Econometric Modeling Of External Migration

Model Construction

The construction of econometric models was preceded with the statistical analysis of all the indices under study.

The tentative assessment of statistical data demonstrated high variation on the resultant indicator. Outliers were analyzed. Smirnov’s criterion was used for calculations. It was established that removal of outliers had lowered the statistical quality of the model. The same could be concluded from the task as the countries’ sample would not demonstrate the real situation in global migration processes without such a big country as the USA.

Table 01 shows the analysis of statistical data for 2015. It demonstrates the matrix of paired correlation coefficients on the factors mentioned above. The following results were obtained: unemployment level influences the number of migrants weakly negatively; favorable migration policy influences the number of migrants weakly positively; there exists multicollinearity among the factors X₁, X₃ and X₄; factors X₁, X₃, X₄, X₅ (on a point of less influence) determine the resultant indicator most of all living standards indicators in the countries correlate (Azose & Raftery, 2018), and, hence, cannot be included into one regression model. To remove multicollinearity the factors X₁, X₃ and X₄ are not used together in the models.

Different variants of multifactor linear and nonlinear models were constructed based on the undertaken analysis. The best models from the constructed ones were chosen for comparison. Let us consider the regression models of the following type:

$Y=b_{\mathrm{0}}+b_{\mathrm{1}}{X}_{\mathrm{1}}+b_{\mathrm{5}}{X}_{\mathrm{5}}+e$ (1)

$Y=b_0+b_1{X}_1+e$ (2)

$Y=b_0+b_1{X}_3+e$ (3)

$Y=b_0+b_1{X}_4+e$ (4)

Estimations of parameters and basic characteristics of the models are shown in Table 02 . Comparative analysis demonstrated that the preference could be given to model (1).

Notes:

* corresponds to significance of coefficient estimation under significance level 10%;

** corresponds to significance of coefficient estimation under significance level 5%;

*** corresponds to significance of coefficient estimation under significance level 1%.

The next step of research was firstly, to compare various models, and secondly, to compare models with the equations:

$$\begin{array}{c}Y=b_{\mathrm{0}}+b_{\mathrm{1}}{X}_{\mathrm{1}}+b_{\mathrm{2}}{X}_{\mathrm{1}}^{\mathrm{2}}+e\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left(\mathrm{5}\right)\\ Y=b_{\mathrm{0}}+b_{\mathrm{1}}{X}_{\mathrm{1}}+b_{\mathrm{2}}{X}_{\mathrm{5}}+b_{\mathrm{3}}Z+e\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left(\mathrm{6}\right)\\ Y=b_{\mathrm{0}}+b_{\mathrm{1}}{X}_{\mathrm{1}}+b_{\mathrm{2}}{X}_{\mathrm{1}}^{\mathrm{2}}+b_{\mathrm{3}}{X}_{\mathrm{1}}^{\mathrm{3}}+e\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left(\mathrm{7}\right)\\ Y=b_{\mathrm{0}}+b_{\mathrm{1}}{X}_{\mathrm{1}}+b_{\mathrm{2}}{X}_{\mathrm{1}}^{\mathrm{2}}+b_{\mathrm{3}}{X}_{\mathrm{1}}^{\mathrm{3}}+b_{\mathrm{4}}{X}_{\mathrm{5}}+e\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left(\mathrm{8}\right)\end{array}$$

The results of assessment of models’ quality (5) - (8) are shown in Table 03 .

The criteria were chosen for the best econometric models among those with estimates of regression coefficients significance under the significance level of not less than 10%. Adjusted determination coefficient and statistical model in whole were taken. Among all the models under statistical significance and quality of fit the best one was model (8), the next model was (7). Table 3 shows that the adjusted determination coefficient in the model (8) is higher than in model (7). The addition of the factor Crime increased the model’s quality.

