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Economics Letters

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economics tetters

A nonparametric analysis of the growth process oflndian cities

Jeff Lucksteada-*, Stephen Devadossb

CrossMark

a University of Arkansas, United States b University of Idaho, United States

highlights

We study the growth process of Indian cities between 1991 and 2011. We analyze Gibrat's and Zipf's laws by applying nonparametric estimation. Gibrat's law holds for largest cities in India. The local Zipf exponent is around one and stable.

article info

Article history: Received 2 July 2014 Received in revised form 21 July2014 Accepted 23 July 2014 Available online 4 August 2014

JEL classification:

Keywords: Gibrat's law Growth process Indian cities Local Zipf exponent

abstract

We examine the growth process of the largest cities in India for the post economic reform period 1991-2011 to analyze Gibrat's and Zipf's laws by applying nonparametric estimation. The results from stochastic kernel, contour plots, and expected growth rate and variance conditional on city size establish that Gibrat's law holds for largest cities in India, i.e., city growth is independent of population size, and the local Zipf exponent is around one and stable. Gibrat's law is also confirmed by the parametric regression of the aggregate relationship of the growth rate on city size.

Published by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction

Two empirical regularities - Zipf's law and Gibrat's law - are observed in city size data (Eeckhout, 2004). According to Zipf's law, cities are inversely proportional to their ranks.1'2 Gibrat's law asserts that city growth rates are independent of city sizes, which is also known as the proportionate growth process because cities, whether large or small, on average grow at similar rates. While

* Corresponding author.

E-mail addresses: jluckste@uark.edu (J. Luckstead), devadoss@uidaho.edu (S. Devadoss).

1 Zipf's law is generally found in the upper tail (large cities) of the size distribution.

2 City size distribution is generally analyzed using parametric estimation based on several distributions such as Pareto, lognormal, general Pareto, and q-

Exponential (Rosen and Resnick, 1980; Krugman, 1996; Anderson and Ge, 2005; Gangopadhyay and Basu, 2009; Luckstead and Devadoss, 2014).

http://dx.doi.org/10.1016/j.econlet.2014.07.022

0165-1765/Published by Elsevier B.V. This is an open access article underthe CC BY-]

parametric methods provide useful tests for Zipf's law,3 they do not provide a robust test for Gibrat's law. Gabaix (1999) laid the foundation to connect the growth process and Gibrat's law to Zipf's law. Based on Gabaix's finding, Ioannides and Overman (2003) empirically verified using nonparametric estimation of the growth process that Gibrat's and Zipf's laws hold for the largest US cities.4 This study examines the growth process for the largest Indian cities by applying nonparametric estimation. India is the second most populous and largest democratic country in the world. Even though migration from rural to urban area was fairly stagnant until 1990 because of the subsistence nature of the economy, population mobility has increased significantly after the economic reforms starting from the early 1990s. This is evident from the demographic

See Urzua (2000) for the Lagrangian test of Zipf's law. 4 Rozenfeld et al. (2011) analyze size distribution of small cities through clustering and find Zipfs law holds in the United States for cities as small as 12,000.

!-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

13 13.5

14 14.5 15 15.5 Log of CitySize

16 16.5

Fig. 1. Scatter plot of growth rate versus city size.

data of urban/rural population mix of 18%/82% in 1960 but 32%/68% in 2012 (The World Bank, 2014). Because of the economic growth and significant expansion of urban population, it is worth studying the growth process of Indian cities.

2. Proportionate city growth

Gangopadhyay and Basu (2009) and Luckstead and Devadoss (2014) find, based on parametric estimation of the size distribution, Indian cities did not follow Zipfs law before the economic reforms in 1990, which is likely due to very limited migration from rural to urban areas because of a lack of employment opportunities in big cities. However, these studies do find that the Indian city size distribution adheres to Zipfs law after the reforms. We focus on the growth process of India's largest cities to test Gibrat's law and examine Zip's law through the local Zipf exponent for the post reform period.

The data for Indian cities above a population of 300,000 for census years 1991,2001, and 2011 are collected from the Census of India (2014). This data provides 117 population growth rate pairs between 1991 and 2001 and 140 pairs between 2001 and 2011, yielding 257 observations of growth rates and corresponding city sizes. Fig. 1 depicts the scatter plot of the city size versus growth rate on a log-log scale. Visual inspection of this plot suggests that the growth rates do not depend on city sizes.

