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P. Phillips, Shuping Shi, Jun Yu (2013)
Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500ERN: Other Econometrics: Applied Econometric Modeling in International Economics (Topic)
Joerg Breitung, Samarjit Das (2005)
Panel unit root tests under cross‐sectional dependenceStatistica Neerlandica, 59
Aurélie Lalanne (2014)
Zipf’s Law and Canadian Urban GrowthUrban Studies, 51
P. Phillips, Yangru Wu, Jun Yu (2007)
Explosive Behavior in the 1990s' NASDAQ: When Did Exuberance Escalate Asset Values?ERN: Hypothesis Testing (Topic)
X. Gabaix (1999)
Zipf's Law for Cities: An ExplanationQuarterly Journal of Economics, 114
J. Córdoba (2008)
A Generalized Gibrat's LawWiley-Blackwell: International Economic Review
M. Pesaran (2003)
A Simple Panel Unit Root Test in the Presence of Cross Section DependenceEconometrics eJournal
S. Ronsse, Samuel Standaert (2017)
Combining growth and level data: An estimation of the population of Belgian municipalities between 1880 and 1970Historical Methods: A Journal of Quantitative and Interdisciplinary History, 50
C. Gengenbach, F. Palm, J. Urbain (2009)
Panel Unit Root Tests in the Presence of Cross-Sectional Dependencies: Comparison and Implications for ModellingEconometric Reviews, 29
P Perron, T Vogelsang (1992)
Nonstationarity and level shifts with an application to purchasing power parityJ Bus Econ Stat, 10
Rafael González‐Val, L. Lanaspa, F. Sanz-Gracia (2014)
New Evidence on Gibrat’s Law for CitiesUrban Studies, 51
MH Pesaran (2007)
A simple panel unit root test in the presence of cross-section dependenceJ Appl Econom, 22
D. Champernowne (1953)
A Model of Income DistributionThe Economic Journal, 63
EW Frees (1995)
Assessing cross-sectional correlation in panel dataJ Econom, 69
D. Dickey, W. Fuller (1981)
LIKELIHOOD RATIO STATISTICS FOR AUTOREGRESSIVE TIME SERIES WITH A UNIT ROOTEconometrica, 49
L Luyckx (2010)
Russische krijgsgevangenen van de nazi's: van Displaced Persons tot vluchtelingen (voor het Sovjetcommunisme)Belgisch Tijdschrift Voor Nieuwste Geschiedenis, XL, 3
YM Ioannides, HG Overman (2003)
Zipf’s law for cities: an empirical examinationReg Sci Urban Econ, 33
Pierre Perron (1989)
The Great Crash, The Oil Price Shock And The Unit Root HypothesisEconometrica, 57
J Breitung, MH Pesaran, L Matyas, P Sevestre (2008)
Unit roots and cointegration in panelsThe Econometrics of Panel Data: Fundamentals and Recent Developments in Theory and Practice
D. Black, V. Henderson (2003)
Urban evolution in the USAJournal of Economic Geography, 3
(2002)
Bones, Bombs, and Break Points: The Geography of Economic Activity
S. Devadoss, Jeff Luckstead (2015)
Growth process of U.S. small citiesEconomics Letters, 135
P. Decker (2011)
Understanding Housing Sprawl: The Case of Flanders, BelgiumEnvironment and Planning A, 43
M. Pesaran (2004)
General diagnostic tests for cross-sectional dependence in panelsEmpirical Economics, 60
Serena Ng, Pierre Perron (1995)
Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation LagJournal of the American Statistical Association, 90
X Gabaix, YM Ioannides, JF Thisse, JV Henderson (2004)
The evolution of city size distributionsHandbook of urban and regional economics
E. Bosker, S. Brakman, H. Garretsen, Marc Schramm (2006)
A Century of Shocks: The Evolution of the German City Size Distribution 1925-1999Economic History
F Caestecker, E Vanhaute, A Grauwels (2011)
Leven en werken in het industriële BelgiëHedendaagse economische geschiedenis van België: een inleiding
J. Eeckhout (2004)
Gibrat's Law for (All) CitiesThe American Economic Review, 94
T. Holmes, John Stevens (2002)
Geographic Concentration and Establishment ScaleReview of Economics and Statistics, 84
Zhihong Chen, Shihe Fu, Dayong Zhang (2013)
Searching for the Parallel Growth of Cities in ChinaUrban Studies, 50
Dan Ben-David, R. Lumsdaine, David Papell (1998)
Unit roots, postwar slowdowns and long-run growth: Evidence from two structural breaksEmpirical Economics, 28
Marcelo Resende (2004)
Gibrat's Law and the Growth of Cities in Brazil: A Panel Data InvestigationUrban Studies, 41
S Karlsson, M Löthgren (2000)
On the power and interpretation of panel unit root testsEcon Lett, 66
Hoyt Bleakley, Jeffrey Lin (2011)
Portage and Path Dependence.The quarterly journal of economics, 127 2
G. Block, J. Polasky (2011)
Light railways and the rural–urban continuum: technology, space and society in late nineteenth-century BelgiumJournal of Historical Geography, 37
A. Lalanne, Martin Zumpe (2019)
La croissance des villes canadiennes et australiennes guidée par le hasard ? Enjeux et mesure de la croissance aléatoire urbaineCanadian Journal of Regional Science
J. Clark, Jack Stabler (1991)
Gibrat's Law and the Growth of Canadian CitiesUrban Studies, 28
DA Dickey, WA Fuller (1979)
Distributions of the estimators for autoregressive time series with a unit rootJ Am Stat Assoc, 74
E. Bosker, S. Brakman, H. Garretsen, Marc Schramm (2005)
Looking for Multiple Equilibria When Geography Matters: German City Growth and the Wwii ShockEconometrics eJournal
Michiel Meeteren, Kobe Boussauw, B. Derudder, F. Witlox (2016)
Flemish Diamond or ABC-Axis? The spatial structure of the Belgian metropolitan areaEuropean Planning Studies, 24
Rafael González‐Val (2010)
The Evolution Of U.S. City Size Distribution From A Long‐Term Perspective (1900–2000)Journal of Regional Science, 50
S Sharma (2003)
Persistence and stability in city growthJ Urban Econ, 53
Glenn Ellison, E. Glaeser (1999)
The Geographic Concentration of Industry: Does Natural Advantage Explain Agglomeration?The American Economic Review, 89
H. Simon (1955)
ON A CLASS OF SKEW DISTRIBUTION FUNCTIONSBiometrika, 42
Rafael González‐Val, L. Lanaspa (2016)
Patterns in US Urban Growth, 1790–2000Regional Studies, 50
E. Glaeser, Jesse Shapiro (2003)
Urban Growth in the 1990s: Is City Living Back?Wiley-Blackwell: Journal of Regional Science
Aurélie Lalanne, Martin Zumpe (2020)
Time-Series Based Empirical Assessment of Random Urban Growth: New Evidence from FranceJournal of Quantitative Economics
BH Baltagi, G Bresson, A Pirotte (2007)
Panel unit root tests and spatial dependenceJ Appl Economet, 22
J Clemente, A Montañés, M Reyes (1998)
Testing for a unit root in variables with a double change in the meanEcon Lett, 59
G. Duranton (2007)
Urban Evolutions: The Fast, the Slow, and the StillThe American Economic Review, 97
B De Meulder (1999)
Patching up the Belgian urban landscapeOase, 52
D. Dickey, W. Fuller (1979)
Distribution of the Estimators for Autoregressive Time Series with a Unit RootJournal of the American Statistical Association, 74
B. Baltagi, G. Bresson, A. Pirotte (2006)
Panel Unit Root Tests and Spatial Dependence
J Eaton, Z Eckstein (1997)
Cities and growth: theory and evidence from France and JapanReg Sci Urban Econ, 27
Juan Chauvin, E. Glaeser, Yueran Ma, Kristina Tobio (2016)
What is Different About Urbanization in Rich and Poor Countries? Cities in Brazil, China, India and the United StatesNBER Working Paper Series
S. Barrios, Luisito Bertinelli, E. Strobl (2006)
Geographic Concentration and Establishment Scale: An Extension Using Panel DataJournal of Regional Science, 46
J. Henderson, H. Wang (2007)
Urbanization and city growth: The role of institutionsRegional Science and Urban Economics, 37
M. Partridge, D. Rickman, Kamar Ali, M. Olfert (2008)
Lost in space: population growth in the American hinterlands and small citiesJournal of Economic Geography, 8
Miquel-Àngel Garcia-López, Adelheid Holl, Elisabet Viladecans-Marsal (2015)
Suburbanization and highways in Spain when the Romans and the Bourbons still shape its citiesJournal of Urban Economics, 85
Rafael González‐Val, Javier Silvestre (2020)
An annual estimate of spatially disaggregated populations: Spain, 1900–2011The Annals of Regional Science
A. Banerjee, Massimiliano Marcellino, C. Osbat (2001)
Testing for PPP: Should we use panel methods?Empirical Economics, 30
