Economics Letters 135 (2015) 12-14

Contents lists available at ScienceDirect

Economics Letters

journal homepage: www.elsevier.com/locate/ecolet

economics tetters

Growth process of U.S. small cities

Stephen Devadossa, Jeff Lucksteadb-*

CrossMark

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

highlights

• Examine Gibrat's law for U.S. lower-tail cities.

• Uses data for the census years 2000 and 2010.

• Utilizes Pareto Tails lognormal distribution to determine the threshold city size.

• The results show that U.S. small cities generally follow Gibrat's law.

article info

Article history: Received 22 April 2015 Received in revised form 14 June 2015 Accepted 19 July 2015 Available online 29 July 2015

JEL classification:

Keywords: Growth process Gibrat's law Lower tail United States

abstract

This study analyzes the growth process of lower tail small U.S. cities for 2000 and 2010 census data using stochastic kernel, contour plot, and nonparametric regression. The results show that Gibrat's law hold for small cities.

© 2015 The Authors. 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/4.0/).

1. Introduction

A few studies have investigated whether large cities exhibit Gibrat's law—growth of cities is independent of their size, also referred to as proportionate growth.1 loannides and Overman (2003) used nonparametric estimation to analyze the growth process of

* Corresponding author.

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

1 Large cities have been shown (see for example Krugman, 1995 and Gabaix,

1999) to follow the empirical regularity of Zipfs law, i.e., the largest city is twice the size of second largest city and thrice the size of the third largest city, and so on. However, small cities in the lower tail do not follow Zipfs law, but do exhibit Pareto behavior as confirmed by Reed (2001) who observed strong power law behavior in US lower tail settlement in 1998. Reed (2002) proposed the double Pareto-lognormal distribution (DPLN) and applied it to settlements in West Virginia and California in the United States and Cantabria and Barcelona in Spain. His results showed Power law behavior in both tails. Similar Power law behavior for US lower tail cities are also found by Giesen et al. (2010); Giesen and Suedekum (2014), and González-Val et al. (2015).

the largest US cities and concluded that these cities conform to Gibrat's law. Luckstead and Devadoss (2014a) have also shown that the largest cities in India adhere to proportionate growth. Luckstead and Devadoss (2014b) have provided strong evidence for Gibrat's law for the upper-tail cities in the world. Eeckhout (2004) examined the full sample of all US places using the 2000 US census data and found the parametric distribution of all US cities to be lognormal. Furthermore, his analysis of the growth process of US cities using 1990 and 2000 census data showed that these cities do follow Gibrat's law. Gonzalez-Val (2010) analyzed Zipfs and Gibrat's law for all US incorporated places for the period 1900-2000. He confirmed Gibrat's law with constant mean growth rate but noncon-stant variance of the growth rate and Zipfs law only for a restricted sample.2 Desmet and Rappaport (2015) found that for the United

2 Partridge et al. (2008) investigated the distance and technology effect of urban agglomeration on population growth in hinterland US counties. Their results show a strong inverse relationship between distance and population growth in these counties, highlighting the effects of technology on spatial distribution.

http://dx.doi.Org/10.1016/j.econlet.2015.07.018

0165-1765/© 2015 The Authors. Published by Elsevier B.V. This is an open access article underthe CC BY-NC-ND license (http://creativec0mm0ns.0rg/licenses/by-nc-nd/4. 0/).

S. Devadoss,J. Luckstead / Economics Letters 135 (2015) 12-14

States, Gibrat's law gradually emerged but was not fully reached, and consequently, the proportionate growth was not the force behind the lognormal distribution of city sizes.

However, such growth process has not been analyzed specifically for the small cities in the lower tail.3 Because both the 2000 and 2010 US Census report data for all places, including all lower-tail cities, it is possible to analyze the growth process. Therefore, the purpose of this study is to examine whether the growth process of US lower-tail cities for the census years 2000 and 2010 follow Gibrat's law.

