A spatio-temporal analysis of migration

Abstract

Migration is a space- and time-dependent phenomenon. Traditional geographical migration models have considered the distance between source and destination countries or have applied suitable normalizations to treat the correlation among migratory flows. To disentangle cross-sectional dependence, spatial correlation is explored in mainly two directions. First, migratory flows from “neighbouring” countries are considered to be directly interconnected. Second, a set for exogenous drivers are allowed to be correlated among the different economic units. Swiss immigration, from 153 source countries from 1981 to 2011, is modelled using a dynamic spatial econometric model able to capture both path-dependency and spatial interactions. An out-of-sample forecasting, performed to assess the model’s accuracy, confirms the crucial role played by the spatial terms over the dynamic ones.

Appendices

Appendices

Maximum likelihood estimators for spatial dynamic panel data with fixed effects

This section presents the quasi-maximum likelihood estimation for the spatial dynamic panel data model with fixed effects when both N and T are large given by Yu et al. (2008).

Starting from a model like the following:

$$\begin{aligned}&y_{it} = \gamma y_{it-1} + \beta ' X_{it-2} + \lambda \sum \limits _{j=1}^{N} w_{ij} y_{jt} + \pi \sum _{k=1}^{K} \sum \limits _{j=1}^{N} w_{ij} y_{jt-1}\nonumber \\&\quad + \rho ' \sum \limits _{j=1}^{N} w_{ij} X_{jt-2} + \alpha _i + \varepsilon _{it}, \varepsilon _{it} \sim iid(0, \sigma ^2), \quad k=1,\ldots ,K \end{aligned}$$

(A.1)

where $w_{ij}$ is the i-th element of the spatial weight matrix W. Equation 8 is subject to the bias typical of dynamic models, the Nickell bias, (Nickell 1981) caused by the correlation between the autoregressive component $y_{it-1}$ and the fixed effect $\alpha _i$. Yu et al. (2008) have derived the following maximum likelihood (ML) function where $\eta = (\beta ', \gamma , \lambda , \pi , \rho ', \sigma ^2)'$, $\zeta =(\beta ', \gamma , \lambda , \pi , \rho ', \alpha '_i)$,

$$\begin{aligned} \begin{aligned} \ln L_{i,T}(\eta , \alpha _i) = - \frac{nT}{2} \ln 2 \pi - \frac{nT}{2} \ln 2 \sigma ^2 + T \ln |S_i (\lambda )| - \frac{1}{2 \sigma ^2} \sum \limits _{t=1}^{T} V'_{it}(\zeta )V_{it}(\zeta ) \end{aligned} \end{aligned}$$

(A.2)

where $V'_{it}(\zeta ) = S_i(\lambda )y_{it} - \gamma \sum \nolimits _{k=1}^{K} y_{it-1} - \beta ' X_{it-2} - \pi \sum _{k=1}^{K} \sum \nolimits _{j=1}^{N} w_{ij} y_{jt-1} - \rho ' \sum \nolimits _{j=1}^{N} w_{ij} X_{jt-2} - \alpha _i$ and $S_i(\lambda ) = I_i - \lambda W$. The maximization of the ML Eq. A.2 gives the estimators $\hat{\eta }_{it}$ and $\hat{\alpha }_i$. If $\hat{\eta }_{it}$ and $\hat{\alpha }_i$ follow a normal distribution, a maximum likelihood estimation (MLE) is required otherwise a quasi-maximum likelihood estimation (QMLE) is needed. $\alpha _i$ is concentrated out from the likelihood function A.2, and the deriving concentrated likelihood function is used for the estimation of the $\eta$ parameters. The first-order conditions of A.2 with respect to $\alpha _i$ are:

$$\begin{aligned} \frac{\partial \ln L_{it}(\eta , \alpha _i)}{\partial \alpha _i} = \frac{1}{\sigma ^2} \sum \limits _{t=1}^{T} V_{it}(\zeta ). \end{aligned}$$

(A.3)

Therefore, the concentrated estimator of $\alpha _i$ given $\eta$ is

$$\begin{aligned} \hat{\alpha }_{i}(\eta ) = \frac{1}{T} \sum \limits _{t=1}^{T} \left( S_i(\lambda )y_{it} - \gamma y_{it-1} - \beta ' X_{it-2} - \pi \sum _{k=1}^{K} \sum \limits _{j=1}^{N} w_{ij} y_{jt-1} - \rho ' \sum \limits _{j=1}^{N} w_{ij} X_{jt-2} - \alpha _i\right) . \end{aligned}$$

(A.4)

