# Modeling Local Variations in Intermarriage

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background and Hypotheses

## 3. International Migration and Intermarriage in Spain

## 4. Data and Variables

#### 4.1. Marriage and Population Register Data

#### 4.2. Construction of Variables

#### 4.2.1. The Dependent Variables

#### 4.2.2. Independent Variables

#### 4.2.3. Spatially Lagged Variables

_{ij}in the n × n spatial weight matrix

**W**, for n the number of municipalities, such that ${{\displaystyle \sum}}_{j=1}^{n}{w}_{ij}{x}_{ij}$ results in a scalar representing a linear combination of values obtained by neighboring observations [45]. In this paper, the spatial weight matrix utilizes an inverse distance function for a 170 km distance band, such that each element (w

_{ij}) is equal to $1/{d}_{ij}^{2}$ for each pair of municipalities i, j located at a distance ${d}_{ij}\le 170$ km and zero otherwise (see [46] for further information). We utilize 170 km as the minimum distance at which every municipality has at least one neighbor.

#### 4.2.4. Size of Municipality

## 5. Model Specifications

#### 5.1. Zero Inflated Poisson (ZIP) with Spatial Effects

- ●
- A Probability Mass Function (PMF), $P\left({y}_{i}=0\right)$, which is used to calculate the probability of observing a zero count.
- ●
- A PMF, $P\left({y}_{i}=k\right)$, which is used to calculate the probability of observing k events, given that $k>0$.
- ●
- A link function used to express the mean rate, $\lambda $, as a function of p regression variables
**X**.

^{th}observation (e.g., a Spanish municipality), and ${\lambda}_{i}={e}^{{x}_{i}\beta}$, for ${x}_{i}$ the value of one of the p explanatory variables of the regression model for the i

^{th}observation and $\beta $ the vector of the regression coefficients. The second part of Equation (1), $P\left({y}_{i}=k\right)$, corresponds with the standard Poisson regression model for ${y}_{i}$ the random dependent variable that denotes the observed count of the i

^{th}observation.

^{th}observation is 0 with a probability ${\varphi}_{i}$ or is a realization of a Poisson random variable, which can also be 0, with a probability ($1-{\varphi}_{i})$. That is, zero observations arise from both the zero-component distribution, $P\left({y}_{i}=0\right)$, and the Poisson distribution, $P\left({y}_{i}=k\right)$. The zero-component distribution is related to model the ‘excess’ or ‘inflated’ zeros that are observed in addition to the zeros that are expected to be observed under the assumed Poisson distribution [48]. When the data set does not have any excess zeros in the dependent variable, the value of ${\varphi}_{i}=0$ and the PMF of the ZIP model reduces to the PMF of the standard Poisson model [49].

**X**, as follows:

**X**, as follows:

#### 5.2. Probit Model for Grouped Data with Spatial Effects

^{th}observation, ${x}_{i}$ is a vector gathering a set of k variables which explains the intermarriage decision and β contains a set of parameters.

## 6. Results

#### 6.1. Modeling the Number of Intermarriages

^{2}with and without the spatially lagged variables. The inclusion of the spatially lagged variables enhances the explanatory capacity of all models, though the improvement is better in the larger municipalities. These results suggest that the influence of the characteristics of neighboring municipalities on intermarriage is higher in larger municipalities than in smaller ones.

#### 6.2. Modeling the Intermarriage Rate

^{2}with and without the spatially lagged variables.

^{2}values are somewhat lower than the pseudo-R

^{2}values of the ZIP models, they are not straightforwardly comparable: while the pseudo R

^{2}is computed as the correlation coefficient between the real and estimated dependent variables, the R

^{2}. is the determination coefficient of an Ordinary Least Squares (OLS) estimation. The relative size of the migrant community in the municipality is the explanatory variable with the highest number of statistically significant coefficients and is the most influential to intermarriage. The relationship between relative size of the migrant community and the intermarriage rate is positive for all sizes.

