specific country trait. Country fixed effects capture the changes in city size distribution due to factors that are invariant over time (e.g. geographical location) and affect a given country in all years. is a vector of country specific characteristics at time _t_ when the Zipf’s coefficient is measured while the vector of cc,tt parameters measures the effect of changes in these variables on city size distribution. XX **_Defining a city_** ββ

# **_Defining a city_**

Gabaix and Ioannides (2004) highlight that there is no universally accepted definition of a city for statistical purposes. Differences in definition of cities and the available data on such arbitrarily and non-uniformly demarcated administrative boundaries worldwide may make international comparisons tricky. The definition of cities would be an issue even if official statistics are reliable because it is based on the statistical authorities’ definition of city boundaries (Soo, 2005). First, statistical authorities’ definition of a city may not necessarily be an economically meaningful definition of ‘city’ (e.g. Rosen and Resnick, 1980). Second, the definition of a city adopted by these authorities across a set of countries is also not uniform, so we cannot really expect a city in, say, Africa, to be similar to the definition of a city in Europe. We use WorldPop’s definition of urban agglomeration in our work to map out spatially concentrated regions.

# **_Estimating size distribution_**

Gabaix and Ioannides (2004) explain some of the pitfalls of measuring city size distribution using the Zipf’s coefficient. The most important challenge is that this rank-size regression is only an approximation and it is reasonably accurate in estimating the Zipf’s coefficient in large samples only, but not recommended for cities, as it is inefficient and biased in small samples. To counter this, we support our results with the following estimates of size distribution of urban agglomerations in robustness checks.

- (i) Some studies prefer using a Gini coefficient for measuring spatial inequality (e.g. Henderson and Wang, 2007) instead of the Zipf’s coefficient. Inspired by Duranton (2007), who suggests significant deviations of actual distribution of city size from the Zipf’s Law, this preference for Gini coefficients is clearly due to the flexibility it allows with respect to the specific size distribution. The Gini coefficient is the mean absolute deviation among all pairs of cities, relative to the mean city size. To check for the robustness of our results on the impact of participation in GVCs on spatial concentration, we use the Jasso-Deaton formulation for calculating the Gini coefficient: 2

- G =<sup>llcc,tt+ 1</sup> − nncc,tt −1 −1�

- cc,tt ii,cc,tt ii,cc,tt

- llGG cc,tt cc,tt cc,tt �RR PP

- where is the sample of cities in country ll ll _c_ and time  �ll _t_ . ii=1 cc,tt

- (ii) Gabaix and Ioannides (2004) recommend using a Hill estimator (Hill, 1975). For a sample of ll

- (ii) cities in country _c_ and time _t_ , with sizes 1 2 the Hill (maximum

- likelihood) estimator under the null of perfect power law is given as: cc,tt nn

- ll PP ≥PP ≥⋯⋯⋯≥PP −1

- = <u>,</u> ∑ l −l cc tt

- ll

- cc,tt nn

- Dobkins and Ioannides (2000) and Soo (2005) use the Hill estimator for estimating Zipf’s Law. γγ� ii=1 PPii,cc,tt PPnn,cc,tt

Both these studies find that the Hill estimate of the Pareto exponent is always smaller than the OLS estimate. Black and Henderson (2003) reconcile the apparent appropriateness of the Hill estimator by suggesting that the reliability of this estimate is dependent on the curvature of the log rank–log population relationship. Soo (2005) deals with a comparison of the Hill and the 6