2. Methodology

To test whether growth rates of the lower tail cities are independent of their sizes, we analyze the stochastic kernel of growth rates conditional on city sizes and then implement nonparametric regression to analyze the conditional mean and variance of growth rates.

Population Census data for US cities for the years 2000 and 2010 is obtained from US Census Bureau (2014). We utilized the Pareto Tails and Lognormal Body distribution4 to estimate the threshold city size for the lower tail, which is log (158) = 5.06 for 2010. We specify the lower tail to be all US cities below the cutoff point. Based on this cutoff, there are 2501 population-growth rate pairs.

For the stochastic kernel, the growth rate for each city is normalized by subtracting the mean and dividing by the standard deviation, as defined in loannides and Overman (2003). The same normalization is applied to city sizes. Fig. 1 presents the scatter plot of the normalized growth rates against the normalized city sizes.5 Inspection of this figure suggests that growth rates are independent of city sizes. We also examine the growth process using a stochastic kernel and contour plot. Fig. 2 depicts the stochastic kernel of the growth rates conditional on city sizes. We use the Gaussian kernel and the L-stage Direct Plug-In method to calculate the bivariate bandwidth (Magrini, 2007). The stochastic kernel gives a three-dimensional representation of the distribution of growth rates for each city size. That is, for a given city size, the cross section of the kernel parallel to the growth rate axis gives the distribution of growth rates at the specified city size. Because the kernel generally runs parallel to the population axis, there is some evidence of Gibrat's law; however, because the distribution of growth rates changes for different city sizes, this figure does not definitively confirm Gibrat's law. The contour plot of the stochastic kernel, given in Fig. 3 also does not provide strong evidence for Gibrat's law because the constant-probability-lines are generally parallel to the population axis, but they are not straight lines. While the scatter plot indicates proportionate growth, the stochastic kernel and contour plots do not provide strong evidence for Gibrat's law. A reason for the stochastic kernel and contour plot not establishing a clear picture for Gibrat's law is that there are multiple observations for a given city size in the lower tail, which can cause inconsistency in the growth rate distribution at different city sizes. Consequently, for a more formal test, we next consider nonparametric regression of both the mean and variance.

Nonparametric regression is used to analyze Gibrat's law because it provides unique estimates of mean growth rates and variances within the range of city sizes in the data. lf these means

-3 -2-1 0 1

Normalized Population

Fig. 1. Scatter plot of normalized growth rate and population.

Fig. 2. Stochastic kernel of growth rate conditional on city size.

Normalized Growth Rate

3 Since the 1990 Census did not report data for "census designated places'', many of the cities in the lower tail were not included in the analysis by Eeckhout (2004).

4 This distribution extends loannides and Skouras's (2013) lognormal-uppertail Pareto by explicitly modeling the lower-tail Pareto and endogenously estimating the lower tail threshold city size, i.e., the transition point (see Luckstead and Devadoss, 2015).

5 One outlier with a growth rate of 3500% was removed from the sample.

Fig. 3. Contour plot of the stochastic kernel.

and variances are constant over the sample, then Gibrat's law is confirmed. For this estimation, the growth rates are normalized for each observation by subtracting the mean, while city sizes are normalized by dividing each city's population by the total population for the lower tail (loannides and Overman, 2003). The

S. Devadoss, J. Luckstead / Economics Letters 135(2015) 12-14

mean and variance are given by

~ 3 Ê

e 2 es

о я (Г

■С t 9

с СО

2 3 4 5 6 Normalized Population

Fig. 4. Conditional mean and scatter plot.

-O.OO5

-0.015

-O.O25

1 2 3 4 5 6 7 8 Normalized Population *io 4

Fig. 5. Mean with 95% confidence bands.