## 7. Discussion and Conclusions

^{2}, is higher in the largest municipalities. However, this result does not exclude finding statistically significant coefficients among the smallest municipalities as well.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Spatial distribution of intermarried couples: (

**a**) Spanish Men married to Migrant Women; (

**b**) Spanish Women married to Migrant Men, 2005–2007. Maps were generated with the software Map Viewer, version 8.7.

Variables | Description | Source |
---|---|---|

Dependent variables: | ||

Intermarriage | Number of marriages in each municipality between Spanish citizens and foreign-national individuals in Spain from 2005–2007. | MNP (Spanish Vital Registration) |

Intermarriage rate | Percentage of marriages between Spanish men/women and foreign-national women/men out of the total number of marriages contracted in each municipality from 2005 to 2007. | MNP (Spanish Vital Registration) |

Independent variables: | ||

Relative group size | Percentage of foreign-national population in the municipality | “Padrón” (Population Register) |

Homogeneity of national origin | $\mathrm{Homogeneity}\mathrm{of}\mathrm{migrant}\mathrm{origin}\mathrm{in}\mathrm{each}\mathrm{municipality}(H=100$$\mathrm{means}\mathrm{high}\mathrm{homogeneity}\mathrm{among}\mathrm{migrant}\mathrm{origin};H\approx 0$ means high heterogeneity among migrant origin). This indicator is the result of the summation of the squared proportion of each origin in the foreign population. | “Padrón” (Population Register) |

Scarcity of men for migrant women | Proportion of foreign-national women who do not have an opposite sex counterpart of the same national origin (%) | “Padrón” (Population Register) |

Scarcity of women for migrant men | Proportion of foreign-national men who do not have an opposite sex counterpart of the same national origin (%) | “Padrón” (Population Register) |

Sex ratio among natives | Total number of Spanish men aged 15 to 64 divided by the total number of Spanish women aged 15 to 64 (%). Log transformation | “Padrón” (Population Register) |

Size of Municipality | |||||||||
---|---|---|---|---|---|---|---|---|---|

Variables | 0–249 | 250–499 | 500–999 | 1000–2499 | 2500–4999 | 5000–9999 | 10,000–24,999 | 25,000–99,999 | >100,000 |

Number of municipalities (n) | 2581 | 1214 | 1082 | 1256 | 712 | 545 | 426 | 236 | 59 |

Total marriages | 1584 | 3027 | 6646 | 21,744 | 32,217 | 52,717 | 96,115 | 152,378 | 239,316 |

Relative group size, avg. % | 3.4 | 4.5 | 5.9 | 6.1 | 7.6 | 7.7 | 10.1 | 11.6 | 9.4 |

Sex Ratio among nationals, avg. | 1.47 | 1.27 | 1.20 | 1.13 | 1.09 | 1.06 | 1.04 | 1.02 | 0.97 |

Spanish Men with Foreign-National Women: | |||||||||

Municipalities with intermarriage * | 146 | 221 | 386 | 747 | 584 | 522 | 425 | 236 | 59 |

Intermarriages | 175 | 281 | 530 | 1484 | 1820 | 3044 | 6282 | 10,915 | 18,441 |

Total marriages | 1584 | 3027 | 6646 | 21,744 | 32,217 | 52,717 | 96,115 | 152,378 | 239,316 |

Proportion of intermarriages | 11.0 | 9.3 | 8.0 | 6.8 | 5.6 | 5.8 | 6.5 | 7.2 | 7.7 |

Migrant women | 752 | 2514 | 9385 | 41,144 | 75,976 | 130,710 | 308,917 | 590,004 | 914,725 |

Homogeneity of national origin, avg. % ** | 60.1 | 45.4 | 40.2 | 34.6 | 30.6 | 28.1 | 25.2 | 22.1 | 18.6 |

Scarcity of men for migrant women, avg. % ** | 56.7 | 42.3 | 30.1 | 22.9 | 18.6 | 16.7 | 13.1 | 12.0 | 12.7 |

Spanish Women with Foreign-National Men: | |||||||||

Municipalities with intermarriage * | 35 | 61 | 134 | 361 | 391 | 421 | 413 | 236 | 59 |

