Top Lights. Bright cities and their contribution to economic development. Richard Bluhm Melanie Krause. November 2017

Transcription

1 Top Lights Bright cities and their contribution to economic development Richard Bluhm Melanie Krause November 2017 Draft Version Do not cite or distribute without express consent Abstract Satellite data on nighttime luminosity are an increasingly popular proxy for economic activity. However, the commonly used satellites suffer from top coding and do not accurately capture the brightness of larger cities. In this paper we study the distribution of top lights. Both a simple model of luminosity within cities and our empirical evidence suggest that top lights can be characterized by a Pareto distribution. This allows us to correct the top-coding of the original stable lights data and present a new annual panel of nighttime lights at a resolution of 30 arc seconds over the period from 1992 to We show that the top 4% of pixels account for 32% of the world s observed brightness. In two applications, we illustrate that this new data challenges some of the perceived wisdom regarding local economic activity. First, the regional light-output elasticity in Germany rises considerably after the correction. Second, primary cities in Sub-Saharan Africa grow as fast as secondary cities, while all African cities are becoming increasingly fragmented. JEL Classification: O10, O18, R11, R12 Keywords: Development, urban growth, nighttime lights, top-coding, inequality Leibniz University Hannover, Institute of Macroeconomics, Königsworther Platz 1, Hannover, Germany; Maastricht University; UNU-MERIT, bluhm@mak.uni-hannover.de Hamburg University, Department of Economics, Von-Melle-Park 5, Hamburg, Germany; E- mail: melanie.krause@wiso.uni-hamburg.de 1

2 1 Introduction Economic activity in countries with low quality statistical data is hard to track. This problem not only plagues studies focusing on developing countries but also those dealing with growth, inequality and comparative development on a global scale. Advanced economies tend to be over-represented in empirical studies, while less developed countries, especially in Africa, are often left out due to a lack of data. This data constraint is so severe that it has been referred to as a statistical tragedy (Devarajan, 2013). Light emissions observed by weather satellites orbiting the earth at night are a way out of this predicament. A fast growing literature in economics uses nighttime lights as a proxy for local economic activity (e.g. see Chen and Nordhaus, 2011; Henderson et al., 2012; Michalopoulos and Papaioannou, 2013; Hodler and Raschky, 2014; Alesina et al., 2016; Lessmann and Seidel, 2017; Pinkovskiy, 2017). Prima facie, the advantages of night lights seem obvious. They are publicly available as an annual panel from 1992 onwards for nearly the whole populated world. They have a high resolution compared to regional accounts data and are measured uniformly across the globe. Night lights can predict output growth at the country level while allowing us to circumvent thorny discussions over adjustments for exchange rates and price levels. If light is a normal good and Engel curves are stable, then night lights are as useful a proxy for income at the national or sub-national level as observed consumption decisions (Donaldson and Storeygard, 2016). One important drawback of this new data, however, is that they are top-coded in larger cities and densely populated areas. Due to sensor saturation, the bustling center of a metropolis often appears no brighter than that of a mid-sized town. Many important questions in urban, regional and development economics cannot be answered correctly when cities are misrepresented. Unlocking the potential of night lights for these types of applications is the overarching contribution of our paper. Geo-referenced images of night lights are typically obtained from the National Geophysical Data Center (NGDC) at the National Oceanic Administration Agency (NOAA). Satellites from the Operational Linescan System of the Defense Meteorological Satellite Program (DMSP-OLS) have been orbiting the earth for some decades with the primary purpose of detecting clouds. As a byproduct, they measure light emissions in the evening hours between 8:30 and 10:00 pm local time around the globe every day. The recorded data are pre-processed 1 and averaged over cloud-free days. The result is an image of annual light intensities from 1992 to 2013 for every 30 by 30 arc seconds pixel of the globe (approximately 0.86 square kilometers at the equator). 2 1 To remove observations of cloudy days and sources of lights which are not man-made, such as auroral lights or forest fires. 2 Areas close to the polar zones (65 degrees south and 75 degrees north latitude) are usually excluded. As these regions are very sparsely populated, the exclusion affects only approximately percent of the global population (see Henderson et al., 2012). 2

3 Light intensities of these so-called stable lights 3 are recorded as digital numbers (DN) from 0 (dark) to 63 (bright). The problem is that the upper end of the scale is easily reached by the brightness emitted by a mid-sized city. This has important ramifications. For instance, the center and the outskirts of large cities are recorded as equally bright (see the middle column in Figure 1). As a result, questions regarding within-city heterogeneity, such as city growth at the intensive margin or city fragmentation, are difficult to address with this data. This comes at a time when many developing economies with poor or infrequent official data experience unprecedented urbanization rates and researchers increasingly turn to night lights to track urban agglomerations, e.g. see Fetzer et al. (2016) and Henderson et al. (2017) on African cities. Moreover, top-coding of brighter cities induces downward bias in measures of regional inequality and artificially raises the speed of convergence between cities of different sizes as well as between urban and rural areas. This is an under-appreciated problem affecting many current research areas, including all light-based estimates of sub-national output or spatial inequalities in developing countries. Globally, only around 3-5% of all lit pixels are affected by top-coding but the scale of the truncation is enormous. Cities like New York or London are easily more than ten times brighter than indicated by the stable lights data. While top-coding tends to affect developed countries more than their less-developed counterparts, we show that it is pervasive. Nearly all primary cities in Africa and most larger cities in Asia hit the top-coding threshold. Within developing countries, large economic hubs are affected particularly strongly (see Johannesburg or New Delhi in Figure 1). Top-coding has received little attention in the related literature so far, largely because of a lack of reliable time-series data on non-saturated lights. Here we exploit that for seven years, additional satellites flew with settings that were less sensitive to light and thus capable of capturing the upper part of the light distribution (Hsu et al., 2015). The resulting radiance-calibrated data are no longer top-coded, but, unfortunately, only available for a few years. Furthermore, noise and measurement error severely limit their comparability over time. 4 Unless interest is restricted to a single cross-section (as in Henderson et al., 2018), simply switching to the radiance-calibrated data will often not be a panacea. Nevertheless, we show that they contain valuable information on the distribution of top lights which we systematically exploit in this paper. In robustness checks, we verify our analysis using a third new data source free of top-coding: the Visible Infrared Imagining Radiometer Suite (VIIRS) available only for This paper is the first to analyze the global distribution of top lights at the pixel 3 We use the terms stable lights and saturated lights interchangeably to refer to the same data. 4 Measurement errors are also present in the stable lights data and affect their reliability in the time series dimension but to a lesser extent. The satellites sensors deteriorate over their lifetime and have to be replaced every couple of years. Comparability of the images across and within satellites can therefore be affected. In panel regressions, one can resort to a combination of satellite and time fixed effects to address this issue. 3

4 Figure 1: Selected cities in 1999, saturated and corrected (a) New York City Transect Saturated Corrected Brightness (DN) Corrected Saturated Longitude Longitude Longitude (b) London Transect Saturated Corrected Brightness (DN) Corrected Saturated Longitude Longitude Longitude (c) New Delhi Transect Saturated Corrected Brightness (DN) Corrected Saturated Longitude Longitude Longitude (d) Johannesburg Transect Saturated Corrected Brightness (DN) Corrected Saturated Longitude Longitude Longitude Notes: Comparison of the saturated and corrected data in four major cities. The left panel compares the brightness (DN) along a longitudinal transect through the brightest point in each city. The middle panel shows a map based on the stable lights data from satellite F The right panel shows the same map using the corrected data presented in this paper. Both data have been binned and the color scales were adjusted so as to be comparable. Dashed lines indicate the transect path. 4

5 level. We make three larger contributions. First, we establish that top lights are well-characterized by a Pareto distribution. A stylized model of light emissions within monocentric cities based on standard assumptions (Zipf s law, urban scaling, and an inner city population density gradient) gives rise to a power law in light emissions at the pixel level above a lower threshold. A battery of empirical tests also point to a heavytailed Paretian distribution with an inequality parameter comparable to top incomes or wealth in the US (e.g. see Atkinson et al., 2011). Second, we extend the distribution of lights using a Pareto tail. At the aggregate level, we offer simple formulas to correct mean luminosity and the Gini coefficient of inequality in lights. At the pixel level, we correct the data using a geo-referenced ranking method, yielding a new annual panel of nighttime lights over the entire period from 1992 to 2013 without top-coded values. Third, in two applications, we analyze the impact of top-coding both in developed and developing countries on i) output elasticities in Germany and ii) the economic importance of primary and secondary cities in Africa. Our top-coding correction makes a substantial difference in virtually all major cities. The top 4% of pixels the average share of pixels which we correct account for 32% of the world s total observed brightness. This is nearly double their original share, underlining their contribution to global economic activity. On average, the global Gini coefficient in lights rises by 9 percentage points after the correction. The first and last column in Figure 1 illustrate the effectiveness of our method and the opportunities this new data opens up. Only after the correction, we can clearly identify the urban cores which are much brighter than the outskirts. Phrased provocatively, ignoring top-coding is tantamount to analyzing development without big cities. The applications reveal important new findings. First, night lights have generally been considered a poor proxy of local economic activity in developed economies, but this has been in part due to top-coding. On the basis of German regional accounts data we show that the subnational light-output elasticity rises considerably after the correction. Second, night lights are without doubt most useful in developing countries with weak statistical capacity. Here our new data suggest that there is no convergence of city sizes in Sub-Saharan Africa. On the contrary, primate cities have grown faster than secondary cities over the period from 1992 to Pushing the data even further, our analysis of inner city lights reveals that African cities have become more fragmented over time, i.e. moving towards a collection of disconnected neighborhoods with low economies of scale. The paper proceeds as follows. Section 2 illustrates the extent of top-coding around the world. Section 3 provides theoretical and empirical evidence for a Pareto distribution in top lights and Section 4 presents our top-coding correction. In Section 5 we discuss the results of the impact of top-coding in two empirical applications. Section 6 concludes. The appendices provide additional proofs, tables, and figures. 5

6 2 Top-coding around the world 2.1 Preliminary evidence The literature typically aggregates the raw lights data to some study area of interest (e.g. grid cell, city, region, or country). To study the distribution of the brightest lights as they recorded in the data, we need to work with the native resolution of 30 arc seconds. On the global scale, this is a formidable computational exercise. Each image contains more than 700 million pixels, about a third of which are on land. We restrict our attention to lit regions of earth, since vast parts of the world are dark or covered by water. To further reduce the computational burden, we conduct large parts of the analysis with a spatial random sample of 10% of pixels within all countries that have a landmass larger than 500 km 2. 5 We also remove areas known to be affected by gas flaring and limit our attention to the world below the arctic circle. The resulting data set contains more than 2 million pixels per year in 194 countries and territories. Table 1: Summary statistics of global lights in 2010 World USA Brazil Israel South Afr. China Netherlands Number of pixels 2,154, ,922 60,310 2,060 18, ,521 6,549 Panel a) Saturated lights (from 0 to 63 DN) Mean light intensity Standard deviation Max. light intensity Gini in lights Panel b) Unsaturated lights (from 0 to DN) Mean light intensity Standard deviation Max. light intensity Gini in lights Share of pixels 55 (saturated) Unsaturated mean if saturated 55 Panel c) Comparison at same locations Notes: The table reports summary statistics using the saturated lights in Panel a) and the unsaturated lights in Panel b), both based on a 10% sample of all lit pixels, where each pixel is arc seconds. Panel c) compares both sources at the pixel level. Table 1 presents a pixel-level comparison using the data from the year 2010, for which both the saturated stable lights and the unsaturated radiance-calibrated data 5 Since we only sample the spatial locations once and then form a panel, the data set includes pixels which were not lit in previous year. We ignore these values in our computations, which leads to the discrepancy between the numbers of pixels in the data set (2,337,100) and those reported in Table 1. 6