We take advantage of a new data set on Belgian cities to test random growth, that is, Gibrat’s law. This unique data set provides annual population estimates for all Belgian municipalities (2680 cities) from 1880 to 1970. The use of panel data meth‑ odology and unit root tests can provide a precise test of Gibrat’s law (a unit root is equivalent to random growth). We run both time series and panel data unit root tests, thus obtaining strong support for random growth in the long term. Results hold when allowing for the presence of one and two structural breaks in the mean, with the timing of the breaks coinciding with some major historical events, such as the World Wars and the economic crisis of 1929–1933. JEL Classification C12 · C22 · N93 · O18 · R11 · R12 1 Introduction Urban growth literature has a long tradition. Why some cities grow while others decline is still an open question, although several theoretical explanations have been proposed. These theories can be summarised into three main drivers of growth * Rafael González‑Val rafaelg@unizar.es Arturo Ramos aramos@unizar.es Samuel Standaert Samuel.Standaert@UGent.be Departamento de Análisis Económico, Facultad de Economía y Empresa, Universidad de Zaragoza, Gran Vía 2, 50005 Zaragoza, Spain Facultat d’Economia i Empresa, Institut d’Economia de Barcelona (IEB), John Maynard Keynes, 1‑11, 08034 Barcelona, Spain Institute on Comparative Regional Integration Studies, United Nations University, Potterierei 72, 8000 Brugge, Belgium Department of Economics, Ghent University, Tweekerkenstraat 2, 9000 Ghent, Belgium Vol.:(0123456789) 1 3 R. González-Val et al. (Davis and Weinstein 2002): the existence of increasing returns to scale, the impor‑ tance of locational fundamentals, and random growth. Each of these drivers of urban growth involves different theoretical mechanisms. The existence of increasing returns suggests the presence of endogenous mecha‑ nisms in city growth that can lead to multiple equilibria (Davis and Weinstein 2002; Bosker et al. 2007), depending on the initial conditions. Locational fundamental theory highlights the role played by geographical characteristics: the presence of a natural harbour, a specific climate, or access to the sea, among many other physical characteristics, can determine cities’ populations (for instance, Ellison and Glaeser (1999) stated that natural advantages can explain at least half of the observed geo‑ graphic concentration in the US). Finally, random urban growth postulates that pop‑ ulation growth in cities is a random variable. Studies testing the influence of increasing returns to scale and locational funda‑ mentals have usually relied on parametric (cross‑sectional or panel data) growth regressions, applying an instrumental variable approach in most cases. The latest advances in this literature have come from the use of plant‐level data (Holmes and Stevens 2002; Barrios et al. 2006) and case studies using an identification strategy of instruments that reveals the influence of some historical events on cities’ growth path (e.g., Bleakley and Lin 2012; Garcia‑López et al. 2015). However, most of these studies have adopted a short‑term perspective, and even panel data analyses have considered a few decades at most. The approach taken in the random growth literature is different. First, from the theoretical point of view, random growth can only hold as a long‑run average, while the influence of other factors, like locational fundamentals and increasing returns, may change (or even disappear) over time. With random urban growth, the growth process of cities tends to be multiplicative and independent of their initial size, a proposition that has become known in urban economics as Gibrat’s law. Several theoretical models (Gabaix 1999; Duranton 2007; Córdoba 2008) were developed to explain the fulfilment of Gibrat’s law in the context of external urban local effects and productive shocks, associating it directly with an equilibrium situation. There‑ fore, city‑level variables can explain temporal variation in growth rates across cit‑ ies, but random growth theory provides an appropriate explanation for the long‑term growth. Second, on the empirical side, although seminal contributions (e.g., Eaton and Eckstein 1997) have also used parametric growth regressions to test Gibrat’s law, since the 2000s, several studies have proposed alternative methodologies to para‑ metric growth models. González‑ Val et al. (2014) reviewed this literature, conclud‑ ing that most studies today use nonparametric estimates of urban growth or unit root tests. Nonparametric estimates of growth have become popular in this literature, Quoting Gabaix and Ioannides (2004, p. 2353), “the casual impression of the authors is that in some decades, large cities grow faster than small cities, but in other decades, small cities grow faster.” Formally,“Gibrat’s Law states that the growth rate of an economic entity (firm, mutual fund, city) of size S has a distribution function with mean and variance that are independent of S” (Gabaix and Ioan‑ nides 2004, p. 2346). 1 3 Urban growth in the long term: Belgium, 1880–1970 providing estimates of growth that vary with the initial population over the entire distribution of city sizes. However, these kernel regressions estimate the uncondi‑ tional relationship between growth and size; city and time fixed effects and any other control variables are omitted. Thus, authors have carried out nonparametric analyses for cross‑sectional data (Eeckhout 2004) as well as for a pool of growth rates from different time periods (Ioannides and Overman 2003; González‑ Val 2010). The use of the panel data methodology and unit root tests in the analysis of urban growth, first suggested by Clark and Stabler (1991), can provide a more precise test of Gibrat’s law. This idea was emphasized by Gabaix and Ioannides (2004, p. 2358), who expected “that the next generation of city evolution empirics could draw from the sophisticated econometric literature on unit roots.” Recently, new methods have been proposed to test random growth using unit root tests. Lalanne and Zumpe (2019, 2020) apply an integrated model selection/unit root test protocol with three different model specifications (pure random growth, random growth with drift, and random growth with drift and trend) to a large sample of high‑quality French decen‑ nial census data, on how the rejection of the random growth hypothesis accounts for less than one third of tested cities. However, some empirical limitations have reduced the spread of these techniques. While several papers applied panel data unit root tests to analyse urban growth (e.g., Black and Henderson 2003; Resende 2004; Henderson and Wang 2007; González‑ Val and Lanaspa 2016), the list of studies looking for unit roots in individual time series of cities’ populations is quite short. Why? Unit root tests need large sample sizes (at least 40 observations) to have reasonable power (Clark and Stabler 1991). However, long time series of year‑by ‑year city populations are usually not available, and studies on the temporal evolution of city sizes have considered decennial census data in most cases. Therefore, the lack of annual data for a sample of cities over a long time period on a consistent basis has limited the use of unit root testing in empirical work. To our knowledge, only five studies consider annual city populations to test Gibrat’s law using unit root tests: Clark and Stabler (1991), Resende (2004), Sharma (2003), Bosker et al. (2008) and Chen et al. (2013). Clark and Stabler (1991) used data on the seven largest cities in Canada from 1975 to 1984 (10 temporal observa‑ tions by city), while Resende (2004) used a panel data set with annual data for 497 municipalities in the state of São Paulo (Brazil) for the 1980–2000 period. Sharma (2003) considered a sample of 100 Indian cities for the period 1901–1991 (90 years), Bosker et al. (2008) used a dataset of 62 West German cities from 1925 to 1999 (except for five missing years during the Second World War), and Chen et al. (2013) considered 210 Chinese cities from 1984 to 2006 (23 temporal observations). Although the efforts of these authors to obtain annual city population data and exploit the properties of the unit root tests fully are worthy, these studies still show an important limitation: they focused on the largest cities. Nevertheless, some stud‑ ies have confirmed the different patterns of growth of small cities (Partridge et al. 2008; Devadoss and Luckstead 2015) and, thus, the behaviour of the largest cities cannot be extrapolated to the whole distribution of cities. Another reason to consider all cities is that some studies found differences in growth patterns between different regions within the same country. Lalanne (2014) studied the hierarchical structure of 1 3 R. González-Val et al. the Canadian urban system, splitting the territory into two parts (east and west), thus allowing for the identification of different dynamics. In this paper, we take advantage of Ronsse and Standaert’s (2017) new data set of Belgian cities. This unique data set provides annual estimates of the population for all Belgian municipalities from 1880 to 1970. Thus, it allows us to carry out a robust long‑term analysis of urban growth because the time dimension is long (90 temporal observations by city) and, at the same time, it contains information for all cities, covering the whole city size distribution. Therefore, as far as we know, this is the most comprehensive test of Gibrat’s law using unit root tests ever