Intermarriages | 38 | 67 | 151 | 509 | 793 | 1377 | 3375 | 6625 | 13,375 |

Proportion of intermarriages | 2.4 | 2.2 | 2.3 | 2.3 | 2.5 | 2.6 | 3.5 | 4.3 | 5.6 |

Migrant men | 165 | 804 | 5054 | 27,733 | 69,861 | 140,283 | 359,471 | 672,837 | 982,838 |

Homogeneity of national origin, avg. % ** | 57.2 | 45.7 | 38.9 | 34.1 | 30.6 | 27.1 | 25.1 | 22.1 | 18.6 |

Scarcity of women for migrant men, avg. % ** | 24.3 | 29.5 | 20.2 | 19.0 | 15.0 | 15.1 | 12.7 | 12.0 | 12.7 |

Size of Municipality | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Variables | Total | 0–249 | 250–499 | 500–999 | 1000–2499 | 2500–4999 | 5000–9999 | 10,000–24,999 | 25,000–99,999 | >100,000 |

Spanish Men with Foreign-National Women: | ||||||||||

Relative group size (G) | 0.036 *** | 0.016 | 0.019 | 0.015 * | 0.012 ** | 0.010 ** | 0.028 *** | 0.019 *** | 0.024 *** | 0.095 ** |

Homogeneity of national origin (H) | −0.062 *** | −0.009 | −0.012 ** | −0.011 ** | −0.013 *** | −0.012 *** | −0.010 *** | −0.007 ** | −0.006 * | 0.007 |

Scarcity of men for migrant women (S) | −0.013 *** | 0.010 ** | 0.009 * | 0.003 | −0.002 | −0.005* | 0.003 | −0.002 | 0.012 | 0.011 |

Sex Ratio among natives (X) | −0.167 *** | −0.001 | 0.005 | −0.011 *** | 0.000 | 0.009 | −0.003 | −0.019 ** | −0.047 *** | −0.265 *** |

Spatially Lagged Variables: | ||||||||||

Relative group size | −0.078 | 0.085 | 0.129 | 0.008 | 0.063 * | 0.114 *** | 0.073 *** | 0.024 | −0.004 | −0.024 |

Homogeneity of national origin | −0.049 ** | 0.037 | −0.005 | 0.004 | 0.005 | 0.003 | 0.012 | −0.004 | −0.029 ** | −0.055 |

Scarcity of men for migrant women | −0.070 | 0.056 | 0.060 | −0.032 | 0.013 | 0.037 ** | 0.023 | 0.006 | −0.023 | −0.108 |

Sex Ratio among natives | 0.014 * | 0.000 | 0.002 | 0.001 | 0.008 *** | 0.014 *** | 0.009 *** | 0.002 | 0.005 | 0.023 * |

Constant | 23.703 ** | −4.859 | −3.703 | 1.366 | −1.164 | −3.066 *** | −0.545 | 4.272 *** | 8.824 *** | 31.977 *** |

Pseudo R^{2} | 0.4103 | 0.266 | 0.299 | 0.290 | 0.420 | 0.481 | 0.479 | 0.380 | 0.383 | 0.583 |

SL Pseudo R^{2} | 0.4299 | 0.277 | 0.300 | 0.303 | 0.427 | 0.525 | 0.515 | 0.385 | 0.451 | 0.694 |

Spanish Women with Foreign-National Men: | ||||||||||

Relative group size (G) | 0.033 *** | 0.122 ** | 0.076 *** | −0.015 | −0.002 | 0.020 *** | 0.027 *** | 0.016 *** | 0.023 *** | 0.022 * |

Homogeneity of national origin (H) | −0.048 *** | 0.002 | −0.001 | −0.005 | −0.004 | −0.016 *** | −0.018 *** | −0.014 *** | 0.011 | −0.001 |

Scarcity of women for migrant men (S) | −0.020 *** | 0.031 *** | 0.007 * | −0.012 ** | −0.002 | −0.008 *** | −0.010 *** | −0.008 *** | −0.018 *** | −0.024 ** |