7 are available. 6 Although not all differences between the saturated and unsaturated lights can be ascribed to top-coding, the comparison gives some insight into its scope. The mean light intensities hardly differ (17.55 vs ), but the difference in scale between the two data sources is striking. 7 The range of the saturated lights ends at 63, while the unsaturated lights do not have a theoretical upper bound. The brightest pixel on earth is recorded at rather than 63. This is reflected in a standard deviation which is three times as high (44.35 vs ) and a Gini coefficient of inequality in lights which is much higher (0.43 vs. 0.61). For reasons that will become obvious in the next subsection, note that about 6% of all saturated pixels in 2010 are in between 55 and 63, while their unsaturated counterparts are, on average, more than twice as bright. Figure 2: Share of pixels between 55 and 63 DN and country characteristics (a) GDP per capita (b) Log population density SGP SGP Share of pixels 55 DN and above BHR HKG EGY TTO ISR BEL BRN DJI PRI ARE NLD COG KWT NOR LUX QAT Share of pixels 55 DN and above BHR BRN EGY TTO BEL ISR DJI PRI ARE GUM AGO QAT NLD KOR PSE RWA LBN MUS BGD COM HKG GDP per capita (in 2011 PPPs) Log population density Notes: Illustration of the systematic bias introduced by top-coding. The data are a 10% representative sample of all non-zero lights in satellite F GDP and population data are from the World Development Indicators. Top-coding affects all countries but not uniformly. The remaining columns of Table 1 report the same summary statistics for selected countries. Big emerging market economies, like South Africa or Brazil, have about 4% to 6% of lit pixels with values of 55 DN or greater, whereas their share is a lot higher in mature economies, such as the US or Israel. In the Netherlands, a high average light intensity coupled with a high prevalence of top-coding generate an artificially low spatial Gini coefficient. This changes when the full scale of the light distribution is considered, then the spatial Gini of night lights nearly doubles. In general, countries which are i) highly developed, ii) 6 The saturated data are averaged across the whole year, while the radiance-calibrated data come from satellite F16, which circled the earth from 11 January to 9 December Note that despite some calibration issues among the different sources, the DNs can be compared since the lower-end of the radiance-calibrated image in part derives from the stable lights data. In fact, Hsu et al. (2015, p. 1865) clearly point this out when they write DNs below saturation of the Stable Lights product and DN EQs of merged fixed-gain imagery can be directly compared to each other. 7

8 small and iii) urbanized are more strongly affected by top-coding than others, but there is substantial heterogeneity between countries (see Figure 2). Although there clearly is a positive relationship between a greater number of bright pixels and income, countries with very different income levels have approximately the same number of top-coded pixels. Similarly, we observe a clear density gradient but that is only part of the explanation. On the one hand, the majority of pixels in high income, high density city states, such as Singapore, Hong Kong, or Bahrain, are top-coded. On the other hand, there are middle income countries with a low population density but a high share of top-coded pixels concentrated in the primate city (such as Egypt). 2.2 The threshold Before analyzing top lights in more detail, we have to define where to put the top-coding threshold. The influence of top-coding has been underestimated in part because much of the literature assumes it only affects pixels with the highest recorded value. Even though the scale of saturated satellites goes up to 63, we have good reason to assume that many pixels with DNs of 62, 61, down to the mid-50s, are subject to top-coding and should be brighter than they are recorded in the data. The rationale behind this is straightforward. In the data construction process the stable lights data have already been averaged twice. First, the DMSP satellites average several higher resolution pixels on-board to reduce the amount of information that needs to be transmitted. 8 Second, the data providers at NOAA process the daily images into one annual composite. As a result, many pixels suffering from top-coding in at least one of the underlying data points or daily images will end up with an average value of less than 63. In fact, Hsu et al. (2015) suggest that this subtle type of top-coding may even start at a DN as low as 35. A statistical approach to the data suggests a more conservative top-coding threshold in the mid-50s, but clearly below 63. If only the saturated lights at 63 DN were subject to top-coding, we would expect the histogram in Figure 3a to show a decreasing shape ending in a spike only at 63 DN. Instead, we observe an increase in the number of pixels from 55 onwards. Further evidence along these lines is provided by Figure 3b. It shows a histogram of the light intensity of all those saturated lights associated with high radiancecalibrated values (above 160 here). There are a large number of pixels with DNs down to the mid-50s which correspond these high radiance-calibrated values and whose full brightness is curtailed. Other years show very similar patterns. 9 8 The OLS system is 1970s technology. It records images at a nominal resolution of 0.56 km (so-called fine resolution) which is averaged on-board into 5 5 blocks to create a 2.77 km (smooth) resolution and then reprojected onto a 30 arc second grid, see satellite-missions/d/dmsp-block-5d. 9 Table B-1 list the percentile values of the radiance-calibrated lights corresponding to saturated lights at 55 DN, 56 DN and so on. The saturated lights at 63 DN have the highest radiance-calibrated values (50% of them are higher than 390 DN). But there is also a significant share of 55 DN lights corresponding 8

9 Figure 3: Histograms of saturated lights in 1999 (a) If saturated DN > 9 (b) If unsaturated DN > 160 Density Density Saturated lights (DN) Saturated lights (DN) Notes: Illustration of the location of the top coding threshold in the saturated data. Panel a) shows a histogram of the F12 satellite in 1999 for all pixels with a DN greater 9. Panel b) shows a histogram of the same satellite only for pixels where the unsaturated light intensity is greater 160 DN. The input data are a 10% representative sample of all non-zero lights in the stable lights and radiance-calibrated data at the pixel level (see Elvidge et al., 2009; Hsu et al., 2015). We consider a DN of 55 to be the approximate start of the range where top-coding plays such a significant role that it swamps the declining part of the histogram. Note that the procedure we suggest will work with any sufficiently high top-coding threshold. 2.3 Comparing both data sources Based on this cursory comparison, the question may arise why switching the saturated with the radiance-calibrated data is not the most desirable, and perhaps simplest, solution. Unfortunately, these data have severe drawbacks making comparisons over time very unreliable. First, the radiance-calibrated data are only available for seven years, as opposed to the annual panel of saturated lights over the period from 1992 to Second, they contain a lot of instability, noise and measurement error. 10 Table B-2 in the Appendix compares the time variation of both data sets. Mean luminosities in the saturated data vary slightly across years and satellites. Between 2.7% and 5.9% of all pixels are at the top of the scale between 55 and 63, more so in later years. There are many years where two satellites were in orbit concurrently, allowing us to gauge to high radiance-calibrated values, for instance, 25% are recorded with 140 DN or brighter. 10 The saturated data have been cleaned of stray lights, fires and other non-permanent light sources by NOAA, while the blended radiance-calibrated images have not. 9

10 the significance of between-satellite errors as well. 11 But this modest variation pales in comparison to the fluctuations of the radiance-calibrated satellites. The maximum luminosity doubles within three years and then decreases again by a similar amount. This would be hard to explain economically. 12 In spite of this noise, the radiance-calibrated data carry important information. The shape of the distribution at the top is a lot more stable. We examine this distribution in more detail below, but note for now that the ranks of maximum light intensities across cities vary much less than actual values. Table B-4 in the Appendix presents the maximum light intensities recorded within a 25 km radius in some of the world s biggest cities. The rank correlation between any two adjacent satellites is substantially higher than the simple correlation (see Table B-5 and Table B-6 in the Appendix). We later exploit precisely this property for the rank-based top-coding correction at the pixel level. 3 A Pareto distribution in top lights Our correction rests on the claim that a Pareto distribution is a reasonable description of top lights. A power law in the tail of nighttime lights has not yet been investigated, but it is an often-found empirical feature in many other data (for an overview see Newman, 2005). Most relevant for us, the population of large cities follows a power law. 3.1 A stylized model of inner city lights We now present a tractable model of light intensities within and across cities to show that the density of light at the pixel level is Pareto, or can at least be closely approximated by a Pareto density (depending on the exact assumptions). The model is novel, in the sense that it anchors the empirically-focused night lights literature in formal urban scaling models and combines them with standard results from urban economics. Our starting point is the size distribution of cities. It is well known that combining Gibrat s (1931) law of homogeneous growth of cities with a lower bound of city sizes leads to a Pareto distribution of large cities (Gabaix, 1999; Eeckhout, 2004). In fact, this particular Pareto distribution with α = 1 is the Zipf distribution (Zipf, 1949). It predicts that city ranks are inversely proportional to their size; for instance, the biggest city in the U.S. (New York) has twice the population of the second-ranked city (Los Angeles) 11 Some degradation of the sensors over time is visible; later recordings of any particular satellite tend to also be the brightest. 12 This variability can mostly be attributed to measurement errors introduced when the different fixed gain images and the stable lights images are merged at NOAA. Not all differences between these two data sources can be attributed to top-coding. Regressing all pixels below 55 DN of the saturated lights on the radiance-calibrated lights, we find a regression coefficient around one-half rather than equivalence, see Table B-3 in the Appendix. This is owed to the lack of on-board calibration, the presence of stray light or fires (Hsu et al., 2015), and geo-location errors (Tuttle, Benjamin T. and Anderson, Sharolyn and Sutton, Paul and Elvidge, Christopher and Baugh, Kim, 2013). 10

11 and three times the population of the third-ranked city (Chicago). Numerous studies have provided empirical evidence for Zipf s law based on U.S. cities or metropolitan areas (Gabaix and Ioannides, 2004; Rozenfeld et al., 2014; Ioannides and Skouras, 2013). Despite heterogeneity in the rank-size parameter in other countries (Rosen and Resnick, 1980; Soo, 2005), it is generally considered a good approximation of the size distribution of large cities (Luckstead and Devadoss, 2014). Evidence in its favor becomes even stronger when cities are defined as natural agglomerations and not by administrative boundaries (e.g. see Small et al., 2011; Jiang and Jia, 2011). Hence, we have: Assumption 1. The size distribution of big cities in terms of their population x follows Zipf s law, i.e. a Pareto distribution with shape parameter α = 1 above some population threshold x c. The CDF of the number of cities with population x is F (x) = 1 ( xc ) α x = α xα c dx = x x c xα+1 x x c x c dx. (1) x2 We are interested in the size of cities in terms of their area, not their population. The urban allometry literature, which deals with the scaling of human-made structures within cities, links the two quantities in the following way (Stewart, 1947; Jones, 1975): Assumption 2. The population x and the area s of a city are proportional, i.e. x s φ. The consensus in literature is that 1 < φ < 2, so that larger cities not only spread out on the plane but also grow into the third dimension (Batty and Longley, 1994). Bettencourt (2013), for example, motivates scaling laws based on a network theory of human interactions using the parameter value φ = 1.5. To derive the distribution of individual pixels y, we still require an assumption on the shape of cities. The standard assumption in the workhorse model of urban economics is monocentricity (Mills, 1967; Amson, 1972). Assumption 3. Cities are monocentric and of circular shape. They consist of rings of unit width from the center to the outskirts. Depending on its area s and therefore its population x, each city has r = π 1/2 x 1/(2φ) rings. 13 Integration by substitution yields the CDF of the number of rings per city r 0 for r < r F (r) = 2φx c π φ r 2φ 1 dr = r 1 x c π φ r 2φ for r >= r 13 For simplicity of exposition, we suppress the proportionality constant in x s φ and assume x = s φ. (2) 11

12 with r = π 1/2 x 1/(2φ) c. See Section A.1 in the Appendix for details. Its density f(r) = x c π φ r 1 2φ for r >= r (3) follows a power law with a shape parameter 2φ 1 > α = 1 for φ > 1. In contrast to Zipf s distribution of city sizes in terms of their population, the distribution of rings has fewer extreme values. Each city with r rings consists of πr 2 rectangular pixels of unit size. The pixels are located at distance d from their respective city center. There are two opposing effects governing how the number of pixels depends on the distance: i) within a given city, the number of pixels increases linearly with d because rings farther from the center contain more pixels, as shown by d dd πd2 = 2πd, and ii) the larger the distance d from the city center, the fewer cities of that size are left, namely only the cities with r d rings. The effect within each city has to be multiplied by the survival function 1 F (r) from eq. (2). Dividing the absolute amount of pixels at each distance d by the total number of pixels in the model yields the density f(d) = 2π φ 1 φ 2π 1 φ φ 1 φ x 1/φ c d for d < d x1 1/φ c d 1 2φ for d d with d = π 1/2 x 1/(2φ) c. See Section A.2 in the Appendix for details. Figure 4a illustrates the pixel density from eq. (4) using x c = and φ = 1.5. For small d < d = 12.2, the density increases with d, while from d onwards the linear increase within each city is dominated by a decreasing number of cities of that size. In the tail the pixel density follows a power law with shape parameter 2φ 1 (here a value of 2). To derive the density of the luminosity or light intensity of each pixel f(l) we need one last assumption on how luminosity within a city relates to distance d to the center. We resort to standard models of population density since differences in light intensity within countries are mostly driven by population (Henderson et al., 2018). A popular choice is the negative exponential function, that is, p(d) = P 0 exp( γd) where p(d) is the population density at distance d from the city center, P 0 p is the density at the center and γ > 0 is a decay parameter so that the city periphery is more sparsely populated or, in our case, lit. The inverse exponential was first proposed in the seminal work of Clark (1951) and can be motivated on the basis of the standard Alonso-Muth-Mills model (Brueckner, 1982). It has since been applied in many empirical studies which typically find γ 0.15 (Bertaud and Malpezzi, 2014). A viable alternative to the exponential distribution is the inverse power function, that is, p(d) = P 0 d a with d > 0 and shape parameter a > 0. It has its conceptual origin in gravitational models of traffic flow (Smeed, 1961; Coleman, 1964; Batty and Longley, 1994). (4) 12