conducted. Moreover, the Belgian case is interesting because of some specific historical char ‑ acteristics of the country. As a relatively young country on the European continent, the Belgian state came into existence following a liberal revolution in 1830. Set up as a parliamentary democracy, headed by a monarch with limited powers, the Bel‑ gian state quickly became a haven for political liberalism in 19th‑century Europe. At the same time, however, the young nation wanted to ensure that it had the most up‑to‑date information about the population living within its borders. Following the newest scientific methods, the state apparatus created a highly developed statistical department, which, whilst not unique, was well ahead of its time. The continuous efforts of this department have led to a richness of statistical data spanning the entire history of the country. When studying the trends of population growth in this small state, surrounded by giants, it is also important to note that Belgium was the first industrialized coun‑ try on the European continent. A combination of technological knowhow, labour, capital and readily available natural resources led to a quick‑paced industrialization process, initially concentrated in the city of Ghent and in the southern part of the country (Caestecker 2015, pp. 101–103). Only after the Second World War did the industrial centre of gravity move to the north, with the arrival of many multinational companies that wanted to make use of the vicinity of sea harbours and the excellent road infrastructure of the region (Witte and Meynen 2006; Ryckewaert 2011). Akin to this industrialization, but equally motived by political interests and goals, Belgium also became the centre of an expansive railway network from the late nine‑ teenth century onwards. The construction of the railway had already begun in the 1830s, but the connections were initially limited to those between major urban and industrial hubs. From the 1880s onwards, however, there was a massive effort to expand the railway network to even the smallest towns. In a country of only 30,528 km , the main rail network grew from 3000 km in 1870 to almost 5000 km in 1910. Concurrently, a 3000 km network of light rail was established that “meander within the landscape, from village to village, collecting as many intermediary passengers and goods as possible” (De Block and Polasky 2011, p.318). The high degree of connectivity resulting from this extensive network – as well as a system of cheap train tickets for workers – ensured a steady flow of labour from the still densely populated countryside to the industrial centres of the country, without the large‑scale urbanization – and accompanying politicization of the labour force LOKSTAT, accessed on 23 September 2020 via https:// lokst at. ugent. be/ lokst at_ over_ doels telli ng. php. 1 3 Urban growth in the long term: Belgium, 1880–1970 – that characterized the industrialization process in Belgium’s neighbouring coun‑ tries (De Meulder et al. 1999). Before the end of the twentieth century, more than one in three Belgians commuted to and from work every day (De Decker 2011). These factors affected the distribution of economic activity and population in Belgium during the course of the twentieth century, and set the grounds for a new long‑term equilibrium of the distribution of population. However, as both industri‑ alization and completion of transport infrastructures took place at the beginning of our sample period (in the late nineteenth and early twentieth centuries) that covers almost one century, we expect to capture not the short‑term transitional periods of higher or lower growth, but rather the long‑term growth that may follow a random growth pattern. The remainder of the paper is organized as follows. In Sect. 2, we describe the data set used. Section 3 explains the different unit root tests carried out, distinguish‑ ing between time series analysis of unit roots allowing for structural breaks and panel data unit root tests, and presents the main results. Section 4 concludes. 2 Data As far as we know, only a few studies have provided long‑term estimates of annu‑ ally and spatially disaggregated populations at the city level: Sharma (2003), Bosker et al. (2008) and González‑ Val and Silvestre (2020), for Indian (100 cities, 1901–1991), West German (62 cities, 1925–1999 period with some gaps) and Span‑ ish cities (49 capital cities, 1900–2011), respectively. In this paper, we use the new data set by Ronsse and Standaert (2017). They constructed a data set of 2680 Belgian municipalities for the period 1880–1970. Their dataset ends in 1970, as this is the last decade in which administrative borders remained relatively constant. Specifically, in 1970 and 1971, the number of munici‑ palities first dropped by 200. This decrease was followed by the main administra‑ tive redrawing of the municipalities in 1976, which reduced the number of munici‑ palities by three quarters. In all, the total number of municipalities dropped from 2675 in 1970 to a mere 596 six years later (Vrienlinck, 2000). Further redrawing of municipal maps followed in 1983 and 2019 and resulted in the current total of 581 municipalities, although this value is set to change again in 2025. Given the impact of these changes on the continuity of their dataset, the authors chose 1970 as the final year. To compose this data set, they combined the population census data, which are collected every ten years, with the yearly data on births, deaths and migra‑ tion from the city population registers (the movement data). The less centralized The unit of analysis of this database are municipalities as defined by their administrative borders. Amalgamating these into metropolitan areas is difficult for many reasons, the main problem being that the entirety of Belgium is described as one big metropolitan area. As noted in Van Meeteren et al. (2016, p. 3) the Flemish region is often described as a ‘nebular city’ and Belgium as an “hourglass‑shaped met‑ ropolitan area whose (Walloon) base is less developed than its (Flemish) roof.” 1 3 R. González-Val et al. nature of the latter, in combination with a tendency to underreport outward migration, means that these series cannot easily be combined without creating breaks in the data with every new census. This is illustrated quite well in Fig. 1, in which the dotted red lines show what the population estimate would be if we added the cumulative changes to the population to the last set of census data. The population in Brussels is overestimated by as many as a 100,000 people in 1900. At other times, especially in the later years of the data set, the quality of the population registers can be much higher. To reconcile the two data series, Ronse and Standaert (2017) used a state‑space approach, which models the mouvement data as a noisy signal of the true change in the population growth. As the mouvement is collected by each city individually, the noisiness of its signal is allowed to differ for each city. Furthermore, the model incorporates information on the changes to the administrative borders of the cit‑ ies, allowing the population data to change more drastically in those years (see e.g. Brussels in 1921 in Fig. 1). The results of the models are also shown in Fig. 1, with the thick black lines showing the estimated population and the blue shaded area its 95% confidence interval. The estimated population data clearly follow the change indicated by the mouvement data as much as possible while remaining consistent with the census data. As we noted, the period of analysis in the dataset was chosen so that the adminis‑ trative borders remain consistent. Included in the sample, however, is the first wave of administrative changes of 1964, that reduced the number of municipalities from 2675 to 2586. In addition to these large‑scale changes, information from Vrienlinck (2000) was used to identify smaller, ad‑hoc changes to the administrative borders of municipalities. As a robustness check, cities with more than a 30% change in land area in the period considered were excluded from the analysis. For illustrative purposes, Table 1 shows the number of cities and the descrip‑ tive statistics in census years. The minimum values of each decade indicate that we are considering all municipalities, without size restrictions, even the smallest units. Although their urban character might be debatable, Eeckhout (2004) suggested con‑ sidering the whole distribution. The mean population grows over time and, at the same time, the number of cities remains stable, which points to an urbanization pro‑ cess. Moreover, although the standard deviation increases from 1880 to 1910, indi‑ cating increasing inequality in city sizes, it remains quite stable in the last decades, which suggests stability in the city size distribution. Municipalities comprise the country’s total land area and, therefore, the entire population. During the period considered (almost a century), there was an impor‑ tant increase (71.2%) in the Belgian total population in our sample, from 5,517,017 in 1880 to 9,443,985. One might expect that such a huge increase also generated important changes in the city size distribution. Figure 2 shows the empirical density functions for three representative periods (1880, 1930 and 1970) estimated using adaptive kernels for our sample of all Belgian municipalities. We define the relative size of a city as the quotient between the city’s population and the contemporary average population. The graph shows the distribution of these cities’ relative sizes in log scale, with the zero value in the x‑axis representing medium‑sized cities. 