Sex Ratio among natives (X) | −0.258 *** | −0.006 | 0.013 | −0.029 *** | −0.018 ** | −0.016 * | −0.011 | −0.015 | −0.036 ** | −0.246 *** |

Spatially Lagged Variables: | ||||||||||

Relative group size (WG) | 0.073 * | 0.021 | 0.183 ** | −0.031 | 0.076 ** | 0.074 *** | 0.021 | 0.050*** | 0.039 ** | 0.194 *** |

Homogeneity of national origin (WH) | −0.050 | −0.085 | 0.016 | 0.000 | 0.015 | −0.018 | −0.010 | −0.006 | −0.030 * | −0.064 |

Scarcity of women for migrant men (WS) | −0.026 | 0.023 | 0.019 | −0.024 | −0.002 | 0.011 | −0.003 | −0.012 | 0.025 * | 0.023 |

Sex Ratio among natives (WX) | 0.012 | −0.001 | 0.033 ** | 0.002 | 0.010 ** | 0.013 *** | 0.005 | 0.004 | 0.002 | 0.021 |

Constant | 30.986 *** | −1.126 | −10.02 ** | 3.611 * | −0.470 | −2.577 ** | 2.445 * | 3.826 *** | 6.579 *** | 26.838 *** |

Pseudo R^{2} | 0.216 | 0.127 | 0.161 | 0.309 | 0.341 | 0.449 | 0.506 | 0.415 | 0.439 | 0.569 |

SL Pseudo R^{2} | 0.363 | 0.148 | 0.190 | 0.317 | 0.354 | 0.459 | 0.512 | 0.473 | 0.553 | 0.709 |

^{2}: correlation coefficient between the real and estimated dependent variables. SL Pseudo R

^{2}: Pseudo R

^{2}for the model with the spatially lagged variables. Standard errors: Huber/White/sandwich robust variance estimator. Source: Self elaboration based on data from the MNP and “Padrón”.

Size of Municipality | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Variables | Total | 0–249 | 250–499 | 500–999 | 1000–2499 | 2500–4999 | 5000–9999 | 10,000–24,999 | 25,000–99,999 | >100,000 |

Spanish Men with Foreign-National Women: | ||||||||||

Relative group size (G) | 0.019 *** | 0.010 | 0.005 | 0.009 *** | 0.013 *** | 0.012 *** | 0.022 *** | 0.018 *** | 0.017 *** | 0.023 *** |

Homogeneity of national origin (H) | −0.004 *** | −2.3 × 10^{−}^{4} | 0.002 | 0.002 | 0.003 ** | 0.001 | 1.0 × 10^{−}^{4} | 0.001 | −0.005 * | 0.006 * |

Scarcity of men for migrant women (S) | 0.006 *** | 0.002 | −2.3 × 10^{−}^{4} | 0.001 | 0.001 | 0.001 | 0.004 *** | 0.004 ** | 0.006 ** | 0.002 |

Sex Ratio among natives (X) | −0.002 | 0.001 | 3.7 × 10^{−}^{4} | 0.010 *** | 0.009 *** | 0.007 ** | 0.006 * | −0.003 | −0.005 | −0.024 *** |

Spatially Lagged Variables: | ||||||||||

Relative group size (WG) | 0.039 *** | 0.054 | −0.081 | 0.008 | 0.052 *** | 0.072 *** | 0.071 *** | 0.057 *** | 0.044 *** | 0.019 |

Homogeneity of national origin (WH) | −0.008 | 0.051 | 0.013 | 0.031 *** | 0.016 *** | 0.004 | 0.011 ** | 0.003 | −0.005 | −0.018 ** |

Scarcity of men for migrant women (WS) | 0.010 ** | 0.007 | 0.022 | −0005 | 0.029 *** | 0.035 *** | 0.030 *** | 0.028 *** | 0.013 | 0.006 |

Sex Ratio among natives (WX) | 0.004 *** | 1.7 × 10^{−}^{4} | −0.015 *** | −0.008 *** | −0.003 ** | 0.002 ** | 0.002 ** | 0.002 | 0.003 ** | 0.004 *** |