Thus, model (8) was chosen as the best one and its regression equation looks like:

$$\hat{Y}=-11.738+1.539X_1-0.0847X_1^2+0.0012X_1^3+0.197X_5$$

The model (8) was tested on heteroscedasticity (Sukhanova, Shirnaeva, & Mokronosov, 2016) with residual graphical analysis, with White and Breushch-Pagan tests, and with rank correlation Spearman test. The testing gave a negative result. However, heteroscedasticity was observed under Goldfeld-Quandt and Glazer tests. Glazer test showed the following type of heteroscedasticity $s\left(e\right)=1.873+0.056X_1$. It was then eliminated with generalized least squares estimation (GLS). It resulted in the model

( $R^2=0.79;R_{adj}^2=0.69$):

$\hat{Y}=-8.829+1.313X_1-0.075X_1^2+\underset{\left(0.0002\right)}{\left(0.717\right)}+0.0011X_1^3+\underset{\left(0.0668\right)}{0.150}{X}_5$. (9)

In comparison with model (8) the number of standard deviations (given in brackets) in model (9) decreased after application of GLS. Estimated regression coefficients became statistically significant under the significance level of 5%. Subsequent test for heteroscedasticity gave a negative result on all tests (White test (test statistic: LM = 11.7632, р-value = P(X² (11) > 11.7632) = 0.3817), Breushch-Pagan test (test statistic: LM = 4.7720, р-value = P(X² (4) > 4.7720) = 0.3115)). Therefore, conditions of Gauss-Markov’s theorem were met.

Based on the final (2015) model (9) the forecast was given. The average number of migrants of 1.550 million people was predicted under the average in sample Gross National Income X₁ = 22.801 thousands of dollars and Crime Index X₅ = 38.380 points. According to the interval forecast the number of migrants will not exceed 4.476 million people with 95% confidence if Gross National Income and Crime Index take equal average values within the sample.

Model Verification Based on Other Statistical Data

Based on the research of external migration within various countries for 2015 statistical data for 2017 were gathered on 21 countries from the previous list (The World Bank, 2017; The World Only, 2018; Numbeo, 2017). The following indicators were taken as the factor indicators influencing the resultant indicator Y, the number of migrants in the country (millions of people): X₁ – Gross National Income (thousands of the US dollars per capita) and X₅ – Сrime Index. The indicators were taken based on the analysis from 2015 data. Various regressions models were constructed. Table 04 demonstrates the best models under statistical significance and fitting.

Based on the comparative analysis, preference was given to the model (10). Regression equation of the model (10) is of the form:

$\hat{Y}=-9.033+0.882X_1-0.054X_1^2+0.0009X_1^3+0.227X_5$.

The results of statistical quality verification were the following: the estimates of regression coefficients are significant under the level of 7% (regression coefficient estimate under X₁ worsens the model’s quality); $R^2{}_{\mathbf{H}}{R_{adj}}^2$ demonstrate good explanatory nature of the model; multicollinearity is absent (Table 05 ). Heteroscedasticity was absent under the graphical analysis of residuals and White test (test statistic: LM = 16.3877, р-value = P(X² (11) > 16.3877) = 0.1273). Breushch-Pagan test also showed the lack of heteroscedasticity (test statistic: LM = 1.6007, р-value = P(X² (4) > 1.6007) = 0.8086). The same was true under the tests of rank correlation by Spearman, Goldfeld-Quandt and Glazer. All that suggested that conditions of Gauss-Markov’s theorem were met.

The forecast was given on the final model (10). The number of migrants will reach 1.704 million people if Gross National Income and Crime Index have the following values on average within the sample, X₁ = 22.656 thousand dollars, X₅ = 35.175 points. According to the interval forecast under Gross National Income and Crime Index that are equal to average values within the sample the number of migrants will not exceed 9.470 million people with 93% confidence.

Overall, final migration econometric models for 2015 and 2017 differ slightly and demonstrate stability in regression coefficients estimates.