1 3 Urban growth in the long term: Belgium, 1880–1970 Fig. 1 The population of the four biggest cities in Belgium from 1880 to 1970. The estimated level of the population is indicated by the thick black line and its 95% confidence interval by the blue shaded area. The source data are plotted using the red crosses (census) and dotted line (population registers). Admin‑ istrative changes to the city borders are indicated by the black asterisks. Source: Ronsse and Standaert (2017) The shape of the city size distribution changed dramatically from 1880 to 1970. In 1880, we can observe a very leptokurtic distribution with a great deal of density concentrated in the central values. However, by 1930, the distribution had lost kurto‑ sis and the concentration had decreased. In 1970, we find a more uneven distribution with heavier tails than in previous periods, indicating the presence of more small and large cities than in earlier years. The temporal evolution of the city size distribu‑ tion suggests a divergence pattern in the growth of Belgian cities, thus apparently discarding the idea random growth. 1 3 R. González-Val et al. Table 1 Summary of the descriptive statistics Year Cities Mean population Standard deviation Minimum Maximum 1880 2580 2138.379 6629.206 25 169,112 1890 2594 2338.707 7877.252 23 224,012 1900 2616 2558.284 8919.119 28 272,831 1910 2628 2824.463 9677.538 24 301,766 1920 2637 2807.940 9627.061 37 302,058 1930 2670 3030.319 10,033.540 35 284,373 1940 2629 3154.683 9926.479 37 267,903 1950 2668 3243.136 9894.130 34 261,412 1960 2662 3447.499 9957.616 27 256,619 1970 2526 3738.711 10,031.170 31 224,543 Descriptive statistics in census years Fig. 2 Empirical density func‑ tions. Adaptive kernels of the relative size of the Belgian cities (ln scale) 3 Methodology and results 3.1 Unit roots in city sizes Our basic hypothesis for the long‑term growth of Belgian cities is random growth (Gabaix and Ioannides 2004; González‑ Val et al. 2014). As mentioned above, ran‑ dom growth can hold as a long‑run average, while the effect of other factors may change or dissipate over time. Traditional theoretical models of random growth (Champernowne 1953; Simon 1955) are based on stochastic growth processes and probabilistic models. Recent urban economic theories (Gabaix 1999; Duran‑ ton 2007; Córdoba 2008) include economic factors driving random shocks (e.g., external urban local effects or productive shocks), and these models are able to reproduce two empirical regularities that are well known in urban economics: Zipf’s and Gibrat’s laws (or the rank‑size rule and the law of proportionate growth, respectively). 1 3 Urban growth in the long term: Belgium, 1880–1970 We follow the methodology proposed by Clark and Stabler (1991), who suggested that testing for random growth is equivalent to testing for the presence of a unit root. Starting from a simple autoregressive (AR) growth model, they assumed that the relationship between the size of a city in time period t and that in time period t − 1 is S = S , (1) it it it−1 where S is the city share of city i defined as the quotient derived from dividing it the city’s population ( Pop ) by the contemporary total population of the coun‑ it try, S = Pop Country pop because, from a long‑term temporal perspective, it it t it is necessary to use a relative measure of size (Gabaix and Ioannides 2004). it is the growth rate of city i over the period t − 1 to period t . This growth rate can be decomposed into two (Clark and Stabler 1991) or three components (Bosker et al. 2008): a random component, a non‑stochastic component relating the current growth rate to a (possibly time‑varying) constant and past growth rates, and the ini‑ tial city size. Then, after some algebra, Clark and Stabler (1991) obtained the fol‑ lowing expression: Δs = c +Θ s + Δs + , (2) it i i it−1 ij it−j it j=1 where s = ln S is the log‑city share of city i in year t, Δs = s − s , c is a it it it it it−1 i constant, is a parameter measuring the influence of past growth rates on current ij city growth and k is the number of lags added to ensure that the residuals, , are it Gaussian white noises. Θ is the key parameter that captures the effect of the initial city size on growth. Random growth would imply Θ = 0 , meaning that the growth of a particular city does not depend on its initial city size. Then, the city share would be a non‑stationary time series, and any sudden shock would have permanent effects on the long‑run level of the population of the city (Davis and Weinstein 2002). This shows that testing for random growth (Gibrat’s law) is equivalent to testing for a unit root in city sizes. Evidence supporting a unit root (if Θ is not significant) means that city i ’s growth rate is independent of the initial size. By contrast, when Θ < 0 , the path of city i will be a stationary process (mean reversion). By using Eq. (2), Clark and Stabler (1991) apply the standard Dickey and Fuller (1979) t‑statistic, failing to reject random growth for the seven largest cities in Canada from 1975 to 1984. To test for the presence of unit roots, we run the Augmented Dickey–Fuller (ADF) test (Dickey and Fuller 1979, 1981). The ADF test for nontr ‑ ending data is carried out by running Eq. (2). Following Ng and Perron (1995), we choose the optimal k using a “generalt ‑ ospecific pr ‑ ocedure” based on the ts ‑ tatistic. The null and alternative hypoth‑ eses are, respectively, H ∶Θ = 0 , H ∶Θ < 0 . If Θ is found to be equal to 0, then 0 i A i i the city share series follows a random walk and, on the other hand, if Θ is found to be significantly smaller than 0, the city share is stationary around c . Lalanne and Zumpe (2019) propose a new test‑protocol for testing three different random growth pro‑ cesses (pure random growth, random growth with drift, and random growth with drift and trend) also based on the ADF statistic. 1 3 R. González-Val et al. Table 2 Results of unit root tests on city shares: All Belgian cities City‑specific tests (N = 2680) Alternative hypothesis Trend stationary Trend stationary Trend stationary with one break with two breaks Significance level (%) % unit root rejected % unit root rejected % unit root rejected 1 1 3 2 5 4 8 3 10 8 15 4 The null hypothesis is in all cases a unit root in city shares. Following the suggestion by Ng and Perron (1995), we choose the optimal number of lagged growth rates to be included in the regression to control for autocorrelation using a “general‑to‑specific procedure” based on the t‑statistic. The maximum lag length to start off this procedure is set at 11 Table 2 (second column) reports a summary of the results of the individual city unit root tests. We find that the null hypothesis of a unit root in the city share is not rejected for most of the cities in the sample. In particular, for 207 of the 2680 Bel‑ gian cities (8%), the unit root is rejected at the 10% level, thus strongly supporting the random growth hypothesis. However, a possible concern with these results is that the overall non‑rejection of the unit root hypothesis may be because the stand‑ ard ADF tests are biased (Perron 1989). It is possible that what we identified as a unit root process could be better modelled as a stationary process around highly permanent shocks, especially when such a long period is considered. In their study of German cities, Bosker et al. (2008) addressed this issue by allowing for the pres‑ ence of a one‑time structural break when testing for unit roots, using the unit root test suggested by Perron and Vogelsang (1992). Here, we follow the same approach. Following Bosker et al. (2008), we estimate additive outlier (AO) models, allow‑ ing for a sudden change in mean (crash model). The AO model is appropriate when the change is assumed to take effect instantaneously (for instance, because