13 Both functions are qualitatively similar for intermediate distances. The inverse power function differs from the exponential function in the city center, where it has higher and more sharply declining values, as well as in the outskirts, where it predicts a moderate density when the exponential distribution is already close to zero, see Figure 4b. High values at the center make the inverse power function particularly suitable for business floor-space models (as recommended by Zielinski, 1980). At the same time, its tail is known to fit the urban fringe well (Parr, 1985). Since we are interested in lights rather than population density, we want to capture both the very bright central business districts in most cities and the more dimly lit but significant suburban sprawl in the outskirts. The latter are typically included when natural rather than administrative boundaries define the city and certainly feature prominently in the footprint of city lights (e.g. see Small et al., 2011). Figure 4: Illustration of distributions (a) Pixel density (b) Light gradient Density Light intensity (DN) Negative exponential Inverse power Distance d Distance d Notes: The left panel shows the pixel density from eq. (4) with x c = and φ = 1.5. The right panel shows the negative exponential distribution with γ = 0.15 as well as the inverse power distribution with a = 0.7, both start at P 0 = Modeling inner city light densities with an inverse power law better captures the light gradient and also turns out to be analytically appealing. Hence, it serves as our baseline case. However, given the popularity of the exponential function, we show in Section A.3 of the Appendix that it also leads an expression for the distribution of top lights which (under plausible conditions) can be approximated by a Pareto density. Assumption 4. Within cities, the light density l(d) follows an inverse power law l(d) = L 0 d a for d > 0 with L 0 > l the maximum luminosity at the center and a > 0 as the decay parameter. Applying the variable transformation from the inverse power function to the pixel 13

14 density in eq. (4) yields f(l) = 2π φ 2π φ 1 φ 1 φ φ 1 x1 1/φ c x 1/φ c } {{ } c where l = L 0 π a/2 x a/(2φ) c. ( (1 2φ)/a L 0 /l) for 0 < l l ( 1/a L 0 /l) for l < l L0, (5) Restricting our interest to the upper part of the light distribution from l onwards, we find our key result in eq. (5): Result 1. Based on Assumptions 1 4, top lights above threshold l follow the Pareto ( 1/a distribution f(l) = c L 0 /l) with shape parameter 1/a. Using a grid-search, we find that for a = 0.7 the inverse power function comes closest to the negative exponential function of the consensus parameter γ = Figure 5 shows the resulting density from eq. (5) with a = 0.7. Figure 5: Illustration of power law in light intensities Density Light intensity (DN) Notes: The graph shows the light density from eq. (5) with a = 0.7, L 0 = 2000, y c =10000 and φ = 1.5. Here the threshold l is located at a luminosity of In sum, a limited set of four stylized assumptions allows us to analytically derive a Pareto distribution in top lights, with shape parameter 1/a, maximum luminosity L 0, and a multiplicative constant. 3.2 Extreme value theory Alternatively, we can motivate a Pareto distribution in top lights purely on statistical grounds using extreme value theory (EVT). EVT deals with the probability distributions 14 The curve-fitting was conducted on the domain for d from 1 to 50, minimizing the squared error between the negative exponential function with γ = 0.15 and the inverse power function with a [0.05, 2]. 14

15 of sparse observations such as threshold exceedances. A key result of this theory is that these quantities observe a Generalized Pareto distribution. More precisely, let X 1, X 2,... be a sequence of independent random variables such as light with common but unknown distribution function F, and let M n = max{x 1,..., X n }. If F satisfies the extremal types theorem (Coles, 2001), so that for large n, P[M n > z] G(z) with G(z) as the Generalized Extreme Value distribution, then, for a high enough threshold u, the distribution of the threshold exceedance P[(X u) > y X > u] is approximately H(y) = 1 ( 1 + ξỹ ) 1 ξ, σ (6) no matter which regular distribution X was drawn from. This means that we will observe a Generalized Pareto distribution with parameters ξ and σ for all lights values above a specified threshold. With ξ = 0, this reduces to the exponential distribution and with ξ > 0 the distribution is Pareto. There is strong evidence that the latter case holds for the lights data. Fitting the Generalized Pareto distribution to various top shares of the light distribution of the seven radiance-calibrated satellites, we find a very good fit with an estimated ξ parameter that is always significantly positive, see Table B-7 and Figure B-3 in the Appendix. 3.3 Empirical tests Given our strong theoretical prior that top lights should be Pareto distributed, it remains to be shown that this is indeed the case. In economics, tests for a Pareto tail were popularized by the top incomes literature (e.g. Piketty, 2003; Atkinson et al., 2011) and the city size literature (e.g. Rosen and Resnick, 1980; Gabaix, 1999; Gabaix and Ioannides, 2004; Rozenfeld et al., 2014). Here we apply a range of methods to the top end of the seven radiance-calibrated satellites and then compare these results to the new (and superior) VIIRS data Visual inspection Any data y above a certain threshold y c which is Pareto distributed with shape parameter α > 0 has the probability density function (PDF) f(y) = αyc α y α 1 for y y c (7) and the survival function ( ) α yc F (y) = for y y c, (8) y 15

16 which gives the probability that the random variable Y is larger than the given value y. Taking logs leads to an equation which is the basis of the often-used Zipf plot. A linear Zipf plot is usually considered evidence in favor of the Pareto distribution, but its relevance is being contested (Cirillo, 2013). The Zipf plots for the top 2% of lights are qualitatively similar to those of the top incomes literature, in that they display linear sections together with some initial curvature and outliers at the end (see Figure B-1a in the Appendix). 15 Another interesting graphical feature of a Pareto tail derives from Van der Wijk s Law. The Pareto distribution is unique in that the average above some level y is proportional to y at all points in the tail, with a factor of proportionality equal to α α 1 > 1. As expected, we observe a linear relationship with a slope above unity (Figure B-1b in the Appendix) Log-rank regressions Log-rank regression are a more rigorous way of testing for a Pareto distribution and firmly rooted in urban economics. For Pareto-distributed observations y i, i = 1,...N, with the survival function from eq. (8), we have rank(y i ) Nyc α y α i, or, in logarithms log rank(y i ) log N α log y c α log y i. Hence, in the regression ( log rank(y i ) 1 ) log N = α 1 log y c + α 2 log y i + ɛ (9) 2 only the Pareto distribution satisfies the null hypothesis that α 1 = α 2 with α 2 < The two parameter estimates are very close to each other in all seven satellites at various different thresholds, i.e. the top 5% to top 1%. The R 2 s are very high ( ) and increase further with the threshold (see Table B-8 in the Appendix). 17 Conducting a restricted Pareto rank-size regression where the null α 1 = α 2 = α is enforced sheds more light on the parameter estimate. It is well-known that the OLS coefficients of such a rank-size regression are biased while the standard errors are underestimated due to the autocorrelation induced by the ranking procedure (Gabaix and Ioannides, 2004; Gabaix, 2009). The more robust Hill estimator ˆα Hill = (N 1) ( N 1 i=1 log y i log y c ) 1 is usually preferred. Under the assumption of a Pareto 15 Working with any top share, from the top 5% to the top 1% gives qualitatively similar results, even if the case for a Pareto distribution tends to be stronger the higher we set the threshold. This is in line with the empirical literature on Pareto applications in other fields. On the other hand, it is also known that Zipf plots often deviate from linearity at the very top since fewer and fewer values are observed at the extremes. Sometimes this is addressed by removing the very top. We use logarithmic bins so that the size of the bins increase by a multiplicative factor (Newman, 2005). 16 We follow Gabaix and Ibragimov (2011) in subtracting one half from the rank to improve the OLS estimation of the tail exponent in the rank regression. 17 Note that formal statistical tests, e.g. a Kolmogorov-Smirnoff test, do not make much sense in huge samples such as ours. Gabaix and Ioannides (2004, p. 2350) capture this nicely: with an infinitely large dataset one can reject any non-tautological theory. The extremely small standard errors lead to overrejections of the null hypothesis unless the empirical value equals exactly the theoretical value. 16

17 distribution, it equals the efficient maximum likelihood estimator and is superior to the OLS estimator in the tail of the fitted distributions (Soo, 2005; Eeckhout, 2009). Table 2: Parameter estimates from rank regressions (Hill estimator) Satellite F6 (96) F6 (99) F6 (00) F6 (03) F6 (04) F16 (06) F16 (10) Average Panel a) Top 5% α (0.0040) (0.0035) (0.0035) (0.0041) (0.0040) (0.0042) (0.0040) Observations 96, , , , ,899 99, , ,955 Panel b) Top 4% α (0.0045) (0.0042) (0.0042) (0.0051) (0.0049) (0.0052) (0.0049) (0.0047) Observations 77,348 93,484 85,482 80,075 85,489 79,590 86,196 83,952 Panel c) Top 3% α (0.0054) (0.0053) (0.0055) (0.0065) (0.0064) (0.0068) (0.0065) (0.0061) Observations 58,011 70,115 64,111 60,058 64,134 59,692 64,647 62,967 Panel d) Top 2% α (0.0077) (0.0078) (0.0085) (0.0095) (0.0093) (0.0100) (0.0097) (0.0089) Observations 38,673 46,742 42,740 40,039 42,756 39,794 43,097 41,977 Panel e) Top 1% α (0.0146) (0.0147) (0.0155) (0.0170) (0.0161) (0.0179) (0.0181) (0.0163) Observations 19,337 23,373 21,372 20,020 21,377 19,898 21,551 20,990 Notes: The table reports the results of a restricted rank regression as in eq. (9) with α 1 = α 2 = α imposed, using the Hill estimator described in the text. The data are a 10% representative sample of all non-zero lights in the radiance-calibrated data at the pixel level, where each pixel is arc seconds. We compare the Hill parameter estimates of the restricted rank regression in Table 2 with the OLS estimates in Table B-9 in the Appendix. 18 In spite of the noise inherent in the unsaturated data, our results remain relatively stable across satellites and thresholds. 19 As expected, the OLS estimator underestimates inequality in the tail. For the top 5% to top 3% of the data, the Hill estimates are in between 1.25 and 1.5 whereas the OLS estimates range from 1.38 to Top lights are therefore more equally distributed than city sizes (Gabaix, 2016), but more unequally distributed than top incomes in most countries (Piketty, 2003; Atkinson et al., 2011). The parameter range 18 The standard errors or the Hill estimator are computed as ˆα Hill / N 3 (Gabaix, 2009). For OLS, we report classical standard errors as well as the adjusted standard errors 2N 1 ˆα (Gabaix and Ioannides, 2004). 19 In theory, with Pareto-distributed data, conducting the estimation with the portion of the distribution above a higher threshold y h c > y c should lead to the same estimated α. In practice, this will often not hold exactly and the results for tail regressions are known to depend on the precise threshold used (see for instance Rosen and Resnick, 1980). Table 2 and Table B-9 suggest the parameter estimates are rather stable for the top 5%-3% but vary more when restricting the sample to the top 1%. 17

18 is comparable to that reported for the top tail of the U.S. income or wealth distribution Other distributions As a robustness check, we pit the Pareto distribution against other plausible candidates, paying particular attention to the log-normal distribution which is commonly used to describe the complete distribution of incomes or city sizes. Figure 6: Discriminant moment ratio plots (a) Top 4% (b) Top 1% Skewness Pareto Area Gray Area Lognormal Area Exponential/ Thin tailed Normal or Symmetric Skewness Pareto Area Gray Area Lognormal Area Exponential/ Thin tailed Normal or Symmetric CV CV Notes: The panels show discriminant moment ratio plots (Cirillo, 2013). The input data are a 10% representative sample of all non-zero lights in the radiance-calibrated data at the pixel level, where each pixel is arc seconds. Figure 6 shows a discriminant moment ratio plot with the coordinate pair of the coefficient of variation (i.e., standard deviation divided by the mean) on the x-axis and skewness on the y-axis (based on Cirillo, 2013). Each parametric distribution has its particular curve of feasible coordinates, so that the area can be divided into a Pareto area, a lognormal area, and a gray area possibly belonging to both. Examining the top 4% of pixels in panel (a), all but one satellite are located in the area of indeterminacy. At higher thresholds, such as the top 1% in panel (b), the evidence in favor of a Pareto distribution becomes unambiguous. 20 from the lognormal area and those of other distributions. All but one satellite are in the Pareto area, far 20 It is well-established from the literature on top incomes or the size of cities that the Pareto distribution is often only a good representation of the very top of the relevant distribution, with slightly decreasing fit as the threshold decreases. Our data suggest that a Pareto distribution of top lights remains a good approximation until we go as low as the top 10% of data. This is more than sufficient for us, because only 3-5% pixels are suffering from top-coding in every year. 18

