J Geograph Syst (2006) 8: 307–316 DOI: 10.1007/s10109-005-0014-5
O R I GI N A L P A P E R
Javier Alvarez Æ Pascal Mossay
Estimation of a continuous spatio-temporal population model
Revised: 20 September 2005 / Accepted: 14 October 2005 / Published online: 2 December 2005 Springer-Verlag 2005
Abstract In this paper we propose a continuous spatio-temporal model that describes population change in a region in terms of population growth, migration drift towards regions with better economic or climate conditions, and population diffusion from more populated to less populated areas. Finite-differences are used to approximate the space and time derivatives. The model is estimated by using population data from the US census corresponding to the period 1790–1910. People tend to migrate from east to west, and to relocate towards regions with lower precipitation levels and more abundant coal and iron resources. Also population growth tends to be larger in zones with higher precipitation levels and higher temperatures. Keywords Population dynamics Æ Migration Æ Diffusion Æ Drift Æ Estimation JEL classification R23 Æ C51
1 Introduction Spatial interaction, discrete choice and hazard-duration models are routinely used in migration research today. This contribution makes an attempt to model regional population change as a continuous spatio-temporal process that involves three components: natural growth/decline (births and deaths), migration drift towards regions that are more attractive, and population
J. Alvarez Æ P. Mossay (&) Departamento de Fundamentos del Ana´lisis Econo´mico, Universidad de Alicante, Alicante, Spain E-mail:
[email protected] J. Alvarez Banco de Espan˜a, Madrid, Spain
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diffusion from more to less densely populated areas. The contribution, thus, lies in the tradition of continuous time and continuous space modelling (see Isard and Liossatos 1979; Beckmann and Puu 1985; Donaghy 2001) on the one side and extends the classical Hotelling diffusion model to explain migration flows in a continuous geographical space on the other (see Hotelling 1921; but also Tobler 1981, 1997). There are only very few empirical investigations with spatial dynamic models. A notable exception is the contribution by Donaghy and Plotnikova (2004) who published the estimates of a model in continuous time and continuous space, obtained with empirical data. This paper demonstrates another viable approach to the estimation of a model in continuous time and continuous space. The structure of the paper is as follows. In the section that follows, theoretical specifications of the model are elaborated, and the model is discretised and operationalised for fitting with historical data from the United States Census Bureau. The data section briefly describes the data used, while the estimates of the model parameters are reported and discussed in the section of Empirical results. Conclusion briefly summarises the major empirical results.
2 A population model In this section we present the spatio-temporal population model. The geographical space consists of a continuum of locations distributed along a line segment, x 2 [0, L]. Here x=0 and x=L correspond, respectively, to the east and west borders of the USA, the geographical space under consideration. Time is denoted by t. We indicate the population density in location x at time t by P(x, t). Consider the infinitesimal regionR ½x; x þ dx: The population inhabiting regionR at time t is P(x, t)dx. Our objective is to determine how the population in regionR varies over the time span dt. For this purpose we decompose the population change ¶tP(x, t)dx into population growth, diffusion, and migration drift. We now come to the description of each of these three components of population change. First, population growth in regionR accounts for new births net of deaths that take place in the region itself. The net population accumulation per unit of time in regionR is described as QðP ðx; tÞÞdx ¼ ðaP ðx; tÞ bP 2 ðx; tÞÞdx;
ð1Þ
where a denotes the growth rate of population and b takes into account some crowding (saturation) consideration. In the absence of crowding, the parameter a corresponds to the difference between the birth and death rates. The effect of crowding is to depress the rate of population growth when population is becoming increasingly large. Second, diffusion accounts for population flows from highly populated regions towards less populated regions, so that the migration diffusion flow / d(x, t) in location x at time t is proportional to d ¶xP(x, t). As a
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consequence, the population flows through the left and right borders of regionR are described by /d ðx; tÞ /d ðx þ dx; tÞ ¼ d@x P ðx; tÞ þ d@x P ðx þ dx; tÞ;
ð2Þ
where d denotes the intensity of the diffusion process across space. Note that if ¶xP(x, t)<0, then population is flowing into regionR through border x. Similarly, if ¶xP(x + dx, t)<0, then population is flowing out of regionR through border x + dx. Third, migration drift is introduced to reflect incentives of people to migrate to preferred locations. We consider people to move in the westward direction at a constant speed V>0, so that the drift flow /a(x, t) in location x at time t corresponds to V P(x, t). As a consequence, the flows through the left and right borders of regionR are described by /a ðx; tÞ /a ðx þ dx; tÞ ¼ V P ðx; tÞ V P ðx þ dx; tÞ;