Constant | −2.606 *** | −3.240 | 0.059 | −2.459 *** | −3.905 *** | −4.251 *** | −4.371 *** | −2.796 *** | −2.154 *** | 0.467 |

R^{2} | 0.318 | 0.059 | 0.038 | 0.088 | 0.098 | 0.127 | 0.315 | 0.383 | 0.460 | 0.653 |

SL R^{2} | 0.359 | 0.139 | 0.180 | 0.147 | 0.118 | 0.161 | 0.350 | 0.419 | 0.525 | 0.724 |

Spanish Women with Foreign-National Men: | ||||||||||

Relative group size (G) | 0.015 *** | −0.021 | −0.013 | 0.005 | 0.010 *** | 0.013 *** | 0.017 *** | 0.013 *** | 0.016 *** | 0.029 *** |

Homogeneity of national origin (H) | 0.002 * | 0.011 ** | 0.009 ** | 0.002 | 0.005 *** | 0.001 | −0.001 | −0.002 | 0.002 | 0.004 |

Scarcity of women for migrant men (S) | −0.003 *** | −0.005 | −0.002 | −4.9 × 10^{−}^{4} | 0.001 | 0.001 | −2.4 × 10^{−}^{4} | −0.002 ** | −0.005 *** | −1.4 × 10^{−}^{4} |

Sex Ratio among natives (X) | −0.010 *** | 0.002 | 0.004 | 0.004 | 0.003 | 0.006 ** | −0.006 | −0.007 | −0.006 | −0.033 *** |

Spatially Lagged Variables: | ||||||||||

Relative group size (WG) | 0.025 *** | −0.110 | −0.077 | −0.002 | 0.012 | 0.009 | 0.012 * | 0.031 *** | 0.028 *** | 0.003 |

Homogeneity of national origin (WH) | −0.010 ** | −0.061 | −0.014 | 0.007 | 0.013 * | −0.004 | −0.002 | −0.003 | −0.010 | −0.025 *** |

Scarcity of women for migrant men (WS) | 0.001 | −0.006 | −0.008 | 0.016 | −5.3 × 10^{−}^{6} | 0.001 | −0.005 | −0.007 | 0.007 | −0.008 |

Sex Ratio among natives (WX) | 0.004 *** | −0.010 | −0.012 ** | −0.007 ** | −0.005 *** | −0.002 * | −1.5 × 10^{−}^{4} | 0.004 *** | 0.002 | 0.002 |

Constant | −1.224 *** | −3.674 | 1.292 | −1.691 * | −2.213 *** | −2.258 *** | −1.198 * | −1.503 ** | −1.552 * | 2.147 ** |

Pseudo R^{2} | 0.301 | 0.214 | 0.180 | 0.046 | 0.178 | 0.204 | 0.224 | 0.322 | 0.489 | 0.554 |

SL Pseudo R^{2} | 0.364 | 0.329 | 0.304 | 0.131 | 0.220 | 0.245 | 0.253 | 0.394 | 0.554 | 0.686 |

^{2}: R

^{2}for the model without the spatially lagged variables. SL R

^{2}: R

^{2}for the model with the spatially lagged variables. Standard errors: Huber/White/sandwich robust variance estimator. Source: Self elaboration based on data from the MNP and “Padrón”.

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**MDPI and ACS Style**

Esteve, A.; Chasco, C.; López-Gay, A. Modeling Local Variations in Intermarriage. *Mathematics* **2022**, *10*, 1106.
https://doi.org/10.3390/math10071106

**AMA Style**

Esteve A, Chasco C, López-Gay A. Modeling Local Variations in Intermarriage. *Mathematics*. 2022; 10(7):1106.
https://doi.org/10.3390/math10071106

**Chicago/Turabian Style**

Esteve, Albert, Coro Chasco, and Antonio López-Gay. 2022. "Modeling Local Variations in Intermarriage" *Mathematics* 10, no. 7: 1106.
https://doi.org/10.3390/math10071106