19 This prompts us to take a closer look at the issue of lognormal versus Pareto. Table B-10 in the Appendix shows the results from separate regressions of the empirical distribution function on the Pareto CDF and the lognormal CDF based on the top 4% of the data. The estimated coefficient for the Pareto CDF is closer to unity and the R 2 is substantially larger than in the lognormal counterpart (0.98 vs. 0.83). Figure B-4 visualizes this difference in fit for the year The lognormal CDF fits the data poorly, while the Pareto CDF is always closer to the empirical distribution The VIIRS data Since October 2011, the first satellite of the Suomi National Polar Partnership Visible Infrared Imagining Radiometer Suite (NPP-VIIRS) has been in orbit. The VIIRS daynight-band (DNB) on-board sensors have a much higher native resolution of 15 arc seconds, are radiometrically calibrated, do not suffer from top-coding, and record a physical quantity (radiance). Although the new system is undoubtedly superior in many respects, comparability with the previous series is limited for at least two reasons: i) the first annual VIIRS composite made available by NOAA refers to the year 2015, so that there is no temporal overlap with the DMSP-OLS series, ii) the VIIRS satellites have an overpass time around midnight, in contrast to the evening hours of the DMSP-OLS satellites, so that it is not entirely clear what kind of production and consumption activity they capture (Elvidge et al., 2014; Nordhaus and Chen, 2015). Hence, we only use these data as a robustness check. To compare the higher resolution VIIRS image to the DMSP data, we resample the raster to the DMSP resolution and then extract radiances of each pixel at the locations of the 10% sample that we have been using thus far. Naturally, there are considerable differences in the scale since the VIIRS-DNB records radiance. 21 The spatial Gini is much higher using the VIIRS data than in the radiance-calibrated data (0.79 vs ) which is owed to their improved sensors and finer resolution. Nevertheless, the top tail of the light distribution essentially exhibits the same properties. Table B-11 in the Appendix repeats the unrestricted and restricted rank regression from earlier tables. The results are remarkably similar. The Hill estimate for the top 5-3%, for example, is Additional tests confirm that in spite of their differences, the VIIRS data are as indicative of a Pareto tail in top lights as the radiance-calibrated data. 21 Note that radiance is measured in nano watt per steradian per centimeter (10 9 W cm 2 sr 1 ). The difference in scale is reflected in the summary statistics of the VIIRS data. The mean is 3.98, the standard deviation is 18.65, and the maximum is

20 4 Correcting for top-coding Based on this evidence, we propose a simple top-coding correction procedure. All lights below the top-coding threshold remain unaltered, while approximately 3-5% of pixels those above the threshold are replaced by a Pareto-distributed counterpart. An appealing feature of this approach is that it keeps most of the data intact, replacing only a small but highly influential amount of pixels. We can correct several aggregate statistics without knowing the location of the affected pixels, but most applications require a pixellevel correction. We still have to settle on which value(s) the shape parameter should take. Our theoretical and empirical arguments strongly point to a value around 1.5. To see this, plug in representative values into the model presented earlier to find α = 1/ Moreover, the empirical estimates presented above indicate a Hill estimates in the range of and an OLS estimates around when the top 3 5% of pixels are considered. The corresponding VIIRS estimate is also slightly larger than 1.5. We thus opt for rule-of-thumb parameter of α = 1.5. Of course, our procedure works with any chosen parameter value and we will make additional versions of our data set with parameters ranging from α = 1.2 to α = 1.8 available online Analytically correcting summary statistics Researchers are often interested in aggregate measures, such as average luminosity or light inequality in a region or a country. There are simple formulas to correct these summary statistics, since lights below and above the threshold are non-overlapping subgroups. Mean luminosity: The top-coding corrected mean luminosity µ of a country or region is simply the weighted average of the bottom and top means µ B and µ T. If the latter is the mean of a Pareto distribution starting at y c, we have α µ = ω B µ B + ω T µ T = ω B µ B + (1 ω B ) α 1 y c (10) where ω B and ω T = 1 ω B are the shares of pixels below and above the threshold It is not strictly necessary to use the same Pareto parameter across time and space. In additional analyses, we assigned the seven satellite-specific estimates to the adjacent saturated satellite-years. This approach gives very similar results overall, albeit with more jumps. Country-specific estimates also lead to similar results, although comparatively darker and poorer countries experience larger corrections. Additional results are available on request. 23 A simple numerical illustration shows how correcting for top-coding drives up the mean luminosity. If top-coding starts at y c = 55, affects 5% of the study area of interest, α = 1.5 and mean luminosity in the non-top-coded pixels is µ B = 10, then the corrected mean luminosity is rather than

21 Spatial Gini coefficients: The overall Gini can be written as the weighted sum of the bottom-share and top-share Ginis (i.e., the within-group Gini) as well as the difference between the top share of total lights minus the top share of pixels (i.e., the between-group Gini), such that G = ω B φ B G B + ω T φ T G T + [φ T ω T ], (11) where the shares of all light accruing to the top and bottom groups are φ B = ω B µ B /µ and φ T = ω T µ T /µ, and G T = (2α 1) 1. Section A.4 derives this result. A greater share of top-coded pixels ω T, brighter top-coded pixels φ T, and a greater spread in the distribution of the top-coded data G T all increase the size of the correction. Below we present examples illustrating these corrections. 4.2 Pixel-level replacement Aggregate statistics are not sufficient when researchers want to work with micro-level data of regions, cities, or grid cells, or plan to divide pixel-level information by population data. Our pixel-level replacement therefore directly replaces the underlying saturated lights value by its counterpart from a Pareto tail. The challenge is to geo-reference the theoretical values from the Pareto distribution so that the brightest pixels actually end up in the centers of dense urban agglomerations. It turns out that there is a straightforward solution. Since we know the location of all pixels, we can rank them according to their radiance-calibrated values and distribute the highest values from the Pareto distribution in the same way. Working with ranks also avoids importing the artificial variability of the radiance-calibrated satellites but preserves the structure of the data. Recall that ranks are more stable than the actual values. Another challenge are outliers at the extreme end of the theoretical distribution. The support of the Pareto distribution is unbounded, so that its highest quantiles with α = 1.5 will yield a handful of values greater than 5000 or even DN, far exceeding the natural limit of man-made light intensity. Our conservative solution is to work with a truncated theoretical Pareto distribution with upper bound 2000 DN. 24 An algorithm for pixel-level replacement: 1. For each of the 34 satellite-years t of saturated data, calculate the number X t of pixels 55 DN to be be replaced. 24 The truncated Pareto has the CDF F (y) = ( 1 ( yc y ) α ) ( / 1 ( y c H ) ) α for y c y H, where an upper bound of H = 2000 does not affect the overwhelming majority of saturated pixels 55 to be replaced, but does ensure realistic values at the very, very top. The upper threshold approximately corresponds to the average maximum in the world s brightest cities, see Table B-4 in the Appendix. 21

22 2. Produce a ranking of these X t pixels based on the radiance-calibrated data associated with the given satellite or the data from the closest year Generate X t theoretical values from a truncated Pareto distribution with the ruleof-thumb α = 1.5, the top-coding threshold y c = 55 and upper bound H = Replace the X t saturated pixels 55 so that the saturated pixel with the i-th highest rank from (2) is replaced by the i-th highest theoretical value. This procedure leads to our top-coding corrected nighttime lights panel at the pixel level over the entire period from 1992 to This is the data which we use in the following illustrations and applications. 4.3 Characteristics of the corrected data Our correction makes a substantial difference whenever larger cities are involved. We return to this issue in the applications, but note for now that, on average, 3.7% pixels above 55 DN account for 17.7% of the total sum of lights observed in the unsaturated data. After the correction, their share almost doubles to 31.94% (which is approximately true in every individual satellite-year). In other words, less than 4% of pixels account for about a third of all economic activity. The impact of the correction on other summary statistics is equally remarkable. Table B-12 in the Appendix reports mean luminosities and global Gini coefficients before and after the correction for each satellite, using both the analytic formulas and the data corrected at the pixel level. Our geo-referenced pixel-level replacement comes close to the analytic solutions but is generally more conservative (due to the fixed upper bound). Mean luminosity increases on average from 12.7 DN to DN and inequality in lights from 0.47 to Table B-13 in the Appendix reports the corresponding statistics in 2010 using a wider range of α parameters as robustness checks. Figure B- 2 in the Appendix plots the time series graph of global inequality in lights from 1992 to 2013, both before and after the top-coding correction, with α values of 1.4 and 1.6 serving as confidence bands for the benchmark case of 1.5. The global distribution of lights became slightly more unequal over the 1990s, remained flat in the first decade of the new millennium and then became temporarily more equal in the aftermath of the global financial crises and great recession. However, this year-to-year variation, in which measurement error might also play a role, is completely swamped by the size of the top-coding correction. 25 Whenever the radiance-calibrated data creates ties, we first try to break those ties using the ranks of the saturated data and then use the ranks of neighboring radiance-calibrated satellites. This produces a near unique ranking each time. We also experimented with other rankings. The results are very similar and available on request. 22

23 The global variation hides a lot of between-country heterogeneity. Figure B-5 in the Appendix illustrates this with a series of scatter plots. Both the correction in the mean and the Gini coefficient are strongly increasing in GDP per capita, weakly in country size and moderately in population density. Numerous developing countries experience sizable corrections (such as Egypt, Paraguay or Mexico). City states, such as Singapore, have large top-coding corrections, as do smaller countries, like Israel and Estonia. Nevertheless, even large countries like the US experience a sizable increase in both mean luminosity (plus 7 DN) and the Gini coefficient (plus 14 percentage points). No single factor captures all the relevant heterogeneity. Instead, the correction is a complex function of the spatial equilibrium, that is, the size, number, and brightness of larger cities in each country. 5 Applications 5.1 Night lights and local GDP Growth Our first application examines the relationship between lights and GDP at the regional and local level. We follow Henderson et al. (2012) who first investigated light-output elasticities at the national level. They calculate average light intensity in each cell that falls on land, weighted by the size of that particular cell, and then run fixed effects regressions of log GDP on their measure of log lights per square kilometer. They report an income elasticity of lights around Comfortingly, Table B-14 in the Appendix shows that we are able to replicate their national-level results. In fact, even at this high level of aggregation, our top-coding corrected data perform slightly better. 26 The main benefit of night lights is that they allow us to study local or regional development which is precisely where top-coding matters most. We first address the perceived wisdom that lights are poor predictors of output in developed economies. We choose Germany as an example of a rich and decentralized country with a wide geographic distribution of economic activity and excellent regional accounts. Lessmann et al. (2015) compile GDP per capita estimates and other statistics at the district level in Germany over the period from 2000 to To this, we add our estimates of lights per capita from the saturated and corrected data. Lessmann et al. (2015) show that the light-output elasticity using the saturated data is substantially lower for German districts than at the country level and even becomes indistinguishable from zero when population and area are used as controls. 26 When we define a rich-and-dense dummy for countries which were richer and more densely populated than the world average in 2000, we cannot reject the null hypothesis that in these countries the relationship between lights per km 2 and GDP is zero in the saturated data. But we are able to reject this hypothesis in the corrected data. We also find that the radiance-calibrated data perform worse than our corrected data, which underlines their inaccuracy and validates our strategy of only relying on them to infer the distribution of pixels at the very top. 23

24 Table 3: Income elasticity of lights, Germany at NUTS-3 level, cross-section Dependent variable: GDP per capita Saturated data Corrected data (1) (2) (3) (4) (5) (6) Lights per capita (0.019) (0.094) (0.093) (0.018) (0.101) (0.101) Population (0.041) (0.042) (0.041) (0.042) Area in km (0.062) (0.061) (0.068) (0.068) Urban (0.043) (0.042) Former FRG (0.022) (0.023) Adjusted-R Regions Notes: The table reports cross-sectional OLS estimates. All columns include a constant (not shown). Lights per capita, population and area are all measured in logs. Urban and former FRG are binary variables. The specifications are variants of ln y i = β ln lights i + x i γ + ɛ i, where x it is a vector of controls. The data have been averaged over the period from 2000 to Robust standard errors in parentheses. Significant at: p < 0.10, p < 0.05, p < Table 3 compares the light output elasticities obtained from cross-sectional regressions using the saturated data (columns 1 to 3) and our corrected data (columns 4 to 6). The informational value of the saturated lights drops as population, area, an urban dummy, and an old federal republic dummy are added to the regression. 27 In fact, columns (2) and (3) suggest that there is little value added in using night lights to predict local output once population and the spatial extent of the region are accounted for. This picture changes completely with the corrected data in columns (4) to (6). The sub-national light output elasticity rises substantially and remains highly significant even in the presence of additional controls. Our corrected data without top-coding correlate much better with local output. Instead of falling, the cross-sectional light-output elasticity reaches levels comparable to its national counterpart a finding that is completely new to the literature. This result carries over to changes in light and output over time albeit with some qualifications. Table 4 repeats the regression from Table 3 using panel data from 2000 to For the saturated data, the inclusion of state and time fixed effects decreases the light-output elasticity to one third of its national counterpart. With the corrected data, most specifications recover results comparable to the cross-sectional estimates if we use within state variation, see columns (4) and (5). The estimates in columns (3) and 27 The former FRG dummy is included, since the East has benefited substantially from public investments in unified Germany and also adopted energy-saving lights differently than the West. 24