2 3 4 5 6 Normalized Population

Fig. 6. Variance with 95% confidence bands.

nonparametric regression equation is

Yi = в (Pi) + st, Vi,

where p gives the relationship between the growth rate yi and city size Pi, and si is the error term. For kernel weight K (•) and bandwidth h, the Nadaraya-Watson regression estimation for the

ß (P) = =■

±K (P—P)

i=1 v 7

â2 (P) =

£ K (p—p)(Y — ß (P))2 £K (pp—p )

for all p in the support of Pi. For the regression analysis, the Gaussian kernel is used and the bandwidth is calculated based on Bowman and Azzalini ( 1 997). Fig. 4 overlays the estimated mean в (p) on the scatter plot of the

normalized growth rate and city size. Based on the nonparametric estimation, the average growth rate is centered on zero and constant for different city sizes. More formally, Fig. 5 graphs the mean growth rate with 95% confidence bands based on 500 bootstrapped samples with replacement. While the mean growth rate increases slightly from about -0.0 1 4 to about -0.0 1 1 , a constant mean is well within the 95% confidence bands. Fig. 6 shows that, while the variance decrease slightly from about 0.115 to 0.110, it is constant within the confidence bands. Because both the mean and variance are constant over the range of normalized population sizes, the results provide statistical evidence in support of Gibrat's law.

In summary, this study examines Gibrat's law using nonparametric estimation, and the results show that US small cities generally follow Gibrat's law.

References

Bowman, A., Azzalini, A., 1997. Applied Smoothing Techniques for Data Analysis. Oxford University Press, London.

Desmet, K., Rappaport, J., 2015. The settlement of the United States, 1800 to 2000: The long transition towards Gibrat's law. J. Urban Econ..

Eeckhout, J., 2004. Gibrat's law for (All) cities. Amer. Econ. Rev. 94 (5), 1429-1451.

Gabaix, X., 1999. Zipfs law for cities: An explanation. Quart. J. Econ. 114 (3), 739-767.

Giesen, K., Suedekum, J., 2014. City age and city size. Eur. Econ. Rev. 71,193-208.

Giesen, K., Zimmermann, A., Suedekum, J., 2010. The size distribution across all cities-double pareto lognormal strikes. J. Urban Econ. 68 (2), 129-137.

González-Val, R., 2010. The evolution of US city size distribution from a long-term perspective (1900-2000). J. Reg. Sci. 50 (5), 952-972.

González-Val, R., Ramos, A., Sanz-Gracia, F., Vera-Cabello, M., 2015. Size distributions for all cities: Which one is best? Pap. Reg. Sci. 94 (1), 177-196.

loannides, Y., Overman, H.G., 2003. Zipfs law for cities: An empirical examination. Reg. Sci. Urban Econ. 33 (2), 127-137.

loannides, Y., Skouras, S., 2013. US city size distribution: Robustly pareto, but only in the tail. J. Urban Econ. 73 (1), 18-29.

Krugman, P., 1995. Development, Geography, and Economic Theory. MIT Press, Cambridge, MA.

Luckstead, J., Devadoss, S., 2014a. A nonparametric analysis of the growth process of Indian cities. Econom. Lett. 124 (3), 516-519.

Luckstead, J., Devadoss, S., 2014b. Do the world's largest cities follow Zipfs and Gibrat's laws? Econom. Lett. 125 (2), 182-186.

Luckstead, J., Devadoss, S., 2015. Pareto tails and lognormal body of US cities size distribution. Working Paper, University of Arkansas.

Magrini, S., 2007. Analysing convergence through the distribution dynamics approach: Why and How? Working Paper No. 13, University Ca'Foscari of Venice. http://core.kmi.open.ac.uk/download/pdf/6234101.pdf.

Partridge, M.D., Rickman, D.S., Ali, K., Olfert, M.R., 2008. Lost in space: population growth in the American hinterlands and small cities. J. Econ. Geogr. 727-757.

Reed, W.J., 2001. The Pareto, Zipf and other power laws. Econom. Lett. 74 (1), 15-19.

Reed, W.J., 2002. On the rank-size distribution for human settlements. J. Reg. Sci. 42(1), 1-17.

US Census Bureau, 2014. Population estimates: Historical data. https://www. census.gov/popest/data/historical/index.html.