After having plugged in $\hat{\alpha _i}$ in Eq. (A.2), the concentrated likelihood becomes

$$\begin{aligned} \ln L_{i,T}(\eta ) = - \frac{nT}{2} \ln 2 \pi - \frac{nT}{2} \ln 2 \sigma ^2 + T \ln |S_i (\lambda )| - \frac{1}{2 \sigma ^2} \sum \nolimits _{t=1}^{T} \tilde{V'}_{it}(\zeta ) \tilde{V}_{it}(\zeta ) \end{aligned}$$

(A.5)

where $\tilde{V}_{it}(\zeta ) = S_i(\lambda ) \tilde{y}_{it} - \gamma \tilde{y}_{it-1} - \beta ' \tilde{X}_{it-2} - \pi \sum _{k=1}^{K} \sum \nolimits _{j=1}^{N} w_{ij} \tilde{y}_{jt-1} - \lambda \sum \nolimits _{j=1}^{N} w_{ij} y_{jt} - \rho ' \sum \nolimits _{j=1}^{N} \sum \nolimits _{k=1}^{K} w_{ij} \tilde{y}_{jt-1}$, $\tilde{y}_{it} = y_{it} - \bar{y}_{T}$, $\tilde{y}_{it-1} = y_{it-1} - \bar{y}_{T}$, $\tilde{y}_{jt-1} = y_{jt-1} - \bar{y}_{T}$, $\tilde{X}_{it-2} = X_{it-2} - \bar{X}_{iT}$, $\bar{y}_{iT} = \frac{1}{T} \sum \nolimits _{t=1}^{T} y_{it}$, $\bar{y}_{iT-1} = \frac{1}{T} \sum \nolimits _{t=1}^{T} y_{it-1}$, $\bar{X}_{iT} = \frac{1}{T} \sum \nolimits _{t=1}^{T} X_{it-2}$.

The result of the maximization of function (A.5) is the QMLE $\hat{\eta }_{it}$ while the QMLE of $\alpha _i$ is the result of the substitution of $\hat{\eta }_{it}$ into expression (A.4) ($\hat{\alpha }_i(\hat{\eta }_{it})$). This procedure allows, in a first step, to concentrate out $\alpha _i$ from the ML function and solve the correlation problem in the estimation of the coefficients embodied in $\eta$. In a second step, the estimated coefficient $\hat{\eta }$ is plugged into Eq. (A.4) in order to derive the estimated fixed effect ($\hat{\alpha }_i (\hat{\eta })$).

Data sources and summary statistic

1.1 Data sources

Data on migration flows to Switzerland stem from the Swiss Federal Statistical Office. For the immigrant population of non-Swiss nationality, the data sources used are PETRA (Statistics for the resident population of foreign nationality), for the period 1981–2009, and STATPOP, from 2010, which report the migrants flows by age and nationality. The economic variables, real GDP and employment have been merged from the Penn Table 8.1. Population variables (total and active) were obtained from the FAOSTAT database [Food and Agriculture Organization (FAO), 2015].¹¹ The just-mentioned original data sources contain missing data for Czechoslovakia, Yugoslavia and USSR before the end of the communist regimes. To compute country-specific values of states belonging to the Soviet Bloc, for the period preceding their independence, specific weights are calculated and applied to the aggregated data available before 1990. Weights for explanatory variables (population, real GDP and employment) were computed in the following steps. First, for the period after the end of the regimes, the variables of the new countries were summed up by year in order to have a sort of ex-Bloc value. Second, the “original” variable was divided by the sum computed in the first step to obtain the relative weight of a state. Finally, the mean weight over the time span was used to weight the aggregate data available for the regimes period. For example, in the case of Czech and Slovakian Republics, from 1992 the yearly sum of real GDP of the two has first been computed. Then, the Czech yearly GDP was divided by the sum computed in the fist step and the same for Slovakia. Lastly, the weight mean was calculated. Migration data, for the eastern countries, show a sharp discontinuity before and after the collapse of communism since they were notably precluding people from migrating. However, this issue does not only concern the migration variables, but also the economic ones which display some breaks due to the downfall.

1.2 Variables construction and summary statistics

The variables used in this study were constructed from the previously described datasets. The description of the variable elaboration is now presented. The real GDP per capita differential was built from the values of real GDP expressed in millions of 2005 US dollars at chained purchasing power parity (PPP). Each value was divided by the total country population. The differential is simply the relation between the Swiss and the origin country’s real GDP per capita. The employment indicator denotes the number of people employed in a country. An employed person is an individual of 15 years or older, engaged even just for 1 h a week, or who had a job or business from which she was temporarily absent. This variable was divided by the labour force of the country in order to obtain the employment differential between Switzerland and the origin country. As for the GDP differential, it is expected that the larger the differential, i.e. Swiss values are higher than the source country ones, the greater the incentives to migrate. The total population for each origin country is introduced in the estimation to control for the population size of the so-called stayers. Intuitively the larger the population in the home state, the larger the potential size of migrants. The model includes also a dummy variable indicating the membership of a country to the bilateral agreements area which are expected to have a positive impact on immigration flows.