25 Table 4: Income elasticity of lights, Germany at NUTS-3 level, Dependent variable: GDP per capita Saturated data Corrected data (1) (2) (3) (4) (5) (6) Lights per capita (0.059) (0.060) (0.008) (0.067) (0.068) (0.008) Population (0.038) (0.039) (0.077) (0.036) (0.039) (0.077) Area in km (0.039) (0.040) (0.045) (0.045) Urban (0.043) (0.043) Time FE Yes Yes Yes Yes Yes Yes State FE Yes Yes No Yes Yes No Municipality FE No No Yes No No Yes Within-R Observations Regions Notes: The table reports panel fixed effects estimates. Lights per capita, population and area are all measured in logs. Urban is a binary variable. The specifications are variants of ln y it = β ln lights it + x it γ + c i + s t + ɛ it where x it is a vector of controls, c i is a vector of state or municipality fixed-effects, and s t are time dummies. NUTS-3-clustered standard errors in parentheses. Significant at: p < 0.10, p < 0.05, p < (6) using the different data sources come very close once we include municipality FEs. This occurs for two reasons. First, the within-district variation is very small to begin with, making it difficult to empirically trace out the light-output relationship. Second, since the urban structure of Germany is very mature, our correction primarily affects the cross-sectional ordering of districts, rather than their ranking over time. In any case, we either substantially improve estimates of the light-output relationship at the local level or at the minimum provide comparable answers. Figure 7 plots the average light intensity per square kilometer of the saturated and corrected series over the average population density of all districts in Germany to better understand the nature of the correction. Clearly, the two series begin to diverge after a density of about a thousand people per km 2. The saturated series display an asymptotic movement towards the top-coding threshold of slightly above 100 DN whereas the corrected series is approximately linear in population density. 28 Another 28 Note that when working with square kilometers rather than arc second pixels, the saturated series in lights per square kilometer is not bounded by 63 DN. Most pixels in Germany are smaller than 1 km 2 but bigger than 500 m 2, hence the theoretical upper bound is slightly above 100 DN, eg. 63 DN/0.5 km 2 = 126 DN/km 2. This occurs because the plate carrée projection keeps the area of each pixel constant in degrees not kilometers. 25

26 Figure 7: Light intensity versus population density in German regions Corrected Stable lights Frankfurt Berlin Munich Average DN per square km Hamburg Frankfurt Munich Berlin Hamburg Population per square km (in 1000s) Notes: Illustration of the value added of the top-coding corrected data for urban economics. The data are cross-sectional averages of lights per km 2 and population density in German NUTS-3 regions over the period from 2000 to notable feature is that the three brightest cities according to the stable lights data are all medium-sized cities with populations (well) below half a million (Herne, Oberhausen and Gelsenkirchen) whereas the corrected data perfectly identifies the three largest and most populated economic centers as the brightest (Frankfurt, Munich and Berlin). In sum, top-coding is a major issue for estimating urban-rural differences, precisely because there the cross-sectional comparison matters most. Using the German data, we establish two fundamental facts that are likely to hold in other settings as well. First, topcoding rises with urbanization. Second, the economic ranking of cities is counter-intuitive in the saturated data but a sensible ranking emerges after the correction. 5.2 Cities in Sub-Saharan Africa Our second and most extensive application deals with urbanization and development in Sub-Saharan Africa. As urbanization is rapidly accelerating across the developing world, the debate on its causes and consequences is receiving renewed attention. Sub-Saharan Africa is a particularly interesting case in point. It is still the world s poorest region but urbanizing at a much earlier level of development than other regions (e.g. see Glaeser, 2014; Castells-Quintana, 2017). 472 million people live in urban areas in Africa as a whole, a number which is set to double within the next 25 years (United Nations, 2014). Despite the tremendous interest in gauging the evolution of African cities, official numbers have proved elusive. Owed to poor data and rapid expansion, more and more papers are 26

27 making use of nighttime lights to measure the economic activity and urban extent of African cities (e.g. see Fetzer et al., 2016; Storeygard, 2016; Henderson et al., 2017). Our analysis focuses on the difference between primary and secondary African cities. It is well-known that African countries have large primacy ratios: most economic activity and population is concentrated in the biggest city (Henderson, 2005; Junius, 1999). This kind of spatial concentration is linked to poor infrastructure and a low level of development (Krugman, 1991; Puga, 1998). Political institutions play an important role. Autocratic countries have higher primacy ratios, while democracies have a more balanced distribution of economic activity (Ades and Glaeser, 1995; Davis and Henderson, 2003). As economies develop, that is, trade, infrastructure and political institutions improve, it is ambiguous whether secondary cities catch up or if primary cities outgrow them further (Duranton, 2008). Fetzer et al. (2016) find that moves towards democracy in Africa lead to higher growth of smaller cities at the expense of the primary city. It is therefore important to get their relative growth rates right. African cities are also known to be crowded and fragmented, typically consisting of a dense core and a scattered collection of small neighborhoods with limited intra-city infrastructure (Lall et al., 2017). This sets them apart from cities in other parts of the world and limits the positive effects of agglomeration. Little is known so far on whether all African cities are structured similarly or whether fragmentation has been changing over time due to a lack of suitable data. Hence, last but not least, we move beyond growth at the city level and use our new data to study fragmentation of primate and secondary cities over time. We first have to identify the urban boundaries and classify cities as either primary or secondary. Applying a DN threshold to continuously lit clusters of night lights accurately detects city boundaries and prevents overestimating the urban extent (e.g. see Small et al., 2011). This is especially relevant in Africa where urbanization has been rapid and administrative boundaries do not reflect the footprint of most cities. To study city growth at the intensive margin, we necessarily have to keep city boundaries fixed over time. Boundaries from earlier satellites identify fewer cities with a smaller urban extent, while later satellites produce a sample selection problem akin to Baumol s fallacy, tautologically generating growth of previously darker areas. In addition, measurement errors imply that two satellites from adjacent years might point to noticeably different city boundaries. We solve these trade-offs by defining cities as contiguously lit areas over the period from 1992 to 1994, i.e. four different satellites at the beginning of the sample. A pixel is counted as part of the contiguous cluster when it is brighter than a threshold of 16 DN in three out of four satellites. 29 The threshold of 16 DN, or 25% of the stable lights detection range, is not chosen arbitrarily. Using a similar procedure on the latest satellites 29 Before applying this procedure we process the light rasters by applying a land mask and setting all known gas flare areas to missing. 27

28 and comparing the detected urban extents with Google Maps images of the respective city, we find this threshold matches the urban footprint in SSA very well. We identify city locations by intersecting these urban extent polygons with all settlement points in the Global Rural-Urban Mapping Project (GRUMP) that fall within 4 km of the urban area. A city polygon receives the name and attributes of the largest settlement (according to their population in 2000) that is located within the expanded urban perimeter. A few polygons that do not coincide with a known settlement are dropped. Finally, we identify the primate city of a country as the city with largest population and define all other cities as secondary. Figure B-6 in the Appendix shows the boundaries and current maps of Johannesburg, Lagos, Luanda and Nairobi. Figure 8: Primary and secondary cities in SSA BWA BEN BDI CAF DJI ERI GMB GHA CIV KEN LBR LSO MDG MWI MLI MOZ NER GNB RWA SLE SOM TGO TZA UGA BFA SWZ ZMB GIN SEN MRT ZWE NAM ETH AGO TCD COD CMR GAB NGA ZAF SDN COG Region Eastern Africa Middle Africa Northern Africa Southern Africa Western Africa Population in mln 4 mln 6 mln 8 mln 10 mln Notes: Illustration of the location of 42 primate and 525 secondary cities in SSA using the urban footprint detection algorithm outlined in the text. Note that we lose Equatorial Guinea due to gas flaring near the capital of Malabo on the island of Bioko. Figure 8 shows a map of our universe of 567 cities and their population in cities, one in each country, are defined as primate, while 525 are secondary cities. The largest and brightest city in the sample is Johannesburg. 30 We track these cities over the period from 1992 to 2013, calculating their total and average light intensities, as well as measures of inner-city variation and fragmentation. 31 Note that our measure of economic activity at the city level will be mostly driven by population density (Henderson et al., 30 Primacy in urban extent and population usually coincide, so that in most cases we could have equally well used the largest urban polygon in each country. 31 We average over satellites so that there are no multiple observations per year. 28

29 2018) and thus primarily describes population growth. We solely focus on city growth at the intensive margin to separate this analysis from the various forms of urban expansion, where new patches are added through leapfrogging or slow sprawl. Table 5: Summary statistics of African cities, Primate Cities Secondary cities Saturated Corrected Saturated Corrected Sum of lights Annual growth Mean luminosity Annual growth Coefficient of variation Annual change (relative) Moran s I (scaled) Annual change (relative) Notes: The table reports a selection of summary statistics for African cities. The data are computed for each city and then averaged across 42 primate and 525 secondary cities. Top-coding affects primary and secondary cities very differently: the correction is much greater in primate cities. Table 5 reports the summary statistics averaged across all cities of the same type. Primate cities emit more light, both in total and on average. The correction strongly raises their light emissions, particularly in later years when more pixels reach the top-coding threshold. Secondary cities, on the contrary, experience a relatively mild correction that is only moderately increasing over time. This has important implications for their relative growth performance. Secondary cities artificially appear to be growing faster than primate cities before the correction a picture that is completely reversed afterwards. The annualized growth rate of primate cities increases by about 0.5 percentage points and now marginally exceeds growth in secondary cities. This stands in stark contrast to existing findings stressing the rise of secondary cities in Africa (e.g. see Fetzer et al., 2016; Storeygard, 2016). Table B-16 in the Appendix presents similar data for each country, listing the primate city, as well as the growth rates of both the primary and all secondary cities. Top-coding plays no role in small primate cities, such as Bissau. However, in Kinshasa, Johannesburg, Luanda, Maputo, Dakar and Khartum, the annualized growth rates increase by 1% or more. 29

30 We underpin these descriptive results with several panel regressions. In each case, we regress the log of average lights emitted by city j in country i at time t, ln lights ijt, on a linear time trend, an interaction of a linear time trend with an indicator for primate cities, P ij, and a battery of fixed or random effects that vary across specifications. We typically include city or country effects and satellite dummies to purge measurement errors. 32 Table 6: Growth regressions for African cities, Dependent variable: Log mean lights Saturated data Corrected data (1) (2) (3) (4) (5) (6) t (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) t P ij (0.002) (0.002) (0.002) (0.003) (0.003) (0.003) Estimation Fixed Fixed Random Fixed Fixed Random City FE or RE Yes Yes Yes Yes Yes Yes Satellite FE No Yes Yes No Yes Yes Country FE No No Yes No No Yes Within-R Observations Cities Notes: The table reports the results of city-level panel regressions using the saturated and top-coding corrected data. The specifications are variants of ln lights ijt = β 1 t + β 2 (t P ij ) + c ij + s t + ɛ ijt where t is a linear time trend, P ij is an indicator for primate cities, c ij is a vector of city and/or country effects and s t contains satellite dummies. Standard errors clustered on the city are in parentheses. Significant at: p < 0.10, p < 0.05, p < Table 6 confirms that primate cities outgrew secondary cities by a significant margin over this period. As expected, all fixed effect specifications using the uncorrected data misleadingly suggest that there is no difference between city types. With the corrected data, the interaction term is significant at the 5% level. Primate cities grow 0.6 percentage points faster on average than secondary cities. This amounts to a sizable difference of over 250% for our entire study period of 21 years. Only the random effects specifications tend to give similar answers for both data sources. Yet, here too, the growth differential between primary and secondary cities increases by 0.5 percentage points to 0.9% using the corrected series. In a nutshell, we see that secondary cities in Africa are not catching up, while primate cities are consolidating their position. Our corrected data also allow us for the first time using lights to study what is going on within the city. As discussed earlier, recent research is pointing towards a particularly 32 Note that, contrary to common practice, our vector of satellite dummies accounts for all the possible combinations of satellites, e.g. the fact that any value for 1993 is an average of satellites F10 and F12 which is different than a value that is solely obtained from either F10 or F12. 30