of war ‑ fare destruction). This model is estimated by way of a two‑step procedure. The first step removes the deterministic part of the series by estimating the regression s = + DU + , (3) it i i t it where DU = 0 if t ≤ TB(the break date) and 1 otherwise. The resulting residuals are then tested for the presence of a unit root by estimating k k = DTB + + c Δ + , (4) it j it−j it−1 j it−j it j=0 j=0 where is the estimated residual from Eq. (3), TB is the break date and DTB = 1 it it if t = TB + 1 and 0 otherwise. Both equations are estimated using OLS for each break year TB = k + 2, ..., T − 1 , with T being the number of observations and k being the truncation lag parameter. The null hypothesis of a unit root is rejected if the t‑statistic for is significant. In this case, the city share will be a stationary time series around a structural break. All but one shock (the break) would cause 1 3 Urban growth in the long term: Belgium, 1880–1970 temporary movements of the city’s share. By contrast, if the t‑statistic for is not significant, the city share will be a non‑stationary time series and any sudden shock will have permanent effects on the long‑run level of the city share. The results of applying the AO‑model to test for a unit root in city shares under the null of a unit root versus stationarity around a possibly shifting mean under the alternative are also summarized in Table 2 (third column). Although the percent‑ ages of rejection of a unit root increase slightly, the results do not vary substantially from those of the ADF test. At the 10% level, the unit root null hypothesis cannot be rejected in favour of a stationary city share with a one‑time break for roughly 85% of the cities (2274 of 2680 cities). Moreover, the breaks are significant in almost all cases; only for 51 cities is the break not significant. Therefore, evidence supporting random growth persists. The previous analysis only captures the single most significant break in each city share series. However, since the period considered is quite long (almost one century) and variables rarely show just one break, we also attempt to determine whether the city share series show a double change in the mean. We use the test developed by Clemente et al. (1998), who based their approach on Perron and Vogelsang (1992), but allowing for two breaks. Formally, (3) and (4) change to: s = + DU + DU + , (5) it i i1 1t i2 2t it and k k k = DTB + DTB + + c Δ + , (6) it 1j 1t−j 2j 2t−j it−1 j it−j it j=0 j=0 j=0 where DU = 1 if t > TB (j = 1, 2) and 0 otherwise. DTB sets equal 1 if t = TB + 1 jt j ijt j and 0 otherwise (j = 1, 2) . TB and TB are the time periods when the mean is being 1 2 modified. Like Clemente et al. (1998), we suppose that TB = T (j = 1, 2) , with j j 0 <𝜆 < 1 , which implies that the test is not defined at the limits of the sample, and that 𝜆 >𝜆 , which eliminates those cases in which breaks occur in consecutive 2 1 periods. To test for the unit root null hypothesis, Eq. (5) is first estimated using OLS to remove the deterministic part of the variable, and then the test is carried out by searching for the minimal pseudo t‑ratio for the = 1 hypothesis in Eq. (6) for all the break time combinations. The null hypothesis of a unit root is rejected if the t‑sta‑ tistic for is significant. In this case, the city share will be a stationary time series around two structural breaks. Most shocks would cause temporary movements of the city’s share, but two shocks (the breaks) would cause permanent effects. Unlike the situation in which the t‑statistic for is not significantly different from zero, the city share will be a non‑stationary time series and any sudden shock will have permanent effects on the long‑run level of the city’s share of the population. We would expect that allowing for the possibility of two endogenous break points could provide further evidence against the unit root hypothesis (Lumsdaine and Papell, 1997; Ben‑David et al. 2003), especially because both breaks are significant in most of the cases: only in 45 and 49 cases (of 2680) are the first and the second break, respectively, not significant. However, the percentage of unit roots rejected 1 3 R. González-Val et al. at the 10% level is lower than that for the one‑break test (see Table 2, fourth col‑ umn). Furthermore, at the 5% and 10% significance levels, the percentage of rejec‑ tion is even lower than that of the ADF test (no‑break scenario). There is no clear pattern in the size of the cities for which the unit root is rejected; their mean popula‑ tion is slightly above the average size for all cities from 1880 to 1890, but it slowly decreases over time, and, over the whole period, these city sizes are on average only 12% below the mean population of all cities. Therefore, on average, these cities tend to be slightly below the mean city size but certainly are not the smallest units in our sample. Regarding the dates of the breaks, Fig. 3 displays the distribution of the breaks’ timing. The y‑axis shows the frequency (i.e., the total number of significant breaks detected) by year (the x‑axis). It shows both the one‑ and two‑break cases; only significant breaks are included in the graph. Across the one‑ and two‑break cases, three distinct major events stand out: the First World War, the economic crisis of 1929–1933 and the Second World War. Most distinctly present in the one‑break case, the Great Depression of the early 1930s, following the crash of the Ameri‑ can stock exchange in 1929, undoubtedly carries with it a lot of explanatory weight (Caestecker 2015, pp. 133–134). As unemployment soared, so did the mobility of those people who were looking for alternative ways to make a living. A decade earlier, the impact of the First World War is clearly visible in both the one‑break and the two‑break first‑break cases. In Belgium, the onset of the war caused an enormous stream of refugees to flee their home towns and settle in other places, both within the Belgian territory and abroad. Some 1.5 million Belgian refu‑ gees left the country to settle temporarily in the Netherlands, France and Great Brit‑ ain (Amara 2008). During and in the aftermath of the war, there was an enormous drop in the Belgian population, caused by a fall in the number of births and a peak in the number of deaths due to both the war and the Spanish flu that directly followed it (Caestecker and Vanhaute 2011, pp. 94–96). That the break in population growth is only visible in the aftermath of the war is probably related to the fact that, during the war, administrative services were severely disrupted. Even though there was no internal displacement on such a large scale during the Second World War, the onset and aftermath of the war present a clear break in popu‑ lation growth, visible both in the one‑break cases and in the two‑break cases’ sec‑ ond break. The onset of the war, however, did cause some people to flee abroad, and, just before the outbreak, many thousands of refugees from Nazi Germany and the occupied territories arrived in Belgium, looking for shelter (Debruyne 2007, pp. 75–76). In the aftermath of the war, as Belgian refugees returned home, thousands of displaced persons found themselves on Belgian territory. Most of them were for‑ mer prisoners of war, brought to Belgium by the Nazi occupants to perform forced labour (Luyckx 2010). The fast economic recovery of Belgium in the immediate aftermath of the war might also have led to internal migration towards the revived industrial centres, where plenty of jobs were to be found (Witte and Meynen 2006, pp. 35–36). Two more important moments of change can only be deduced in a meaningful way from the two‑break case. In the two‑break case, for the first break, we can see a clear break around the turn of the nineteenth century. The last two decades of 1 3 Urban growth in the long term: Belgium, 1880–1970 Fig. 3 Distribution of the timing of the breaks that century heralded a period of economic crisis, resulting in growing unemploy‑ ment and poverty. In these turbulent times, thousands of people moved to another town, where they hoped to find a more secure income. Thousands of others wan‑ dered around the country, looking for work. Finally, Belgium also experienced a 1 3 R. González-Val et al. demographic boom at the end of the nineteenth century, due to a spectacular rise in life expectancy (Caestecker and Vanhaute 2011, pp. 89–94). This certainly forms part of the explanation for the sudden break in population growth that we can dis‑ cern at the fin de siècle. The final break in growth, which we can only meaningfully discern in the two‑ break case, as the 2nd break, is noticeable around the year 1955. For this change in the pattern, we can find no immediate explanation. The 