ð3Þ
where V indicates the intensity of migration drift. Consequently—as a response to westward migration drift—population is flowing into regionR through border x. Similarly, population is flowing out of regionR through border x + dx. Now we can obtain the change of population in regionR over time span dt by summing the expressions as given by relations (1), (2), and (3) @t P dx ¼QðP Þdx þ /d ðx; tÞ /d ðx þ dx; tÞ þ /a ðx; tÞ /a ðx þ dx; tÞ ¼QðP Þdx d@x P ðx; tÞ þ d@x P ðx þ dx; tÞ þ V P ðx; tÞ V P ðx þ dx; tÞ: By using Taylor expansions and dividing by the size of the region dx, we get the following law governing the evolution of population over space and time @t P þ @x ðV P Þ ¼ QðP Þ þ @x ðd@x P Þ:
ð4Þ
The evolution law for population as given by Eq. 4 identifies three components of population change in a given location: population growth, diffusion, and migration drift towards preferred regions. The modelling of diffusion, which goes back to the heat equation of Laplace, has been introduced in a population modelling context by Hotelling (1921), and in biological models by Skellam (1951). The aggregate effect of diffusion is to relocate individuals from highly populated areas towards less populated areas. It turns out that diffusion can be modelled as resulting from the interaction of individual random migration decisions. To illustrate this point, consider some spatial distribution of population and assume that at each period of time each individual moves locally westward, eastward, or stay, with equal probability (1/3). The temporal evolution of the spatial distribution of population can described by a pure diffusion force (see Murray 2004). Diffusion may also reflect heterogeneous deterministic individual behaviour by incorporating idiosyncrasies in location taste on behalf of individuals (see Mossay 2003). The magnitude of d is informative about the strength of the diffusion effect.
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Incorporating climate and economic conditions in the empirical analysis, we will rely on a specification that allows the migration drift to depend on some explanatory variables. For this purpose, we construct a utility function U(x) that expresses the utility of being located at a distance x from the east coast U ðxÞ ¼ gx þ l0 ZðxÞ;
ð5Þ
where Z(x) is a vector of explanatory variables including climate and economic variables; g and the vector l are parameters. The utility function refers to the standard of living in some location and is the driving force of population migration. In order to decompose the effect of each of the possible variables on migration, we have supposed that the utility function is separable (i.e. linear) in the explanatory variables Z. Individuals relocate in the direction providing them the largest utility level, so that the adjustment speed V will be given by V ðxÞ ¼ @x U ¼ g þ l0 @x ZðxÞ:
ð6Þ
The main difference with the basic model is that the adjustment speed is not anymore constant across locations, instead it depends on the explanatory variables Z. Such a formulation for migration drift, has been used by Sonnenschein (1982) to describe the spatial relocations of firms towards higher profit locations, and by Mossay (2003) to model the migration of labour towards higher real wage regions. The evolution law for the population distribution is now given by @t P þ @x ½ðg þ l0 @x ZÞP ¼ QðP Þ þ @x ðd@x P Þ:
ð7Þ
It may be rewritten as 2 ZP þ @x Z@x P ¼ QðP Þ þ @x ðd@x P Þ: @t P þ g@x P þ l0 @xx
ð8Þ
In order to discretize Eq. 8, the line segment [0, L] is divided into N cells of size Dx, and discrete time is denoted by t Dt, where Dt is the time step. We indicate the population value P(i Dx, t Dt) by Pi,t, for i=0, 1, 2, ..., N and t=0, 1, 2, ..., T. Next, finite differences are used to approximate the time and spatial derivatives appearing in Eq. 8 Pi;tþ1 Pi;t þ OðDtÞ; Dt Pi;t Pi1;t Piþ1;t þ Pi1;t 2Pi;t 2 P¼ þ OðDx2 Þ þ OðDxÞ; @xx @x P ¼ Dx Dx2 Zi Zi1 Ziþ1 þ Zi1 2Zi 2 @x Z ¼ Z¼ þ OðDx2 Þ: þ OðDxÞ; @xx Dx Dx2 @t P ¼
ð9Þ
By using theses expressions, we obtain a discretised approximation of Eq. 8
Estimation of a continuous spatio-temporal population model
Pi;tþ1 Pi;t Pi;t Pi1;t 0 Ziþ1 þ Zi1 2Zi Pi;t : þg þl Dt Dx Dx2 Zi Zi1 Pi;t Pi1;t þ Dx Dx Piþ1;t þ Pi1;t 2Pi;t ¼ QðPi;t Þ þ d þ OðDx; DtÞ: Dx2
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ð10Þ
By substituting expression Q as given by Eq. 1, we have the following equation that will be used in the empirical analysis Pi;t Pi1;t Pi;tþ1 Pi;t 2 ¼ ðaPit bPit Þ g þ Dt Dx l0 2 ðZiþ1 þ Zi1 2Zi ÞPi;t þ ðZi Zi1 ÞðPi;t Pi1;t Þ ð11Þ Dx d þ 2 Piþ1;t þ Pi1;t 2Pi;t þ OðDx; DtÞ: Dx This equation implies that the population change can be decomposed into four components. The first one refers to the natural growth of population with a factor to control for potential crowding effects. The second term captures the westward migration drift. The third term reflects migration drift driven by better climate or economic conditions. Finally, the last term corresponds to the diffusion effect.