31 fragmented structure of African cities. We compute two indices for each African city and every year from 1992 to 2013 to study this phenomenon: the coefficient of variation (CV) and Moran s I (Moran, 1950). The CV is a simple inequality measure capturing the variation of light intensities across an entire city. A low (high) value indicates big (small) inner-city differences. Moran s I takes the precise location of the pixels within a city into account. It is a measure of spatial autocorrelation often used in regional science, but has not been used to study city fragmentation so far. We find it very suitable for this purpose. Positive values of Moran s I indicate that pixels are surrounded by others of similar brightness (positive autocorrelation), while negative values would reflect heterogeneous areas of dissimilar pixels (negative autocorrelation). 33 A monocentric city in which luminosity slowly decreases from the center to the outskirts will have a higher Moran s I than a checkered city in which bright and dimly lit areas take turns, even at a comparable CVs. Only both measures together provide a complete picture. Primate cities are brighter, substantially more fragmented than secondary cities, and therefore also experience stronger corrections. The correlation between the CV (Moran s I) for primate cities in the corrected and saturated data is only about 0.46 (0.29), while it is above 0.98 for secondary cities. As Table 7 shows, no single city occupies the same rank in the top 10 lists of both data sources and fragmentation measures. Cities such as Luanda, Kinshasa and Nairobi have a bright center surrounded by similarly bright areas with a slow decay towards darker outskirts. This is reflected in a high inner city variance (high CV) together with comparatively high degree of spatial autocorrelation (high Moran s I). Cities like Johannesburg are structured very differently. As one of the largest urban agglomerations in the world, Johannesburg encompasses several smaller cities and sub-centers. In line with this, we clearly observe numerous smaller bright clusters surrounded by darker areas in our data (see Figure 1). Johannesburg has the highest CV after the correction (doubling from to 0.67) but such a low Moran s I that it does not make the top 10 list. It is much more fragmented than the other cities. Over time, African cities have become both less unequal in terms of light and more fragmented (see Table 5), but the magnitude of this observed development is influenced by top-coding. Table B-17 and Table B-18 in the Appendix run the same regressions as before, but with the city-level CV or Moran s I as the dependent variable. Pixels within ever brighter primate cities appear too similar in the saturated data, suggesting a decline in inner city variation that is not as pronounced in the corrected data. There the difference 33 Moran s I is defined as I = N i j w ij(x i x)(x j x) S 0 i (x i x) 2 where N are the number of pixels in each city, w ij are elements of an inverse distance weight matrix, S 0 is the sum of all w ij, x i or x j is the pixel-level luminosity and x is mean luminosity. We work with a scaled version of Moran s I to make cities consisting of different numbers of pixels comparable, that is, we subtract its expected value under the null hypothesis of no spatial correlation: I = I E[I] = I 1 N 1. 31

32 Table 7: Coefficient of variation and Moran s I, African cities in 2013 City Saturated statistic City Corrected statistic Panel a) Top 10 cities with highest Coefficients of Variation Mogadishu Johannesburg Harare Luanda Bangui Mogadishu Mbabane Lusaka Asmara Alkhartum Bujumbura Kinshasa Johannesburg Harare Kinshasa Windhoek Antananarivo Gaborone Lusaka Bangui Panel b) Top 10 cities with highest Moran s I Nairobi Luanda Mogadishu Kinshasa Kinshasa Nairobi Maseru Dakar Lusaka Niamey Conakry Mogadishu Gaborone Maputo Antananarivo Lusaka Bangui Gaborone Niamey Djibouti Notes: The table summarizes information about the inner-city variation of lights in African cities. Panel a) lists the ten primate cities with the highest coefficient of variation based in terms of saturated and corrected data. Panel b) does the same based on Moran s I. between primary and secondary cities halves and the negative trend in secondary cities is no longer significant. We also observe a downward trend in spatial autocorrelation, that is, an increase in fragmentation. Interestingly, this development is uniform across primary and secondary cities once the correction is applied another finding which has so far escaped the literature. This speaks to two very different types of development. In primate cities, sub-centers are forming and growing relative to the core, so that the overall variance decreases over time while the city fragments. In secondary cities, development is also patchy but without fundamentally changing the city-wide distribution of economic activity relative to the central business district. Again, Johannesburg is a notable exception. It has become substantially less fragmented over time (as shown in Figure B-7 in the Appendix). It is important to emphasize that there is nothing tautological about these results in our correction; if anything the procedure provides a lower bound of primate city growth and fragmentation. Top-coded pixels that become brighter over time move up in the global ranking and thus receive higher theoretical values. The correction does not care if 32

33 these pixels are located in primary or secondary cities, or if they are surrounded by other bright pixels. This is precisely how Johannesburg bucks the trend of other primate cities. 6 Concluding remarks While satellite data of nighttime lights are an increasingly popular proxy for economic activity, they suffer from top-coding and severely underestimate the brightness of most cities. The key contribution of this paper is to provide a solution to this problem which unlocks the full potential of nighttime lights for the study of all city-related questions. Our solution rests on the claim that top lights can be characterized by a Pareto distribution. We support this conjecture in two ways. First, a model of luminosity emitted by large cities suggests that plausible assumptions directly lead to a power law in light emissions. Second, a battery of empirical tests indicates that a Pareto distribution is a sound representation of the data. Naturally, other parametric or non-parametric approaches are possible, but we find it most appealing to directly link the distribution of bright lights to both Zipf s law of cities and to the standard tail extrapolation problem. On this basis, we propose a correction which can be applied in two very different ways. Both involve the replacement of top-coded values by a Pareto tail. At the aggregate level, we provide simple formulas for the top-coding corrected mean and Gini coefficient of inequality in lights. At the pixel level, we use a geo-referenced ranking procedure to replace the top-coded pixels with their theoretical counterparts. This yields a new global panel of high-resolution images over the period from 1992 to The new data set lends itself to numerous applications. To provide a glimpse into its potential utility, we explore this data in two directions. We first benchmark the data in a high-income, high-quality data setting. Using Germany as a case study, we show that the corrected data better predicts local economic activity than the existing data and that it produces a meaningful cross-sectional ranking of Germany regions. Our second application focuses on urban primacy in Sub-Saharan Africa. Even though most African countries are barely affected by top-coding, it turns out that the correction is important enough to overturn previous insights on city growth. Contrary to earlier studies, we show that primary cities are solidifying their position vis-à-vis secondary cities by growing at a premium of about 0.5% a year at the intensive margin. Finally, we push the data even further to study inner city differences. We find that the city level variance of the average primary city in Sub-Saharan Africa has decreased over time, but that primate and secondary cities have both become more fragmented within. This naturally leads to the question of the causes and consequences of these developments. We only take this analysis as a first step, but hope that this new data opens the door to further research on inner-city differences and city growth, as well as many other promising applications. 33

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37 Appendix A Additional proofs and derivations A.1 Deriving the CDF of the number of rings Note that r = π 1/2 x 1/(2φ) implies x = π φ r 2φ and dx = 2φπ φ r 2φ 1 dr. Substituting these definitions into eq. (1) and integrating yields the CDF of the number of rings per city as presented in eq. (2) of the main text r [ F (r) = 2φx c π φ r 2φ 1 dr = 2φx c π φ 1 ] r r 2φ r 2φ r 0 for r < r = π 1/2 x 1/(2φ) c = 1 y c π φ r 2φ for r >= r = π 1/2 x 1/(2φ) c. A.2 Deriving the density of pixels (A-1) Start with the distribution of the number of pixels. At distances d < d, the amount d of pixels increases linearly in d as rings farther away contain more pixels: dd πd2 = 2πd. Beyond d, the effect within each city has to be multiplied by the survival function 1 F (r) from eq. (2), as there are fewer and fewer cities of such size. Denoting the number of cities as M, the absolute amount of pixels N as a function of d is 2πdM for d < d = π 1/2 xc 1/(2φ) P (d) = 2π 1 φ Mx c d 1 2φ for d d = π 1/2 x 1/(2φ) c. The total number of pixels, N, can be obtained by integration (A-2) d [ ] d [ ] 1 N = 2πdMdd + 2π 1 φ Mx c d 1 2φ dd = 2πM 0 d 2 d2 + 2π 1 φ 1 Mx c 0 2 2φ d2 2φ d ( ) = πm y1/φ cπ + π1 φ My c y 1/φ 1 φ c = Mx 1/φ c + 1 φ 1 π φ 1 Mx1/φ c = φ φ 1 Mx1/φ c. (A-3) Dividing eq. (A-2) by N yields the density, f(d), shown in eq. (4). A.3 An exponential distribution of light in cities This part derives the distribution of top lights if lights within a city follow a negative exponential function. The following replaces assumption 4 from the main text. Assumption 5. Within cities, the light density l(d) follows an exponential function l(d) = L 0 exp( γd), where L 0 l is the density at the center and γ > 0 is a decay parameter. 37

38 Contrary to our baseline case this does not directly generate a Pareto distribution in lights. The distribution depends on L 0, the maximum luminosity of the center of the city. We consider three cases. In each case, we focus on the light density f(l) conditional on L 0. Using the variable transformation of eq. (4) together with d = 1 γ ln(l 0/l) yields 2π φ f(l L 0 ) = 2π φ 1 φ 1 φ φ 1 x1 1/φ c x 1/φ c } {{ } c where l = L 0 exp( γ d). [ 1 2φ 1 ln(l γ 0/l)] for l l 1 ln(l 0 /l) for γ l < l < L 0, (A-4) This density increases for dim luminosities (at the fringes of the largest cities) and decreases from l onwards. The turning point and maximum, l, corresponds to the minimum size, d, of each city in terms of distance from center. At higher luminosities, there are fewer and fewer pixels as these are the ones located in ever smaller rings closer to the center. To derive the marginal density f(l), we have to make assumptions about the distribution of maximum luminosities L 0 across cities. Case 1: L 0 is a constant, so that all cities large and small are equally bright in the center. This is unrealistic, but mathematically simple. The marginal density of lights f(l) equals the conditional density f(l L 0 ). Analyzing the top end of the distribution c ln(l 0 /l) for l < l < L 0 we observe a near linear decrease with a slope of c d dl ln(l 0/l) 1 l for l close to L 0. There is no power law. The assumption of a constant L 0 across all cities generates too many pixels with the highest luminosities. Case 2: L 0 follows a Pareto distribution across cities so that some city centers are much brighter than others. Empirical evidence points in this direction (see Table B-4). We are only interested in the upper part of the density in eq. (A-4). 34 If L 0 follows the Pareto density with α = 1 so that f(l 0 ) = L min /L 2 0, with L min as the minimum center luminosity, we have the joint density of l and L 0 as f(l, L 0 ) = f(l 0 )f(l L 0 ) = c L min L 2 0 ln(l 0 /l) for l max < l < L max and L 0 > l. (A-5) The marginal density, f(l), is found by integrating L 0 out of the above ] Lmax Lmax [ 1 ln l ln L0 1 f(l) = cl min ln(l l L 2 0 /l)dl 0 = cl min 0 L 0 l [ 1 = cl min l 1 ( ln(lmax /l) + 1 ) ], (A-6) L max 34 Note that the threshold l depends on the random variable L 0. Hence, we restrict our analysis to the area of the density starting at the highest possible threshold value l max, corresponding to L 0 = L max. 38

39 which holds for l max < l < L max. Case 3: Alternatively, an intermediate case for L 0 is a uniform distribution between values L min and L max with density f(l 0 ) = (L max L min ) 1. Following the same steps as in the second case, the marginal density is Lmax c c [ ( f(l) = ln(l 0 /l)dl 0 = L 0 ln L0 ln l 1 )] L max L max L min l L max L min l c [ ( = l + L max ln(lmax /l) 1 )], L max L min (A-7) which holds for l max < l < L max. Cases 2 and 3 are heavy-tailed distributions which differ mathematically from the simple Pareto. But given their heavy tails, they may be approximated by a Pareto. As shown in Figure A-1, this works particularly well for case 2 with a Pareto distribution of α = 1.5, while for case 3 the Pareto distributions with α = 1.2 works reasonably well. Result 2. Based on Assumptions 1 3 and 5, as well as a sufficient variation of maximum luminosities across cities, it follows that top lights above a threshold l can be approximated by a Pareto distribution. In sum, when inner city lights follow a negative exponential function, a Pareto distribution of top lights does not arise analytically in the three cases considered here. However, depending on the exact assumptions made about differences in the maximum brightness across cities, the resulting distribution is heavy-tailed and can be approximated by a Pareto distribution. Figure A-1: Approximating the theoretical densities with Pareto distributions (a) Case 2 and Pareto with α = 1.5 (b) Case 3 and Pareto with α = 1.2 Density Case 2 Pareto Density Case 3 Pareto Light intensity (DN) Light intensity (DN) Notes: The left panel shows the pixel density from eq. (A-6) with x c = 10000, φ = 1.5 and γ = 1.5. The Pareto distribution with α = 1.5 is scaled to fit. The right panel shows the pixel density from eq. (A-7) with x c = 10000, φ = 1.5 and γ = 1.5. The Pareto distribution with α = 1.2 is scaled to fit. 39