1950s were a period of slow economic growth in Belgium, characterized by relatively high unemployment. Com‑ pared with the following years, the Golden Sixties, the mid‑1950s can be seen as a last moment of high mobility before a long decade of relative stability, connected to a boom in economic growth and a historically low unemployment rate (Witte and Meynen 2006). Finally, we run several robustness checks. First, we consider a sub‑sample of cities. The geographic boundaries of cities could change over such a long time period and, thus, in some cases, the city growth or the structural break may only be reflecting the change in city boundaries. We have information about administrative changes to the boundaries of Belgian cities and the size of the municipalities from 1865 to 1970 (every 10 or 15 years). Using this historical information, we can cal‑ culate the change in land area by city and, following Glaeser and Shapiro (2003), we can set a threshold for the change in land area and exclude all cities above it. In particular, we exclude all cities with more than a 30% change (positive or negative) in land area, thus reducing the sample size to 2476 cities. This correction elimi‑ nates extreme cases in which the city in 1880 is very different from the city in 1970. Then, we rerun all the unit root tests; the results are reported in Table 3. The first conclusion is that the results are quite similar to those shown in Table 2, which indi‑ cates that the issue of changing in boundaries was not driving our main results. The second implication of these results is that the support for a unit root in city shares is even stronger when we exclude changing boundary units; although the percent‑ ages of unit roots rejected for the no‑break scenario test are the same as in Table 2, when one or two breaks are allowed, these percentages decrease slightly to very low figures. For instance, only in 1% of the cases is the unit root rejected at the 5% confi‑ dence level under the two‑break specification. Second, we re‑run the time series analysis considering the generalised supre‑ mum ADF (GSADF) test statistic for explosive behaviour proposed by Phillips et al. (2011, 2015). This recursive test does not assume exogenous structural breaks. For all the cities, the percentage of unit roots rejected at the 5% level increases up to the 36% (976 cities out of 2680). Nevertheless, the main result holds, as we cannot reject a unit root for most cities (64%). Alternative values of the threshold yield similar results, as the change in land area in most cities is negligible: for 82% of our Belgian cities, the change in land area is lower than 1%, for 86% the change is lower than 5%, and for 89% the change in land area is lower than 10%. 1 3 Urban growth in the long term: Belgium, 1880–1970 Table 3 Robustness check: Results of unit root tests on city shares excluding cities with more than 30% of change in land area City‑specific tests (N = 2476) Alternative hypothesis Trend stationary Trend stationary Trend stationary with one break with two breaks Significance level (%) % unit root rejected % unit root rejected % unit root rejected 1 1 2 1 5 4 7 1 10 8 14 3 The null hypothesis is in all cases a unit root in city shares. Following the suggestion by Ng and Perron (1995), we choose the optimal number of lagged growth rates to be included in the regression to control for autocorrelation using a “general‑to‑specific procedure” based on the t‑statistic. The maximum lag length to start off this procedure is set at 11 3.2 Panel unit root test Most theories proposed for the underlying mechanisms that govern the city size distribution are dynamic, and they make predictions of how particular cities will behave in a panel. To take full advantage of the panel dimension of our data set, we also test for a unit root in a panel. Some authors have also tested for the presence of a unit root using growth equa‑ tions and panel data (Black and Henderson 2003; Resende 2004; Henderson and Wang 2007; Chen et al. 2013). Nevertheless, this approach has some problems and limitations (Gabaix and Ioannides 2004; Bosker et al. 2008). As Chen et al. (2013) point out, panel unit root tests with short temporal dimension can suffer from low power. Although our time dimension is higher than that in previous studies (91 tem‑ poral observations for most cities), it may be still a low number, in contrast to the number of cities in the panel. Our annual data overcome one of the common limitations in this literature, namely the use of census data and decade‑by ‑decade city populations, but an econo‑ metric issue persists: the presence of cross‑sectional dependence across the cities in the panel can give rise to estimations that are not very robust (González‑ Val and Lanaspa 2016). Cross‑sectional dependence implies that the cities are interdepend‑ ent. The causes of cross‑sectional dependence in the errors can be the presence of common shocks and unobserved components that ultimately become part of the error term, spatial dependence and idiosyncratic pair‑wise dependence in the dis‑ turbances with no particular pattern of common components or spatial dependence (Baltagi et al. 2007). The econometric literature has established that the panel unit Some authors (Lalanne and Zumpe 2019) argue that panel unit root testing is far from being a panacea for the low test power problem. Indeed, as shown by Karlsson and Löthgren (2000), what most contrib‑ utes to the increase of test power is the extension of the panel’s temporal dimension. Consequently, the only extension of the cross‑sectional dimension, as commonly practised in the urban growth literature, is not a viable solution. 1 3 R. González-Val et al. root and stationarity tests that do not explicitly allow for this feature among indi‑ viduals present size distortions that can lead to misleading inference (Banerjee et al. 2005). Moreover, Baltagi et al. (2007) examined the performance of several panel unit root tests under spatial dependence, and found that tests assuming cross‑section independence perform better. Therefore, following González‑ Val and Lanaspa (2016), among all tests avail‑ able in the literature especially developed to deal with this issues (Breitung and Das 2005; Breitung and Pesaran 2008; Gengenbach et al. 2010), we use Pesaran’s (2007) test for unit roots in heterogeneous panels with cross‑sectional dependence. The test of the unit root hypothesis is based on the t‑ratio of the OLS estimate of b in the following cross‑sectional augmented Dickey–Fuller (denoted by CADF) regression: Δs = a + b s + c s + d Δs + e , it i i i,t−1 i t−1 i t it (7) where s is again the log‑city share of city i in year t, a is the individual city‑spe‑ it i −1 cific average growth rate and s is the cross‑section mean of s , s = N s . To t it t jt j=1 eliminate cross‑dependence, standard Dickey–Fuller (or augmented Dickey–Fuller) regressions are augmented with the cross‑section averages of lagged levels and the first differences of the individual series, such that the influence of the unobservable common factor is asymptotically filtered. The null hypothesis assumes that all series are non‑stationary, and Pesaran’s CADF is consistent under the alternative that only a fraction of the series is stationary. Nevertheless, the test does not allow for struc‑ tural breaks in the series. Unfortunately, due to computational limitations, we cannot run the test using the full sample of cities; we must restrict our analysis to a sub‑sample of the largest cities considering groups from the 50 to the 500 largest cities. Nevertheless, this sample of largest cities represents around two‑thirds of the total country population in all years. Black and Henderson (2003) and González‑ Val and Lanaspa (2016) highlighted one potential issue of sample selection in a dynamic framework: if the sample of cities is defined according to the largest units in the latest year, the analy ‑ sis may be biased because these are the “winning” cities, namely those that have presented the highest growth rates over time. To deal with this potential problem, we define the groups of cities using two samples including the largest cities in the first (1880) and last (1970) years. Although the latter group may only include winning cities, the former group can also include information about cities that were impor‑ tant in 1880 but declined over time. Table 4 shows the results of applying Pesaran’s panel unit root test. Panel A reports the results for the sample of largest cities in 1880. We find that the null hypothesis of a unit root is not rejected in most of the cases. The unit root is only rejected in two cases, for the groups including the 50 and 100 largest cities in 1880, when three lags and a trend are added. Panel B shows the results using a sample of the largest cities in 1970. In this case, the unit root hypothesis is not rejected in Baltagi et al. (2007) concluded that Pesaran and Phillips–Sul tests seem to be the least affected by spa‑ tial dependence among units. The test is calculated using the ‘pescadf’ Stata package. 