Comparison of Internal and International Migratory Schedules Using Optimization Algorithms for Rogers and Castro Models

Joaquín Recaño, Center of Demographic Studies - UAB
Javier Sebastian Ruiz-Santacruz, Center of Demographic Studies - UAB

The study of migratory patterns is important in determining the structure and composition of migrants in the various cities and countries of destination. This can be done by estimating the models proposed by Rogers and Castro (1981) and by the detailed information of the population. There are several studies that treat the problem from the internal migration and its calendars and population projections (Rees 1977; Rogers y Raymer, 2009, Bates J y Brachen I, 1982; Liaw and Nagnur, 1985; Kawabe, 1990), leaving outside the relation with international migration and having troubles with the selectivity of the initial values in the optimization process. The main goal of this study is to estimate the Rogers and Castro equations by solving the initial values problem, which guarantees that the values resulting from the optimization are influenced as little as possible by these values. The data used are the National Institute of Statistics of Spain and correspond to the standardized migration rates for the year 2015 by sex. The results presented in this example are the estimates obtained from the parameters for various types of migration such as interregional, interprovincial and international. Here, differences between the peak ages of labour migration as well as the post-retirement ages are observed. Subsequent work has to do with the refinement of the algorithm, as well as the use of the same for the calculation of temporal series of the parameters that uses information from sources as IPUMS and administrative registries of more countries, thus making comparisons on the performance of various groups of emigrants.


Study migratory patterns is important to determine the structureand composition provide to migrants in different countries and citydestination. In demography, the initial research of Rogers and Castro in 1981show descriptions of migratory schedules and other social behaviours relatedwith the weight of labour migration through parameters estimations of a non-linealfunction that can be of 7, 9, 11 or 13 parameters expressed as the sum ofseveral exponential functions. In the estimation, the optimization algorithms developedup to now, begin with initial values studied in the original article and areobtained for data in the Stockholm area, what induces the algorithm to find abiased solution by the initial values. There are several uses for the modelschedules mainly in the analysis of internal migration and populationprojections (Rees 1977; Rogers y Raymer, 2009, Bates J y Brachen I, 1982; Liawand Nagnur, 1985; Kawabe, 1990), leaving a lack of information of the behaviourshowed by international migration. The objective of this study, is to realizethe estimation of the equations of Rogers and Castro solving the problem ofchoosing the initial values, which would guarantee that the resultant values ofthe optimization are influenced as little as possible.

Data and Methodology

For this example there are used inputs obtained of the Spanish NationalInstitute of Statistics (INE) and correspond to the age-period rates calculatedfor man and women in 2015. The estimation methodology is realized by theresampling of parameters of the selected model, in this case using the model of11 parameters, following a general algorithm: 1. obtaining a random sample ofsize one from a uniform distribution of each parameter, using range similar toStockholm parameters, 2. then there is realized the optimization to obtainingthe finest parameters estimated according to the minimization of the error functionbetween the estimated theoretical and the empirical data and 3. It is chosenthe best one among n repetitions and there is estimate a confidenceinterval with the results obtained. Optimization is made with function nlminbof R software which minimises the error function between theoreticalfunction and empirical data, this allows calculate in detail each of componentsof the optimization.


Results obtained shows the model estimation by resampling process andthe optimization with one thousand repetition for this example. The figure 1present the simulation curves (in blue), the empiric data (yellow), and the optimalcurve (red) for the out-migration of man and women in Spain in 2015. Estimationin Table 1, shows important differences for example with regard to the parametermu2 which indicates the age peak of labour force age, being higher forinternal (interregional and interprovincial) than international migration forboth male and female.




Figure 1. Estimated schedule with standardized rates for differenttypes of out migration man (left) and women (right). Source: Own elaboration.


Table 1. Estimated parameters with 1000 repetition for severaltypes of mobility (International, inter-regional, inter-provincial) by sex (m:male, f: female) Source: Own elaboration.


Expected Results and further work

First, we will focus on the sensibility of initial valuesextending number of repetitions and opening the range of the initial parameterswhere it could be combinations of the values that cannot estimate a gradientand leads to an error. The results of these study will be used to determine therange of initial values for next studies and in the same way it will includeworks done with the model of 13 parameters. The comparisons are made for theinternal and international out-migration of emigrants in Spain but then theidea is to expand it to a bigger countries group using IPUMS data and Europeanadministrative registers due to lack of literature especially in which islinked the relation between internal and international. Other studies will bedone on the affection of sub-registration in lower ages as cero or one for interregionalan interprovincial migration.



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Rogers, A. and Raymer, J. 1999. Estimating the regional migrationpatterns of the foreign-born population in the United States: 1950-1990. MathematicalPopulation Studies. Vol 7 (3), pp: 181-216.

Rogers, A., Castro, L. and Lea, M. 2005. Model MigrationSchedules: Three alternative linear parameter estimation methods. MathematicalPopulation Studies. Vol 12 (1), pp: 17-38.

Wilson, T. 2010. Model migration schedules incorporating studentmigration peaks. Demographic Research. Vol 23 (8), pp: 191-222.



Presented in Session 1234: Internal Migration and Urbanization