Next we provide a more rigorous analysis of the relationship between growth rates and city sizes by considering the distribution of growth rates at every city size and the stability of the expected growth rate and variance. We first visually examine the growth process by estimating the stochastic kernel of growth rates conditional on city size and analyzing the corresponding contour plots. For this kernel regression, following Ioannides and Overman (2003) and Eeckhout (2004), we normalize the city size data by computing the difference between city sizes and the mean city size and dividing by the standard deviation for each year. We also apply this normalization to the growth rates. Fig. 2(a) presents the estimated stochastic kernel5 in a three-dimensional graph. The stochastic kernel gives the distribution of growth rates conditional on city sizes, which can be visualized by taking a cross section of the kernel at each city size parallel to the growth rate-kernel plane. For all city sizes, the distribution is fairly constant with a stable mode and variance. The distribution is skewed just below zero, and the ridge line down the center of the kernel is parallel

to the population axis. Cross sections of the kernel at each growth rate parallel to the population-kernel plane illustrate that the probability is fairly constant across city sizes. Fig. 2(b) presents the contour plots, which are a vertical projection of the stochastic kernel. Each contour line represents a constant probability of a given growth rate. The contour lines are both parallel to each other and to the population axis. The stable growth rate distribution and parallel contour plots provide strong evidence that growth rates are independent of city sizes, adhering to Gibrat's law. Both figures show a peak in the kernel just below a population and growth rate of zero and a slight bend in the kernel at a city size above 7 (this is likely due to a lack of observations in the extreme large city sizes), which implies that the proportionate growth may not hold perfectly as elaborated by Gabaix.

Next we consider the relationship between the expected growth rate and city size using nonparametric regression, which does not depend on parametric assumptions on functional form or the error structure:

gi = p (Si) + eu

where gi is the growth rate of city i and p (Si) approximates the true relationship between Si and gi. The Nadaraya-Watson regression estimates of the mean and variance are

E * ( ^ ) gi

p (s) =- and

n ( c )

i=1 v '

±K (^)(gi - P (s))2

i=1 v '

a2 (s) =

i=1 v '

where K (•) is a kernel and h is the bandwidth for all s in the support Si. Nonparametric regression allows us to estimate both the local mean p (s) and variance a2 (s) around point s based on the continuous smoothing function specified by K (•). The local mean and variance estimates provide further insight into the size independence of both mean and variance and thus Gibrat's law. Furthermore, this method estimates the local average growth rate for each city size as opposed to the aggregate relationship offered by parametric methods.

To estimate p and a in Eq. (2), following Ioannides and Overman (2003), we normalize the city size data as city i's share of the total population for a given year and the growth rate as the difference between city i's growth rate and the average growth rate for a given year. Because of the generality of the functional form and error structure, nonparametric regression requires a large number of observations for all ranges of city sizes to obtain unbiased estimates. However, the small number of observations in the extreme upper tail of cities leads to inaccurate estimates and very high variance. Consequently, we restrict the sample by excluding the extreme largest cities that are greater than 2% of the normalized population (which excludes 16 observations) and cities with abnormal normalized growth rates above 200% (which excludes 2 outliers).

Figs. 3(a) and (b) plot the estimated mean and variance6against the normalized population. We also present the 95% confidence bands generated using 500 bootstrap samples performed with replacement. Panel (a) shows that the mean growth rate is constant

5 Forthe stochastic kernel estimation, we apply the Gaussian kernel and calculate the bivariate bandwidth using the L-stage Direct Plug-In method (Magrini, 2007). We thank Stefano Magrini for providing code for estimating the stochastic kernel.

6 For the Nadaraya-Watson regression, we employ the Gaussian kernel and follow Bowman and Azzalini (1997) to calculate the optimal bandwidth h which specifies the scale of smoothing by the kernel K.

Normalized Growth Rate

Fig. 2. Stochastic kernel and contour plots.

0.005 0.01 0.015

Normalized Population

0.005 0.01 0.015

Normalized Population

Fig. 3. Mean growth rate and variance with 95% confidence bands.

around -0.02 across different city sizes. These estimates are well within the confidence bands, confirming the stability of the mean growth rates. This provides statistical evidence that Gibrat's law cannot be rejected for the largest cities in India. Panel (b) illustrates that the variance declines modestly from 0.09 to 0.05 as the city size increases, but lies well within the 95% confidence bands. Furthermore, a straight line of constant variance can be drawn between upper and lower confidence bands, indicating that we cannot reject size independence of the variance. This result provides further statistical evidence that Gibrat's law holds for Indian cities. Our findings of proportionate growth for largest cities are consistent with the nonparametric results of loannides and Overman (2003) for US cities and parametric results of Glaeser et al. (1995) for US cities and Eaton and Eckstein (1997) for cities in France and Japan.