1 3 Urban growth in the long term: Belgium, 1880–1970 any case, even at the 10% significance level. Overall, both panels present strong evi‑ dence supporting a unit root in these samples of large cities, in line with the results obtained using time series analysis in the previous section. As a robustness check, we re‑estimated the test using 500 random samples of cities. For each sample, we draw 500 cities without replacement from the full sample of 2680 Belgian cities regardless of their size. We could not reject the null hypothesis of a unit root in any case for any model specification. Again, these results (not shown) strongly support random growth. Finally, in Table 5 we report the results from the Pesaran’s (2004) test for cross‑ sectional dependence (CD) for the same subsamples considered in Table 4. Results using the Frees’ (1995) test (not shown) are similar. The null hypothesis of cross‑ sectional independence is rejected in all cases, confirming the need for using a panel unit root test controlling for cross‑sectional dependence. 4 Conclusions We examine urban growth in Belgium from 1880 to 1970 using unit root tests to check random growth in the long term. Our results add to the scarce literature on unit root testing in city sizes (Clark and Stabler 1991; Sharma 2003; Bosker et al. 2008), and, to our knowledge, this is the most comprehensive test of Gibrat’s law using unit root tests ever carried out as we consider annual data for all municipalities. Using both time series and panel data unit root tests, we obtain strong valida‑ tion of the random growth hypothesis, that is, Gibrat’s law, which implies that urban growth is independent of the initial city size. This evidence supports a multiplicative growth process of cities in Belgium, and this kind of growth is consistent with many theoretical urban economics models (Gabaix 1999; Eeckhout 2004; Duranton 2007; Córdoba 2008). Nevertheless, even if city shares follow a unit root, this growth process is compatible with a degree of convergence in the evolution of city growth rates; that is, with some kind of mean‑reverting component (Gabaix and Ioannides 2004). The long‑term pattern of random growth does not imply that the city size distri‑ bution has remained static over the years. On the contrary, a unit root implies that all shocks have had permanent effects on the city share, and, in particular, when allow ‑ ing for structural breaks, we find that exogenous historical shocks had a permanent effect on city shares: the timing of the structural breaks coincides with some major historical events, such as the World Wars and the economic crisis of 1929–1933. The alternative explanations for random growth considered in the literature are, basically, locational fundamental theories and increasing returns to scale (Davis and Weinstein 2002). Although our results are not specifically a test of random growth versus locational fundamentals or random growth versus increasing returns to scale, the strong support obtained for random growth clearly cast some doubts on the rel‑ evance of the two alternative theories in the Belgian case in the long‑term. These results are available from the authors upon request. 1 3 R. González-Val et al. 1 3 Table 4 Panel unit root tests, Pesaran’s CADF statistic Sample size Augmenting lag (1) Augmenting lags (2) Augmenting lags (3) Constant Constant and trend Constant Constant and trend Constant Constant and trend Panel A: Top cities in 1880 Top 50 − 0.726 (0.234) − 1.289 (0.099) − 0.970 (0.166) − 1.320 (0.093) − 1.407 (0.080) − 2.278 (0.011) Top 100 1.675 (0.953) 0.679 (0.751) 0.453 (0.675) − 0.746 (0.228) − 0.643 (0.260) − 2.262 (0.012) Top 200 6.926 (1.000) 5.630 (1.000) 6.549 (1.000) 3.733 (1.000) 5.127 (1.000) 1.499 (0.933) Top 300 9.250 (1.000) 9.177 (1.000) 9.021 (1.000) 7.913 (1.000) 7.515 (1.000) 5.734 (1.000) Top 500 8.932 (1.000) 13.427 (1.000) 7.512 (1.000) 12.331 (1.000) 5.481 (1.000) 9.941 (1.000) Panel B: Top cities in 1970 Top 50 1.729 (1.000) 1.419 (0.922) 2.257 (0.988) 1.050 (0.853) 1.582 (0.943) − 0.437 (0.331) Top 100 1.689 (0.954) 4.425 (1.000) 2.854 (0.998) 3.977 (1.000) 2.178 (0.985) 2.630 (0.996) Top 200 1.597 (0.945) 9.409 (1.000) 2.963 (0.998) 9.514 (1.000) 3.392 (1.000) 8.897 (1.000) Top 300 1.984 (0.976) 14.565 (1.000) 3.065 (0.999) 15.202 (1.000) 3.592 (1.000) 14.973 (1.000) Top 500 3.851 (1.000) 17.179 (1.000) 4.054 (1.000) 18.160 (1.000) 4.201 (1.000) 18.911 (1.000) Annual panel data from 1880 to 1970. The null hypothesis assumes that all series are nonstationary. Pesaran’s (2007) Z t test statistic (pvalue) Urban growth in the long term: Belgium, 1880–1970 Table 5 Pesaran’s CD test for Sample size CD test statistic p‑value cross‑section dependence in panel data Panel A: Top cities in 1880 Top 50 52.83 0.000 Top 100 67.49 0.000 Top 200 64.21 0.000 Top 300 72.73 0.000 Top 500 125.72 0.000 Panel B: Top cities in 1970 Top 50 53.10 0.000 Top 100 114.19 0.000 Top 200 219.41 0.000 Top 300 311.54 0.000 Top 500 345.66 0.000 Annual panel data from 1880 to 1970. The null hypothesis assumes cross‑section independence. Pesaran’s (2004) CD test statistic and p‑value This evidence is consistent with previous research for other countries. To give some examples, González‑ Val et al. (2014) found that random growth (i.e., Gibrat’s law) holds for most cities in the US, Spain, and Italy, and Chauvin et al. (2017) concluded that Brazil and the US both appear to adhere broadly to Gibrat’s law, but China and India do not. As random growth is a long‑term pattern, it involves that the city size distribution reaches a dynamic steady state. For this reason, Chauvin et al. (2017) explained that Gibrat’s law holds in Brazil and the US because both are moderately sized places, which have long been largely urban, while China and India are much larger, and many of their cities are newer. In terms of our study using Bel‑ gian cities, it is natural that random growth emerges as the pattern of growth in the long‑term, considering similar arguments. First, Belgium is a small‑sized country. Second, as explained in the Introduction, both the industrialization and the comple‑ tion of transport infrastructures took place at the beginning of the sample period (in the late nineteenth and early twentieth centuries), implying that urbanization of the country began in the early twentieth century. Third, as in many European countries, most Belgian cities date from ancient times, so the urban system is consolidated. However, our results raise new research questions. Our analysis captures the long‑term pattern of growth, but omits the dynamics towards steady state. In the short‑term, agglomeration economics and locational fundamentals may have been important in the transition to spatial equilibrium. In Chauvin et al. (2017) aspects such as mobility, rent‑earnings relationships or skill‑related factors can influence the steady state configuration. Further study on these economic factors could help to improve our knowledge about urbanization in Belgium and shed some light on the population dynamics in countries currently experiencing the urbanization process. Acknowledgements The authors benefited from the helpful comments and suggestions from Wenzheng Li, Ilyes Boumahdi and two anonymous referees. An earlier version of this paper was presented at the 7th International conference on Time Series and Forecasting (Gran Canaria 2021), at the 68th North America 1 3 R. González-Val et al. Meetings of the Regional Science Association International (virtual conference 2021), at the 61st Con‑ gress of the European Regional Science Association (virtual conference 2022) and at the XLVII Reunión de Estudios Regionales (Granada 2022), and all the comments made by participants are much appreci‑ ated. All remaining errors are ours. Funding Open Access funding provided thanks to the CRUE‑CSIC agreement with Springer Nature. Min‑ isterio de Ciencia e Innovación and Agencia Estatal de Investigación,MCIN/AEI/10.13039/501100011033 (projects ECO2017‑82246‑P, PID2020‑114354RA ‑I00, PID2020‑112773GB‑I00), Gobierno de Aragón (project S39_20R, ADETRE research group), and ERDF. 