40 A.4 Deriving the correction of the Gini coefficient Following Mookherjee and Shorrocks (1982) we begin by defining the Gini coefficient over multiple groups as G = 1 2N 2 µ = 1 2N 2 µ = k y i y j (A-8) i j y i y j + y i y j k i Nk j N k i N k j / N k ) 2 µ k µ G k + 1 y 2N 2 i y j. µ k j / N k ( Nk N i N k (A-9) (A-10) where G K is the within group Gini coefficient of group k. The second term is a measure of group overlap including their between group differences. Perfect separation (no overlap between groups) implies i N k j N h y i y j = N k N h µ k µ h. Hence, we can simplify equation eq. (A-10) to G = k ( ) 2 Nk µ k N µ G k + k h N k N h 2N 2 µ µ k µ h. (A-11) With two bottom and top groups k, h {B, T } (where µ T > µ B ) and some algebra, this becomes G = ( NB N ) 2 ( ) 2 µ B µ G NT µ T B + N µ G T + [ (NT N ) 2 µ T µ N T N ]. (A-12) Now define the population (pixel) shares below and above the threshold as ω B and ω T, where ω T = 1 ω B and the group s share of all income (light) as φ B = ω B µ B µ and φ T = ω T µ T µ to obtain eq. (11) from the main text. 40

41 Appendix B Additional Tables and Figures Table B-1: Percentiles of unsaturated values at given saturated values in 2000 Sat. DN / Rad-Cal CDF 5% 25% 50% 75% 95% 99% Notes: The table reports values from the cumulative distribution function of the radiance calibrated lights which are associated with a given saturated value (from 55 to 63). For instance, 25% of the radiance-calibrated values associated with a saturated value of 61 DN, are below The data are a representative 10% sample for the year

42 Table B-2: Summary statistics of the saturated and unsaturated data Year Saturated Unsaturated Satellite Mean St.Dev. Gini % 55 Rad-Cal Mean St.Dev. Gini Max Notes: The table reports summary statistics using a 10% sample of the saturated and unsaturated lights at the pixel level, where each pixel is arc seconds. There are a number of years when two satellites of the stable lights product flew concurrently, so that there are 34 satellite-years between 1992 and The radiance-calibrated data are only available for the following observation periods: 16 Mar Feb 97 (1996), 19 Jan Dec 99 (1999), 03 Jan Dec 00 (2000), 30 Dec Nov 2003 (2003), 18 Jan Dec 04 (2004), 28 Nov Dec 06 (2006), and 11 Jan 10 9 Dec 10 (2010). 42

43 Table B-3: Regression of saturated on unsaturated data Year Unsaturated (0.0002) (0.0003) (0.0002) (0.0002) (0.0001) (0.0002) (0.0004) Constant (0.0045) (0.0054) (0.0044) (0.0043) (0.0036) (0.0037) (0.0066) R Notes: The table reports OLS estimates of a regression of all pixels smaller than 55 DN of the saturated data on their counterpart in the unsaturated data in all those years for which both data sources are available. Standard errors are in parentheses. The data are a 10% random sample of lights at the pixel level, where each pixel is arc seconds. 43

44 Table B-4: Maximum light intensities in 30 selected cities over time City Average Beijing Berlin Bogota Brussels Cairo Calgary Casablanca Damascus Dhaka Dubai Edinburgh Foshan Istanbul Jakarta Johannesburg London Los Angeles Manila Moscow Mosul Mumbai Nairobi New York Paris Rio de Janeiro Seoul Shanghai Sydney Tel Aviv Tokyo Notes: The table report the maximum light intensity in DN recorded within 25 km radius of the city center in selection of cities. The input data are the radiance-calibrated lights. City locations are from Natural Earth s vectors of major populated places. 44

45 Table B-5: Correlation matrix of maximum city lights Years Notes: The table reports correlations between the maximum light intensities recorded within 25 km radius of the city center of 988 world cities with more than 500,000 inhabitants. Table B-6: Rank correlation matrix of maximum city lights Years Notes: The table reports rank correlations between the maximum light intensities recorded within 25 km radius of the city center of 988 world cities with more than 500,000 inhabitants. 45

46 Table B-7: Fitted Generalized Pareto distributions, varying thresholds Satellite F6 (96) F6 (99) F6 (00) F6 (03) F6 (04) F16 (06) F16 (10) Average Panel a) Top 5% ln σ (0.0062) (0.0051) (0.0053) (0.0052) (0.0050) (0.0051) (0.0049) (0.0052) ξ (0.0056) (0.0043) (0.0044) (0.0042) (0.0040) (0.0040) (0.0038) (0.0043) Threshold Observations 96, , , , ,899 99, , ,954 Panel b) Top 4% ln σ (0.0066) (0.0055) (0.0055) (0.0057) (0.0054) (0.0055) (0.0052) (0.0056) ξ (0.0057) (0.0044) (0.0044) (0.0045) (0.0042) (0.0043) (0.0039) (0.0045) Threshold Observations 77,348 93,484 85,481 80,075 85,489 79,589 86,195 83,952 Panel c) Top 3% ln σ (0.0069) (0.0060) (0.0060) (0.0063) (0.0061) (0.0062) (0.0058) (0.0062) ξ (0.0056) (0.0047) (0.0045) (0.0049) (0.0047) (0.0047) (0.0042) (0.0048) Threshold Observations 58,010 70,112 64,110 60,057 64,133 59,691 64,646 62,966 Panel d) Top 2% ln σ (0.0079) (0.0070) (0.0073) (0.0076) (0.0075) (0.0075) (0.0070) (0.0074) ξ (0.0060) (0.0053) (0.0055) (0.0057) (0.0058) (0.0056) (0.0050) (0.0056) Threshold Observations 38,673 46,742 42,740 40,039 42,755 39,795 43,097 41,977 Panel e) Top 1% ln σ (0.0109) (0.0099) (0.0103) (0.0108) (0.0109) (0.0108) (0.0100) (0.0105) ξ (0.0082) (0.0074) (0.0078) (0.0082) (0.0086) (0.0082) (0.0074) (0.0080) Threshold Observations 19,337 23,371 21,370 20,019 21,378 19,897 21,548 20,989 Notes: The table reports parameter estimates from fitted the Generalized Pareto distribution shown in eq. (6). The input data are a 10% representative sample of all non-zero lights in the radiance-calibrated data above the defined threshold at the pixel level, where each pixel is arc seconds. 46

47 Table B-8: Unrestricted rank regressions Satellite F6 (96) F6 (99) F6 (00) F6 (03) F6 (04) F16 (06) F16 (10) Average Panel a) Top 5% y i (0.0012) (0.0012) (0.0013) (0.0013) (0.0013) (0.0014) (0.0015) (0.0013) y c (0.0014) (0.0015) (0.0016) (0.0016) (0.0015) (0.0017) (0.0018) (0.0016) R Observations 96, , , , ,899 99, , ,955 Panel b) Top 4% y i (0.0015) (0.0014) (0.0016) (0.0016) (0.0014) (0.0017) (0.0018) (0.0016) y c (0.0017) (0.0017) (0.0019) (0.0018) (0.0016) (0.0019) (0.0020) (0.0018) R Observations 77,348 93,484 85,482 80,075 85,489 79,590 86,196 83,952 Panel c) Top 3% y i (0.0019) (0.0018) (0.0019) (0.0019) (0.0016) (0.0020) (0.0021) (0.0019) y c (0.0022) (0.0020) (0.0022) (0.0021) (0.0018) (0.0022) (0.0024) (0.0021) R Observations 58,011 70,115 64,111 60,058 64,134 59,692 64,647 62,967 Panel d) Top 2% y i (0.0025) (0.0023) (0.0022) (0.0023) (0.0018) (0.0024) (0.0025) (0.0023) y c (0.0029) (0.0025) (0.0025) (0.0025) (0.0020) (0.0026) (0.0027) (0.0025) R Observations 38,673 46,742 42,740 40,039 42,756 39,794 43,097 41,977 Panel e) Top 1% y i (0.0039) (0.0031) (0.0031) (0.0030) (0.0027) (0.0032) (0.0029) (0.0032) y c (0.0043) (0.0034) (0.0034) (0.0033) (0.0029) (0.0035) (0.0031) (0.0034) R Observations 19,337 23,373 21,372 20,020 21,377 19,898 21,551 20,990 Notes: The table reports OLS results obtained from the unrestricted rank regressions eq. (9) at various relative thresholds. The input data are a 10% representative sample of all non-zero lights in the radiancecalibrated data above the defined threshold at the pixel level, where each pixel is arc seconds. Standard errors are in parentheses. 47

48 Table B-9: OLS estimates from restricted rank regressions Satellite F6 (96) F6 (99) F6 (00) F6 (03) F6 (04) F16 (06) F16 (10) Average Panel a) Top 5% α (0.0008) (0.0009) (0.0009) (0.0010) (0.0009) (0.0011) (0.0011) (0.0010) [0.0060] [0.0054] [0.0054] [0.0065] [0.0063] [0.0066] [0.0062] [0.0060] Observations 96, , , , ,899 99, , ,955 Panel b) Top 4% α (0.0010) (0.0010) (0.0011) (0.0012) (0.0011) (0.0013) (0.0013) (0.0011) [0.0069] [0.0065] [0.0066] [0.0079] [0.0077] [0.0081] [0.0077] [0.0073] Observations 77,348 93,484 85,482 80,075 85,489 79,590 86,196 83,952 Panel c) Top 3% α (0.0014) (0.0013) (0.0014) (0.0014) (0.0013) (0.0015) (0.0016) (0.0014) [0.0084] [0.0084] [0.0086] [0.0102] [0.0100] [0.0107] [0.0102] [0.0095] Observations 58,011 70,115 64,111 60,058 64,134 59,692 64,647 62,967 Panel d) Top 2% α (0.0019) (0.0017) (0.0018) (0.0018) (0.0015) (0.0019) (0.0020) (0.0018) [0.0120] [0.0122] [0.0132] [0.0147] [0.0143] [0.0154] [0.0151] [0.0138] Observations 38,673 46,742 42,740 40,039 42,756 39,794 43,097 41,977 Panel e) Top 1% α (0.0030) (0.0025) (0.0025) (0.0025) (0.0021) (0.0026) (0.0025) (0.0025) [0.0227] [0.0225] [0.0237] [0.0259] [0.0242] [0.0271] [0.0276] [0.0248] Observations 19,337 23,373 21,372 20,020 21,377 19,898 21,551 20,990 Notes: The table reports the results of a restricted OLS rank regression with α 1 = α 2 = α imposed. Classical standard errors are reported in parentheses and adjusted standard errors computed via 2/N ˆα are reported in brackets. The data are a 10% representative sample of all non-zero lights in the radiancecalibrated data at the pixel level, where each pixel is arc seconds. 48

49 Table B-10: Regression of the EDF on theoretical CDFs, top 4% Satellite F12 (96) F12 (99) F12 (00) F14 (03) F14 (04) F16 (06) F16 (10) Average Panel a) Pareto CDF on RHS Slope (0.0003) (0.0004) (0.0005) (0.0004) (0.0004) (0.0005) (0.0005) (0.0004) Constant (0.0002) (0.0002) (0.0003) (0.0002) (0.0003) (0.0003) (0.0003) (0.0003) R Panel b) Lognormal CDF on RHS Slope (0.0014) (0.0016) (0.0018) (0.0014) (0.0014) (0.0014) (0.0015) (0.0015) Constant (0.0011) (0.0013) (0.0015) (0.0011) (0.0011) (0.0011) (0.0012) (0.0012) R Notes: The table reports results of a regression of the empirical distribution function (EDF) on the Pareto or lognormal CDF, using the top 4% of the data. The data are a 10% representative sample of all non-zero lights in the radiance-calibrated data at the pixel level, where each pixel is arc seconds. 49