Table 6 reports the summary statistics of the non-transformed variables included in the empirical model. Figure 4 shows above the density of immigration in the original scale and below in the transformed scale (hyperbolic sine transformation, see Sect. 4).

Table 6 Summary statistics

Fig. 4

Distribution of the original (upper plot) and transformed (bottom plot) dependent variable immigration. Densities are computed with kernel density estimates with bandwidth selector given by SCO (2008) using factor 1.06

Spatial error model

In Sect. 3, it was outlined how spatial effects can be modelled either through the deterministic or the stochastic part of the model. I argued the implementation of the first type of models, which consistently mirrors the equilibrium Eq. (5) in Sect. 2. Despite that, as a robustness check, I will also test the performance of the second ones since they are known to deliver more flexible results. The dynamic spatial error model (SEM) estimates take the following form:

$$\begin{aligned} \begin{array}{l} y_{it} = \gamma y_{it-1} + \beta ' X_{it} + \alpha _{i} \varepsilon _{it}, \\ \varepsilon _{it} = \psi \sum \limits _{j=1}^{N} w_{ij} \varepsilon _{jt} + \varepsilon _{it}, \quad \varepsilon _{it} \sim iid(0, \sigma ^2). \end{array} \end{aligned}$$

(A.6)

According to Chapter 4, Sect. 4 of Elhorst (2014), within the category of dynamic spatial models, the spatial coefficients of model (8) must be restricted to zero ($\lambda =0, \pi =0, \rho =0$) in order for the model to ensure identification. Contrary to the SDM, for which long- versus short-term direct and indirect effects needed to be distinguished, this is not the case with the SEM either in its static or dynamic version. In fact for this class of spatial models the total long-term marginal (direct) effects are simple the $\beta$s. Table 7 only reports the results for the static SEM since the dynamic SEM contains an explosive non-stationary root ($\gamma > 1$).

Table 7 Static spatial error models

The spatial coefficients are positive and significant for all the specifications suggesting that migration from different countries reacts similarly to any internal or external shock. The RMSE and the MAPE are the second best among the spatial models (Table 5), after the spatial dynamic model ($\gamma =0$) ones, and far better than the other static, as well as the non-spatial models. In particular, the comparison between the static SEM and the dynamic non-spatial model seems to underline how controlling for cross-sectional dependence is fostering the prediction power even more than controlling for serial correlation no matter the definition of $\mathbf {W}$.

Table 8 Regression results spatial models with time interactions: contiguity weighting matrix

Table 9 Migration rates regression results

Regression results with time interactions

Spatial models have been conceived for the use of geographical weighting matrices constant over time. A key needs of this type of modes is the exogeneity of the matrix, which might be undermined when geography starts to be substituted by time-dependent interdependencies. Nevertheless, a concern may arise about the invariance of the spatial weighting matrices, which implicitly assume the importance of a specific definition of proximity has not changed over time. In the present context, for example, it might be argued that the influence exerted by geographical contiguity has evolved with the advent of globalization, the upcoming faster communication tools or the lowering for travel costs, jut to mention but a few. Hence, I test a potential evolution the spatial effects taking the case of the contiguity weighting matrix by interacting the spatial coefficients by a quinquennial categorical variable. Table 8 reports the results for the spatial exogenous terms. Compared to the results shown in Table 5, a weak evidence of the decreasing importance of contiguity can be traced and it seems to be particularly evident in the static model in which no term is controlling for the path dependency (see for example the decrease in significance of the employment differentials). In fact, with the exception of the interactions of the EU variable, significant coefficients are estimated for the first quinquennial (1983–1987). For example, the spatial employment differentials, which are significant for all the models in Table 5, are here, respectively, a 5% more significant, but only for the time span 1983–1987 and up to 1997 for the static model. Exclusively the results for the contiguity weighting matrix are reported since they are the only ones showing little evidence about the change in importance of $\mathbf {W}$ while for the other cases (i.e. language, common legal origins and belonging to the same colonial empire spatial matrices) no differences across 5 years periods are found.

Regression results migration rates

One of the reasons motivating this paper was the critiques towards the use of migration rates, defined in terms of the number of leaves over the number of stayers in a country, as a dependent variable. In this section, I report the results of model (6), both for the static ($\gamma = 0$) and dynamic version, where $y_{it}$ is the migration rate from each origin to Switzerland. As a measure of accuracy, the out-of-sample mean average percentage error (MAPE) is computed as it can be compared with models with different dependent variables (Table 9). In this case, the choice of migration rates as a dependent variable significantly diminishes the model accuracy with respect to the choice of the number of migrants.