3 The data In this section, we describe the main characteristics of the dataset used to estimate the model. We rely on population data for S=47 continental US states as available from the Bureau of Census (1909) for the period 1790– 1910 (Dt=10 years, T=13). In order to construct the population levels on a regular spatial grid, the US territory is divided into N=32 equal-size zones from the East to the West coasts. The size of each of these zones is Dx=100 miles (see Fig. 1). Of course, we need to allocate population among the defined zones so that the population proportion of a state, that is assigned to a zone, just corresponds to the proportion of the state area that makes actually part of that given zone. For this purpose, we construct a 32·47 matrix M whose entries mij indicate the proportion of a given state j that makes part of a given zone i. For instance, looking at Table 1 (where we only include some rows and columns of the matrix), we see that 33% of Maine state is part of the first zone while the 67% left is part of the second zone. Also, a 88% of Vermont state is part of zone 2, while only a 5% of California state is part of the last zone. By using the matrix of weigths M, we can obtain the population level Pi,t for each zone i 2 (1, ..., 32) at date t 2 (1, ..., Ti)
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Pi;t ¼
S X
mi;j pj;t ;
j¼1
where pj,t corresponds to the population data for state j 2 (1, ..., 47) at date t. The number of periods Ti differs across zones since for some zones located in the western part, the population data is only available for the last periods of our sample. In order to avoid problems related to the division of a given state into two new ones, we aggregate West Virginia and Virginia from 1870 on as if the old’ Virginia had remained one single state.
Fig. 1 Discretised US map
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Table 1 Matrix of weights Zone\State
Maine
Vermont
...
Oregon
Nevada
California
1 2 3
33% 67% 0% .. . 0% 0% 0%
0% 88% ..12% . 0% 0% 0%
... ... ..... . ... ... ...
0% 0% ..0% . 26% 16% 1%
0% 0% ..0% . 2% 0% 0%
0% 0% ..0% . 31% 27% 5%
30 31 32
In addition to the population data, we construct variables of climate conditions and economic prospects for each zone. In particular, our climate measures consist of the yearly mean temperature and rain precipitation. They are given for US states during the period 1931–1990. These variables are reported by the U.S. National Oceanic and Atmospheric Administration (NOAA) and collected from the U.S. National Climate Data Center web site. In order to distribute the data on our discretised map, we apply a weight matrix to the climate data for US states. So as to construct the economic variables we follow some previous studies on population growth that use variables of natural resources as a proxy for economic prospects, see e.g. Beeson et al. (2001). The measures we use, refer to the production of iron and coal for US states. This information was obtained from the Report on the Mining Industries of the 1880 US Census of Mining (1886).
4 Empirical results In this section, we describe the main empirical results of the paper. We estimate different versions of our theoretical model as given by Eq. 11 Gitþ1 ¼ aPit þ bPit2 þ cDRit þ k0 Yit þ dDit þ eitþ1 ; i ¼ 2; . . . ; N 1; t ¼ 1; . . . ; T 1;
ð12Þ
where Git+1 denotes the 10-year population change in zone i, (Pi,t+1 Pi,t)/ Dt, and Pit is the population in zone i at period t. The variable DRit corresponds to the westward migration drift and is given by (Pi,t Pi-1,t)/Dx. The vector Yit relates to the migration drift driven by climate and economic conditions. It is given by [(Zi+1+Zi-1 2 Zi)Pi,t+(ZiZi-1) (Pi,t Pi-1,t)]/ Dx2, where the vector Zi includes the variables of climate and natural resources of zone i. The variables considered are the average temperature, the average level of rain precipitation, and the production of iron and coal (in tons). Finally, the diffusion effect denoted by Dit, is given by [Pi+1,t+Pi-1,t 2 Pi,t]/Dx2. In the first column of Table 2 we provide estimates for a specification that includes exponential population growth and westward migration as the only drift term. This implies that coefficients b and k are restricted to be zero in Eq. 12. Both the growth and drift effects are significant. The drift term has the expected sign. Note that given our theoretical model, a negative
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Table 2 Estimation of the population model, 1790–1910 Variable
Model 1
Model 2
Model 3
0.021 (31.47) –
0.027 (12.77) 1.78·109 (2.82) 9.648 (0.38) 0.723 (2.21) –
0.029 (13.08) 2.00·109 (3.11) 85.473 (3.03) 0.495 (1.25) 3.00·104 (1.79) 1.467 (0.46) 2.66·106 (1.14) 1.39·105 (1.15) –
Model 4
0.020 (6.97) 6.89·1010 P (1.07) Diffusion 34.727 97.078 (1.57) (3.20) Drift to the West 0.761 0.016 (2.76) (0.04) 2.65·104 Drift rain – (1.78) Drift temp – – 0.008 (0.00) 5.59·106 Drift coal – – (2.25) 2.84·105 Drift iron – – (1.81) Rain – – 0.013 (2.18) Temp – – – 92.39 (1.54) Coal – – – 0.001 (1.48) Iron – – – 0.004 (0.83) 0.852 0.867 0.873 0.906 R2 The number of observations is 259. In parentheses, t-ratios computed from standard errors robust to heteroskedasticity P
2
