in the south-eastern US.

Another point worth noting is that structural changes occur to the growth path of many cities. This is not surprising since transition to a market economy and rapid urbanisation in China are accompanied by dramatic institutional changes and government interventions, such as the Western region development plan and economic policies favourable to the Pearl River delta and Yangtze River delta regions. Our research shows that it is important to take into account structural changes when studying city size distribution and city size dynamics for developing countries.¹³

Finally, our findings provide a possible linkage to reconcile random growth theory and locational fundamental theory. Davis and Weinstein (2002) conclude that the transitory effect of large shocks to city size is a rejection of random growth theory and supportive evidence for the locational fundamental theory. Our findings suggest that city size growth can be random in the short run but that the long-run path may be determined by locational fundamentals.

5. Conclusion

Four types of urban growth theories are closely related to parallel growth: the endogenous growth theory predicts deterministic, parallel growth; the random growth theory and hybrid growth theory imply random, parallel growth in the long run; and the locational fundamentals theory suggests that cities with similar locational fundamentals grow parallel. However, little empirical evidence is available so far for parallel growth. This paper focuses on identifying the parallel growth patterns of Chinese cities. Given the fact that China is still in a period of rapid urbanisation, even though we apply rigorous time-series econometric techniques, we can only tentatively conclude that overall Chinese city growth does not follow parallel growth. However, once we allow for growth trend and structural change, we find that a small number of cities with certain common group characteristics, such as similar location advantages or policy regime, do exhibit parallel growth. For example, the super large-sized capital cities of Guangzhou and Shanghai grow parallel; the north-eastern steel industry cities of Anshan and Liaoyang grow parallel; and Shenzhen and Xiamen, both in special economic zones, also grow parallel.

Our findings suggest that location fundamentals may have a persistent impact on urban growth. Furthermore, if we extend the concept of locational fundamentals to any location-specific factor, including natural resource endowment, transport accessibility and government interventions, then our findings provide some evidence supporting the locational fundamentals urban growth theory. The future extension would be to collect locational fundamental characteristics data for Chinese cities and apply strategies such as those in Beeson et al. (2001) to test directly the locational fundamentals theory.

Footnotes

Acknowledgements

The authors would like to thank Andrew Cumbers, two anonymous referees, Richard Arnott, Michel Dimou, Yuming Fu, Li Gan, Arthur Lewbel, Zhijie Xiao, Yves Zenou, the participants of the 2006 Chinese Economist Society annual conference, the 2006 AREUEA–Asian Real Estate Society international conference, the 2007 European Regional Science Association conference and the seminar at Southwestern University of Finance and Economics for very helpful comments.

Notes

Funding Statement

Zhihong Chen acknowledges financial support from the National Natural Science Foundation of China (grant 71101030). Dayong Zhang acknowledges financial support from “Project 211(Phase III)” of the Southwestern University of Finance and Economics, Chengdu.

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