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References Amara M (2008) Des Belges à l’épreuve de l’exil: les réfugiés de la Première Guerre mondiale: France, Grande‑Bretagne, Pays‑Bas, 1914–1918. Editions de l’Université de Bruxelles, Bruxelles Baltagi BH, Bresson G, Pirotte A (2007) Panel unit root tests and spatial dependence. J Appl Economet 22:339–360 Banerjee A, Massimiliano M, Osbat C (2005) Testing for PPP: should we use panel methods? Empirical Economics 30:77–91 Barrios S, Bertinelli L, Strobl E (2006) Geographic concentration and establishment scale: an extension using panel data. J Reg Sci 46(4):733–746 Ben‑David D, Lumsdaine R, Papell DH (2003) Unit root, post‑war slowdowns and long‑run growth: evi‑ dence from two structural breaks. Empir Econ 28(2):303–319 Black D, Henderson V (2003) Urban evolution in the USA. J Econ Geogr 3(4):343–372 Bleakley H, Lin J (2012) Portage and path dependence. Q J Econ 127(2):587–644 De Block G, Polasky J (2011) Light railways and the rural–urban continuum: technology, space and soci‑ ety in late nineteenth‑century Belgium. J Hist Geogr 37(3):312–328 Bosker EM, Brakman S, Garretsen H, Schramm M (2007) Looking for multiple equilibria when geogra‑ phy matters: German city growth and the WWII shock. J Urban Econ 61:152–169 Bosker EM, Brakman S, Garretsen H, Schramm M (2008) A century of shocks: the evolution of the Ger‑ man city size distribution 1925–1999. Reg Sci Urban Econ 38:330–347 Breitung J, Das S (2005) Panel unit root tests under cross‑sectional dependence. Stat Neerl 59:414–433 Breitung J, Pesaran MH (2008) Unit roots and cointegration in panels. In: Matyas L, Sevestre P (eds) The Econometrics of Panel Data: Fundamentals and Recent Developments in Theory and Practice, 3rd edn. Springer Publishers, Berlin, pp 279–322 Caestecker F (2015) Hoe de mens de wereld vorm gaf. Academia Press, Gent Caestecker F, Vanhaute E (2011) Leven en werken in het industriële België. In: Grauwels A et al (eds) Hedendaagse economische geschiedenis van België: een inleiding. Academia Press, Gent Champernowne D (1953) A model of income distribution. Econ J LXIII:318–351 Chauvin JP, Glaeser E, Ma Y, Tobio K (2017) What is different about urbanization in rich and poor coun‑ tries? Cities in Brazil, China, India and the United States. J Urban Econ 98:17–49 Chen Z, Fu S, Zhang D (2013) Searching for the parallel growth of cities in China. Urban Stud 50(10):2118–2135 Clark JS, Stabler JC (1991) Gibrat’s law and the growth of Canadian Cities. Urban Stud 28(4):635–639 1 3 Urban growth in the long term: Belgium, 1880–1970 Clemente J, Montañés A, Reyes M (1998) Testing for a unit root in variables with a double change in the mean. Econ Lett 59:175–182 Córdoba JC (2008) A generalized Gibrat’s Law for Cities. Int Econ Rev 49(4):1463–1468 Davis DR, Weinstein DE (2002) Bones, bombs, and break points: the geography of economic activity. Am Econ Rev 92(5):1269–1289 Debruyne E (2007) Gedoogbeleid in al zijn gedaanten Joodse vluchtelingen en België (januari 1933‑sep‑ tember 1939). In: Van Doorslaer, R. (eds.), Gewillig België Overheid en Jodenvervolging in België tijdens de Tweede Wereldoorlog. Brussel: SOMA De Decker P (2011) Understanding housing sprawl: the case of Flanders. Belgium Environment and Plan‑ ning A 43(7):1634–1654 De Meulder B et al (1999) Patching up the Belgian urban landscape. Oase 52:78–113 Devadoss S, Luckstead J (2015) Growth process of U.S. small cities. Econ Lett 135:12–14 Dickey DA, Fuller WA (1979) Distributions of the estimators for autoregressive time series with a unit root. J Am Stat Assoc 74(366):427–481 Dickey DA, Fuller WA (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49(4):1057–1072 Duranton G (2007) Urban evolutions: the fast, the slow, and the still. American Economic Review 97(1):197–221 Eaton J, Eckstein Z (1997) Cities and growth: theory and evidence from France and Japan. Reg Sci Urban Econ 27(4–5):443–474 Eeckhout J (2004) Gibrat’s Law for (All) Cities. Am Econ Rev Am Econ Assoc 94(5):1429–1451 Ellison G, Glaeser EL (1999) The geographic concentration of industry: does natural advantage explain agglomeration? Am Econ Rev Papers Proc 89(2):311–316 Frees EW (1995) Assessing cross‑sectional correlation in panel data. J Econom 69:393–414 Gabaix X (1999) Zipf’s law for cities: an explanation. Quart J Econ 114(3):739–767 Gabaix X, Ioannides YM (2004) The evolution of city size distributions. In: Thisse JF, Henderson JV (eds) Handbook of urban and regional economics. Elsevier Science, Amsterdam, pp 2341–2378 Garcia‑López M‑ A, Holl A, Viladecans‑Marsal E (2015) Suburbanization and highways in Spain when the Romans and the Bourbons still shape its cities. J Urban Econ 85:52–67 Gengenbach C, Palm FC, Urbain JP (2010) Panel unit root tests in the presence of cross‑sectional depend‑ encies: comparison and implications for modelling. Economet Rev 29:111–145 Glaeser EL, Shapiro J (2003) Urban growth in the 1990s: is city living back? J Reg Sci 43(1):139–165 González‑ Val R (2010) The evolution of US city size distribution from a long term perspective (1900– 2000). J Reg Sci 50(5):952–972 González‑ Val R, Lanaspa L (2016) Patterns in U. S. urban growth (1790–2000). Reg Stud 50(2):289–309 González‑ Val R, Lanaspa L, Sanz‑Gracia F (2014) New evidence on Gibrat’s law for cities. Urban Studies 51(1):93–115 González‑ Val R, Silvestre J (2020) An annual estimate of spatially disaggregated populations: Spain, 1900–2011. The Annals of Regional Science, forthcoming Henderson V, Wang HG (2007) Urbanization and city growth: the role of institutions. Reg Sci Urban Econ 37(3):283–313 Holmes TJ, Stevens JJ (2002) Geographic concentration and establishment scale. Rev Econ Stat 84(4):682–690 Ioannides YM, Overman HG (2003) Zipf’s law for cities: an empirical examination. Reg Sci Urban Econ 33:127–137 Karlsson S, Löthgren M (2000) On the power and interpretation of panel unit root tests. Econ Lett 66:249–255 Lalanne A (2014) Zipf’s law and Canadian urban growth. Urban Studies 51(8):1725–1740 Lalanne A, Zumpe M (2019) La croissance des villes canadienne et australienne guidée par le hasard Enjeux et mesure de la croissance urbaine aléatorie. Can J Reg Sci 42(2):123–129 Lalanne A, Zumpe M (2020) Time‑series based empirical assessment of random urban growth: new evi‑ dence for France. J Quant Econ 18(4):911–926 Luyckx L (2010) Russische krijgsgevangenen van de nazi’s: van Displaced Persons tot vluchtelingen (voor het Sovjetcommunisme). Belgisch Tijdschrift Voor Nieuwste Geschiedenis, XL 3:489–511 Ng S, Perron P (1995) Unit root tests in ARMA models with data dependent methods for the selection of the truncation lag. J Am Stat Assoc 90:268–281 Partridge MD, Rickman DS, Ali K, Olfert MR (2008) Lost in space: population growth in the American hinterlands and small cities. J Econ Geogr 8(6):727–757 1 3 R. González-Val et al. Perron P (1989) The great crash, the oil price shock and the unit root hypothesis. Econometrica 57:1361–1401 Perron P, Vogelsang T (1992) Nonstationarity and level shifts with an application to purchasing power parity. J Bus Econ Stat 10:301–320 Pesaran MH (2007) A simple panel unit root test in the presence of cross‑section dependence. J Appl Econom 22:265–312 Pesaran, M. H., (2004). General diagnostic tests for cross section dependence in panels. IZA Discussion Paper No. 1240. Phillips PCB, Shi S, Yu J (2015) Testing for multiple bubbles: historical episodes of exuberance and col‑ lapse in the S&P 500. Int Econ Rev 56(4):1043–1077 Phillips PCB, Wu Y, Yu J (2011) Explosive behavior in the 1990s NASDAQ: when did exuberance esca‑ late asset values? Int Econ Rev 52(1):201–226 Resende M (2004) Gibrat’s law and the growth of cities in Brazil: a panel data investigation. Urban Stud‑ ies 41(8):1537–1549 Ronsse S, Standaert S (2017) Combining growth and level data: an estimation of the population of Bel‑ gian municipalities between 1880 and 1970. Hist Method J Quant Interdiscip Hist 50(4):218–226 Ryckewaert M (2011) Building the economic backbone of the Belgian welfare state: infrastructure, plan‑ ning and architecture 1945–1973. Uitgeverij, Rotterdam Sharma S (2003) Persistence and stability in city growth. J Urban Econ 53:300–320 Simon H (1955) On a class of skew distribution functions. Biometrika 42:425–440 Van Meeteren M, Boussauw K, Derudder B, Witlox F (2016) Flemish diamond or ABC‑ Axis? The spa‑ tial structure of the Belgian metropolitan area. Eur Plan Stud 24(5):974–995 Vrielinck, S., (2000). De territoriale indeling van Belgie (1795–1963): bestuursgeografisch en statistisch repertorium van de gemeenten en de supracommunale eenheden (administratief en gerechtelijk): met de officiele uitslagen van de algemene volkstellingen. Vol. 1. Leuven University Press, 2000 Witte E, Meynen A (2006) De geschiedenis van België na 1945. Standaard Uitgeverij, Antwerp Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 1 3
The Annals of Regional Science – Springer Journals
Published: Mar 1, 2024
Keywords: C12; C22; N93; O18; R11; R12
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