50 Table B-11: Rank regressions based on the VIIRS data in 2015 Unrestricted regressions Hill estimates Restricted regressions Panel a) Top 5% y i α α (0.0013) (0.0065) (0.0011) y c [0.0100] (0.0015) R Observations 59,633 59,633 59,633 Panel b) Top 4% y i α α (0.0013) (0.0079) (0.0011) y c (0.0015) R Observations 47,705 47,705 47,705 Panel c) Top 3% y i α α (0.0014) (0.0101) (0.0012) y c [0.0153] R Observations 35,780 35,780 35,780 Panel b) Top 2% y i α α (0.0017) (0.0140) (0.0013) y c [0.0206] (0.0019) R Observations 23,854 23,854 23,854 Panel e) Top 1% y i α α (0.0033) (0.0226) (0.0024) y c [0.0314] (0.0036) R Observations 11,927 11, Notes: The table uses the VIIRS data to repeat three regressions which were conducted with the radiancecalibrated data before: the unrestricted OLS rank regression, restricted Hill regression, and the restricted OLS rank regression. Classical standard errors are reported in parentheses and adjusted standard errors computed via 2/N ˆα are reported in brackets. he data are a 10% representative sample of all non-zero lights in the radiance-calibrated data at the pixel level, where each pixel is arc seconds. 50

51 Table B-12: Satellite level statistics of the top-coding correction Satellite Top Top share (light) Mean luminosity Gini coefficient share Unadj Adj Unadj Form Pixel Unadj Form Pixel (pixels) Adj Adj Adj Adj F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Average Notes: The table reports summary statistics of the global lights data before the top-coding correction and after the analytical, formula-based correction at the aggregate level (Section 4.1) as well as the pixel-level correction (Section 4.2). Column 1 reports the share of pixels above 55 DN, Column 2 and 3 the share of lights emitted by these top pixels respectively in the unadjusted and adjusted data set. Columns 4-6 and 7-9 report the mean luminosity and Gini coefficient, respectively for the unadjusted data, the analytical, formula-based correction ((10) and (11)) and the pixel-level corrected data. All corrections use α = 1.5 and y c = 55 for the Pareto tail. 51

52 Table B-13: Global mean and Gini coefficient using different parameters, 2010 Unadj. Analytical Corrected Value using Pareto Alpha of Mean luminosity Spatial Gini Notes: The table computes the top-coding corrected mean and Gini coefficient of global inequality in lights for the year 2010 with different α parameters based on eq. (10) and eq. (11) with y c = 55. The input data are a representative 10% sample of non-zero lights from satellite F Table B-14: Income elasticity of lights from , country-level GDP in 2005 PPPs GDP per capita in 2005 PPPs (1) (2) (3) (4) (5) (6) Saturated Corrected Radiance Saturated Corrected Radiance Lights per km (0.067) (0.064) (0.054) Lights per capita (0.056) (0.055) (0.047) Within-R Observations Countries Note(s): The table reports panel FE estimates. Lights per capita and lights per km 2 are measured in logs. All columns include country and time fixed-effects. The specifications are variants of ln y it = β ln lights it ) + x it γ + c i + s t + ɛ it where x it is a vector of controls, c i is a vector of country fixed-effects, and s t are time dummies. Country-clustered standard errors in parentheses. Significant at: p < 0.10, p < 0.05, p <

53 Table B-15: Income elasticity of lights from , interactions, country-level GDP in 2005 PPPs GDP per capita in 2005 PPPs (1) (2) (3) (4) (5) (6) Saturated Corrected Radiance Saturated Corrected Radiance Lights per km (0.066) (0.063) (0.058) Richdense Lights per km (0.036) (0.034) (0.063) Lights per capita (0.056) (0.055) (0.052) Richdense Lights per capita (0.044) (0.041) (0.068) F -test of sum (p-value) Within-R Observations Countries Note(s): The table reports panel FE estimates. Lights per capita and lights per km 2 are measured in logs. All columns include country and time fixed-effects. The specifications are variants of ln y it = β 1 ln lights it +β 2 (ln lights it D i )+x it γ+c i+s t +ɛ it where D i is a dummy for rich and dense countries, x it is a vector of controls, c i is a vector of country fixed-effects, and s t are time dummies. Country-clustered standard errors in parentheses. Significant at: p < 0.10, p < 0.05, p <

54 Table B-16: Annualized growth rates of cities in Africa, Country Primate city Annual growth (primate) Secondary Annual growth (secondary) Saturated Corrected cities Saturated Corrected Angola Luanda Benin Cotonou Botswana Gaborone Burkina Faso Ouagadougou Burundi Bujumbura Cameroon Douala CAF Bangui Chad Ndjamena Congo (D.R.) Kinshasa Cote d Ivoire Abidjan Djibouti Djibouti Eritrea Asmara Ethiopia Addis Abbeba Gabon Libreville Ghana Accra Guinea Conakry Guinea- Bissau Bissau Kenya Nairobi Lesotho Maseru Liberia Monrovia Madagascar Antananarivo Malawi Blantyre Mali Bamako Mauritania Nouakchott Mozambique Maputo Namibia Windhoek Niger Niamey Nigeria Lagos Congo (R.) Brazzaville Rwanda Kigali Senegal Dakar Sierra Leone Freetown Somalia Mogadishu South Africa Johannesburg Sudan Alkhartum Swaziland Mbabane Tanzania Daressalaam The Gambia Banjul Tongo Sokode Uganda Kampala Zambia Lusaka Zimbabwe Harare Notes: The table reports lists the name of the primary city as well as the number of secondary cities for each Sub-Saharan African country, as defined in the text. The annualized growth rate in mean lights is computed from 1992 to 2013, for both the saturated and the top-coding corrected lights. For primary cities, these growth rates refer to the primary city of the country. For secondary cities, the reported growth rate is an average of the secondary cities in the country. 54

55 Table B-17: Trends in the CV for African cities, Dependent variable: Coefficient of Variation Saturated data Corrected data (1) (2) (3) (4) (5) (6) t (0.000) (0.001) (0.001) (0.000) (0.001) (0.001) t P ij (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) Estimation Fixed Fixed Random Fixed Fixed Random City FE Yes Yes Yes Yes Yes Yes Satellite FE No Yes Yes No Yes Yes Country FE No No Yes No No Yes Within-R Observations 11,700 11,700 11,700 11,700 11,700 11,700 Cities Notes: The table reports the results of city-level panel regressions using the saturated and top-coding corrected data. The specifications are variants of CV ijt = β 1 t + β 2 (t P ij ) + c ij + s t + ɛ ijt where t is a linear time trend, P ij is an indicator for primate cities, c ij is a vector of city and/or country effects and s t contains satellite dummies. Standard errors clustered on the city are in parentheses. Note that we lose 33 cities which consist of less than 3 pixels, the minimum we require to calculate inner city fragmentation. Significant at: p < 0.10, p < 0.05, p < Table B-18: Trends in Moran s I for African cities, Dependent variable: Moran s I Saturated data Corrected data (1) (2) (3) (4) (5) (6) t (0.010) (0.038) (0.038) (0.010) (0.038) (0.038) t P ij (0.048) (0.048) (0.046) (0.034) (0.034) (0.032) Estimation Fixed Fixed Random Fixed Fixed Random City FE or RE Yes Yes Yes Yes Yes Yes Satellite FE No Yes Yes No Yes Yes Country FE No No Yes No No Yes Within-R Observations 11,691 11,691 11,691 11,691 11,691 11,691 Cities Notes: The table reports the results of city-level panel regressions using the saturated and top-coding corrected data. The specifications are variants of I ijt = β 1 t + β 2 (t P ij ) + c ij + s t + ɛ ijt where t is a linear time trend, P ij is an indicator for primate cities, c ij is a vector of city and/or country effects and s t contains satellite dummies. Note that we lose 33 cities which consist of less than 3 pixels, the minimum we require to calculate inner city fragmentation. Standard errors clustered on the city are in parentheses. Significant at: p < 0.10, p < 0.05, p <

56 Figure B-1: Zipf plot and Van der Wijk s plot (a) Zipf y in DN 1 F(y) (b) Van der Wijk u E[y y > u] Notes: Popular graphical tests for an approximate Pareto distribution in top lights. Panel a) shows the Zipf plot for the top 2% of all pixels. The figure uses logarithmic binning to reduce noise and sampling errors in the right tail of the distribution (see Newman, 2005). There are about 100 bins in the tail, where the exact number depends on the range of the input data. Panel b) demonstrates Van der Wijk s law, which states that the average light above some value u is proportional to u, this is E[y y > u] u. Here, too, the data is the top 2% of all pixels. The input data are a 10% representative sample of all non-zero lights in the radiance-calibrated data at the pixel level obtained from Hsu et al. (2015). Figure B-2: Global Gini coefficient in lights before and after the correction Spatial Gini Coefficient alpha = 1.7 alpha = 1.5 alpha = 1.3 original Notes: Illustration of the global top-coding correction. The figure shows global inequality in lights calculated by eq. (11) using the specified Pareto shape parameters. For years when more than two satellites flew concurrently, the values were averaged. 56

57 Figure B-3: Generalized Pareto CDF versus EDF, unsaturated, 2010 Density Empirical EVT Model Light intensity in DN Notes: Illustration of Generalized Pareto CDF fitted to the data and the empirical distribution function (EDF). The EDF and Generalized Pareto CDF are fitted to the top 4% of unsaturated lights in The input data are a 10% representative sample of all non-zero lights in the radiance-calibrated data at the pixel level, where each pixel is arc seconds. Figure B-4: Pareto and lognormal CDF versus EDF, unsaturated, 2010 CDF Empirical Lognormal Pareto Light intensity in DN Notes: Illustration of the difference between the Pareto and lognormal CDFs fitted to the data and the empirical distribution function (EDF). Note that the lognormal distribution was fitted to the whole distribution rather than the tail because of its unimodal shape, while the Pareto distribution is estimated only on the tail. For comparison, we adjust the CDFs so that they all start at the top 4% of unsaturated lights in The input data are a 10% representative sample of all non-zero lights in the radiancecalibrated data at the pixel level, where each pixel is arc seconds. 57

58 Figure B-5: Size of the correction and country characteristics (a) Mean correction vs. GDP per capita (b) Gini correction vs. GDP per capita ISR BHR Size of correction (in DN) CHL KOR TTO JPN SAU PRI GBR BEL USA NLD NOR OMN CHE BRN LUX Size of correction (in Gini) KOR ISR SAU HKG ARE SGP KWT CHL JPN MEX USA NOR BRN NLD OMN CHE LUX QAT GDP per capita (in 2011 PPPs) GDP per capita (in 2011 PPPs) (c) Mean correction vs. log area (d) Gini correction vs. log area ISR BHR Size of correction (in DN) TTO KOR PRI BRN BEL GBR CHL EGY ARG EST VUT USA NLD FIN JOR NOR ESP BOL LBY LVA MYS PRY MEX PSE AFG BLR OMN CHN PRT SWE BRA CAN RUS AUS JPN AGO SAU Size of correction (in Gini) SGP HKG QAT KWT ISR ARE KOR SAU AGO JPN CHL ARG MEX TTO EST GBR USA ESP PRI VUT FIN BOL BLR PRY CANRUS BRN BEL CHN LVA EGY JOR NOR MYS KAZ TKM LBY BRA AUS Log land area Log land area (e) Mean correction vs. log density (f) Gini correction vs. log density ISR BHR Size of correction (in DN) SAU AGO PRI CHL BRN GBR BEL ARG EGY EST VUT USA FIN NLD BOL JOR ESP LBY PRY NOR LVA MEX MYS AFG PSE OMN BLR CHN BRA SWE CAN PRT MUS BGD TTO JPN KOR Size of correction (in Gini) SAU ARE QAT KWT AGO ARGCHL JPN MEX EST USA BOL ESP GBR TTO VUT CAN RUS FIN PRY PRI BLR BRA BRN EGY CHN LBY KAZ LVA BEL JOR MYS TKM NOR PSE MUS BGD ISR KOR HKG SGP Log population density Log population density Notes: Illustration of how the correction of means and Gini coefficients, based on eq. (10) and eq. (11), correlate with log GDP per capita (PPP), log land area, and log population density. The data are a 10% representative sample of all non-zero lights in satellite F GDP and population data are from the World Development Indicators. For display purposes, the left panels exclude countries with a correction of mean luminosity larger than 20 DN, these are Singapore, Hong Kong, Qatar, Bahrain, Kuwait, and the UAE. These countries are included in all panels on the right. 58

59 Figure B-6: Urban extents of selected cities in SSA (a) Johannesburg (b) Lagos (c) Luanda (d) Nairobi Notes: Illustration of the urban extents detection algorithm presented in the text. Comparison of urban footprint with current Google maps images. Note that urban extents based on current satellites are larger, in line with built-up structures on the various maps. Google Maps images are used as part of their fair use policy. All rights to the underlying maps belong to Google. 59