Next we consider the aggregate relationship between growth rate and size by estimating the parametric equation

?ii . 0Pm + Pu .

— = a + ß----h £

where P01 and P11 are city size populations for the census years 2001 and 2011. The estimated results of this equation are <5 = 1.214 (61.149) and j = 6.014E - 09 (0.624), where t-statistics are in parentheses.7The estimated coefficient for j

As in Eeckhout (2004), we perform two robustness checks. First, using the base

year (2001) as the independent variable, we confirm the growth rate does not depend on size, as the estimate of the slope of —4.75e-10 is insignificant with a t-value of —0.046. Second, a regression of the logarithm of gross growth between 2001 and 2011 on size in 2001 yields a coefficient estimate of —8.21e—10 which is still insignificant as the t-value is —0.10.

is insignificant from zero, which clearly indicates that growth rates are independent of the population size, confirming the proportionate growth results of the nonparametric estimation. lt is important to observe that this parametric method estimates the aggregate relationship between growth and size for the entire support of the city size distribution, whereas the nonparametric method estimates the growth rate for each city size over the entire distribution. The intercept provides a net growth estimate of 21.4% for these largest lndian cities between the years 2001 and 2011, which translates into an annual growth rate of 1.96% (computed from (1 + g)10 = 1.214, where g is the annual growth rate). This relatively high growth rate of cities signifies high birth rates in lndia and greater migration from rural to urban areas as a result of economic reforms and employment opportunities in large cities.

3. The local Zipf exponent

Gabaix (1999) established the interconnection between the size distribution and the growth process of cities by showing that if Gibrat's law holds, the size distribution converges to the Zipf law. City sizes follow Zipf's law if the normalized city size S exhibits the Pareto CDF

F (S) = 1 -

where c is the scale parameter and the shape parameter 0 is equal to 1. Zipf's law is generally found in the upper tail of city sizes. Gabaix further extended the analysis to account for cases where cities grow randomly with expected growth rate and standard deviation as shown in:

= p(St) dt + a (St) dWt,

0.005 0.01 0.015 0.02

Normalized Population

Fig. 4. Local Zipf exponent with 95% confidence bands.

where p (St) and a (St) are respectively the expected growth rate and standard deviation as a function of size, and Wt is a geometric Brownian motion. The local Zipf exponent Q (S) can then be expressed, using the Kolmogorov equation, as8

Q(S) = 1 - 2

p(S) , 9a2 (S)/a2 (S)

a 2 (S)

where p (S) is normalized as the difference between the growth rate of city size S, g (S), and the mean growth rate, g .The local Zipf exponent is therefore a function of both the mean and standard deviation, which can explain the deviation of Q (S) from 1. A high relative growth rate implies the distribution will decay less quickly causing Q to be below one, and a large variance leads to a higher range of city sizes and a flatter distribution. Ioannides and Overman (2003) empirically implemented Gabaix's theoretical results by applying the nonparametric estimation of the mean and variance given in (2) to calculate the local Zipf exponent (6). Thus, the local Zipf exponent for size s can be computed using the estimated results of p (s) and a (s).

Fig. 4 plots local Zipf exponent and confidence intervals against normalized population. The local Zipf exponent is above one for smaller cities, hovers around one for medium-range cities, and is below one for large cities. However, for all these cities, the local Zipf exponent falls within the 95% confidence bands. Thus, we can conclude that Indian cities follow Gibrat's law and the local Zipf exponent is fairly stable, confirming Zipf's law for the growth

process of Indian cities. These results also corroborate the findings of Luckstead and Devadoss (2014) who demonstrate that the size distribution of the largest cities in India follows Zipfs Law for the years after the economic reforms in the early 1990s.

4. Conclusions

This study considers the growth process of the largest cities in India to analyze Gibrat's and Zipf's law by applying nonparametric estimation. The results from stochastic kernel, contour plots, and expected growth rate and variance conditional on city size establish that Gibrat's law holds for largest cities in India, i.e., city growth is independent of population size, and local Zipf exponent hovers around one and is stable. Gibrat's law is also confirmed by the parametric regression of the aggregate relationship of the growth rate on city size.

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