Foreigners versus natives in Spain: different migration patterns? Any changes in the aftermath of the crisis?

Abstract

The main aim of this study, which takes Spanish provinces over the periods 2004–2007 and 2008–2013 as case study, is threefold: first, to test whether labor factors affect to a greater extent foreigners than natives when it comes to migrating; second, to detect changes in migration patterns over the crisis period; third, to unveil nonlinearities in the relationship between migration and wages. To do so, an extended gravity model, combined with a methodology that identifies endogenous thresholds to nonlinear effects, is estimated. The results support that the role played by labor factors is more important for foreigners than natives, especially before the outbreak of the economic crisis. The results also indicate that the relative size of the service sector and, to a lesser extent, climate conditions have gained importance as attraction factors for natives over the crisis, while the opposite happens for foreigners. Therefore, evidence clearly supports the idea that business cycle modifies the decision making of migrants. Finally, some nonlinearities in the effect of expected wages on migration are found regardless of the group and/or time frame considered.

Appendices

Appendix A

See Tables A.1 and A.2.

Table 3 Variables and definitions

Table 4 Descriptive statistics of variables

Appendix B: the threshold regression model

The method of threshold selection (Hansen 1999) provides us with the threshold value(s) that detects nonlinearities in a target variable. The general model proposed by Hansen (1999) would be as follows:

$$\begin{aligned} y_{ij,t} =\left\{ {{\begin{array}{ll} \beta _{1} x_{ji,t} + \varepsilon _{ij,t}, &{}\quad q_{ji,t} \le \gamma \\ \beta _2 x_{ji,t} + \varepsilon _{ij,t}, &{}\quad q_{ji,t} > \gamma \\ \end{array} }} \right. \end{aligned}$$

(B.1)

where $q_{ji,t}$ is the threshold variable and $\gamma$ is the threshold parameter.

In our case, $q_{ji,t} =w_{ji,t-1}^e$, $y_{ij,t} = m_{ij,t}$ and $x_{ji,t} =[w_{ji,t-1}^e, hc_{ji,t-1} ,hp_{ji,t-1} ,\hbox {agr}_{ji,t-1} , \hbox {const}_{ji,t-1} , \hbox {ser}_{ji,t-1} ,\hbox {clim}_{ji,t-1} ,d_{ij}]$. As can be seen, the two “regimes” are characterized by different slope coefficients ${\beta _1}$ and ${\beta _2}$.

The threshold parameter is unknown, so we should carry out an estimate. To do so and avoid the possibility that the potential threshold $({\hat{\gamma }})$ sorts too few observations into each “regime”, first of all, a grid search over the potential values of the threshold variable choosing, in our case, a $5\%$ trimming is performed to exclude extreme values.

Then, by considering the remaining observations, this method takes different partitions which constitute the potential threshold values $({\hat{\gamma }})$. If we denote ${\hat{\beta }}_1 ( \gamma )$ and ${\hat{\beta }}_2 ( \gamma )$ the corresponding estimates in each partition, it is possible to compute the sum of squared errors $S_1 ( \gamma )$ conditionally to a value of $\gamma$ as follows:

$$\begin{aligned} S_1 ( \gamma )=\sum \limits _{i=1}^N \sum \limits _{j=1}^N \sum \limits _{t=1}^T {\hat{\varepsilon }}_{ij,t}^2 \left( \gamma \right) \end{aligned}$$

(B.2)

The threshold estimate $({\hat{\gamma }})$ is obtained by minimizing the concentrated sum of squared errors:

$$\begin{aligned} {\hat{\gamma }} =\hbox {ArgMin}_{\gamma } \, S_1 ( \gamma ) \end{aligned}$$

(B.3)

Departing from the single-threshold model reported in (B.1), the null hypothesis of no threshold effect $(H_0 : \beta _1 =\beta _2 )$ could be tested with a standard test. Denoting the sum of squares of the linear model as $S_0$, the approximate likelihood ratio test of $H_0$ can be expressed as:

$$\begin{aligned} F_1 =\frac{S_0 -S_1 \left( { {\hat{\gamma }}} \right) }{ {\hat{\sigma }}^{2}} \end{aligned}$$

(B.4)

where ${\hat{\sigma }}^{2}$ denotes a convergent estimate of $\sigma ^{2}$. The difficulty here is that the threshold parameter $\gamma$ is not identified under the null hypothesis. Thus, the asymptotic distribution of $F_1$ is not standard and, in particular, it does not correspond to a chi-squared distribution. To test for the nonlinearity hypothesis, we use bootstrap simulations to compute the p value of the distribution (Hansen 1999).

Then, if the p value associated with $F_1$ led to rejecting the linear hypothesis, we would have to discriminate between one and two thresholds. The corresponding likelihood ratio statistic is the following:

$$\begin{aligned} F_2 =\frac{S_1 ( {{\hat{\gamma }}} )-S_2 \left( {{\hat{\gamma }}_1 ,{\hat{\gamma }}_2 } \right) }{{\hat{\sigma }}^{2}} \end{aligned}$$

(B.5)

where ${\hat{\gamma }}_1$ and ${\hat{\gamma }}_2$ refer to the threshold estimates of a double-threshold model and $S_2 \left( {{\hat{\gamma }}_1 ,{\hat{\gamma }}_2 } \right)$ denotes the corresponding residual sum of squares. Similarly, if the bootstrap p value associated with $F_2$ led to rejecting the null hypothesis of one threshold, we then would have to discriminate between two and three thresholds, and so forth.

As it is shown in the paper, in our case, $F_1$ statistic seems to confirm the existence of nonlinearities in wages, while $F_2$ statistic does not support the existence of a double threshold. So our final model is that of one threshold.