coefficient on the drift variable implies a migration drift in the westward direction. The implications of our estimates is that the natural growth of population is estimated to be 2% per year, while the annual displacement of an individual is found to be 0.7 mile towards the west. In the second column of Table 2 we show the estimates corresponding to a logistic population growth model. This second model includes the variable P2it as an additional explanatory variable. Notice that the westward migration drift effect remains significant and of the same magnitude than in the exponential growth model. Moreover, the growth rate is estimated to be 2.7% with a negative crowding effect which is statistically significant. The diffusion effect is positive though not statistically significant. In the third column of Table 2 we estimate a third model including the climate and natural resource variables as motives to migrate. The equation includes some variables denoted by (drift rain), (drift temp), (drift coal), and (drift iron), which are the variables forming the vector Yit. Notice that by incorporating these variables in the regression, the statistical significance of the diffusion variable increases. The westward migration has the expected sign though it is not statistically significant. However, the migration drift driven by the precipitation variable is marginally significant. The positive coefficient associated with this variable suggests that people move towards regions with lower precipitation levels. Note that given Eq. 11, negative
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values of the drift parameters k imply a migration drift towards regions with higher values of the variable Z. The explanatory variables in Eq. 12 are endogenous if fixed-effects corresponding to a zone enter into the error term of the equation. The endogeneity comes from the fact that regressors depend on time and space lags of the population variable which are correlated with the fixed-effects. In that case, OLS estimation may provide biased estimates of the true coefficients. In order to control for some specific effects of zone, in the fourth column of Table 2, we estimate a fourth model including climate and natural resource variables as explanatory variables in level. The magnitude of the growth rate decreases and is approximately 2%. The westward migration drift is not statistically significant. However, the sign of the migration drift driven by the climate and natural resources can be interpreted in the following manner. Individuals tend to migrate towards regions with more abundant coal and iron resources, and with lower precipitation levels. Moreover, among the variables in level that appear in the equation, only the precipitation variable is statistically significant. The corresponding coefficients reflect that population growth tends to be larger in zones with higher precipitation and temperature. Interestingly, the precipitation level in a region affects the local change of population via two channels. The lower the precipitation level in a region, the lower the population growth in that region, and at the same time, the larger the migration flows towards that region. There is some evidence of spatial and temporal autocorrelations in the OLS residuals. The first- and second-order spatial autocorrelations are 0.78 and 0.51, respectively. In order to evaluate the validity of our estimates, some alternative specifications have been considered. When including time and space lags of the population variable P up to the second order as explanatory variables, the spatial autocorrelations decline (to 0.64 and 0.37, respectively). At the same time, no dramatic difference was observed in the coefficient estimates. While the strategy of including additional temporal and spatial lags allows to reduce spatio-temporal autocorrelation, it leads to the problem of reducing the sample size (number of observations) and gives rise to an empirical specification that is not anymore fully consistent with our theoretical model. 5 Conclusion In this paper, we have extended the original population model by Hotelling (1921) so as to include the migration drift driven by better climate, or economic conditions. This has allowed us to view regional population change as a process that involves at least three components: natural growth/decline (births and deaths), migration drift towards regions that are more attractive, and population diffusion from more to less densely populated areas. We have discretised and operationalised the model for fitting with historical data in the context of the US westward migration. Migration drift has the expected sign (e.g. westward oriented), and is towards more attractive regions (i.e. regions with more favourable climate and economic conditions). Individuals tend to migrate from east to west, and to relocate to regions with more
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favourable climate conditions (i.e. lower precipitation levels) and economic prospects (i.e. more abundant coal and iron resources). While individuals tend to migrate to regions with lower precipitation levels, it is interesting to note that natural growth was also found to be lower in these regions. Acknowledgements We would like to thank the editor Manfred Fischer as well as three anonymous referees for their valuable comments that have helped to improve the clarity and quality of the paper. Financial support from the Spanish Ministerio de Ciencia y Tecnologı´ a, under ERDF project SEJ2004-08011ECON, is acknowledged.
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