# Testing Taylor’s Law in Urban Population Dynamics Worldwide with Simultaneous Equation Models

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### Literature and Logical Framework

## 2. Methodology

#### 2.1. Empirical Data

#### 2.2. Econometric Model

^{2}.

## 3. Results

^{2}mostly above 0.98. Differences in the adjusted R

^{2}between OLS linear and quadratic specifications are relatively weak. Considering the linear TL specification, slope coefficients were positive in all macro-regions, ranging between 1.098 for North American metropolitan agglomerations and 1.832 for India, and 1.805 for Middle East metropolises. Coefficients included in this range were observed for metropolises in the remaining geographical areas. TL models performed highly (adjusted R

^{2}> 0.98) for metropolises in Africa, China, India, Latin America, the Middle East, and North America. Models for European and Russian metropolitan agglomerations totalized a lower R

^{2}, 0.908 and 0.944, respectively.

^{2}= 0.975) and Russia (adjusted-R

^{2}= 0.976), while impacting positively (but less intensively) the goodness-of-fit of models estimated for the remaining seven macro-regions. Interestingly, models referring to European and Russian metropolises displayed a negative linear coefficient (−37.8 for Europe, −13.1 for Russia) together with a positive quadratic term (6.8 for Europe, 2.7 for Russia). China also showed a similar distribution of coefficients, with coefficient signs fully coherent with Europe and Russia, but with a smaller coefficient intensity. As a matter of fact, the linear coefficient for China was −1.4, and the positive quadratic term was only 0.6. All the other models showed completely different regression signs and coefficient values. Comparable signs and a similar intensity in regression coefficients delineate coherent spatial patterns of Taylor’slLaw in common between Europe, Russia, and China. Based on these data, such macro-regions can be considered as having a millenary history of urban development largely affecting the metropolitan hierarchy reflected in a peculiar, quadratic form of Taylor’s law. Interestingly, looking at the coefficients’ signs and intensity, a gradient from Europe to Russia and China can be delineated, likely indicating a sort of ordering from ‘old demographic’ urban systems with a consolidated, long development to systems with slightly more recent development. Based on such results, China is positioned in-between macro-regions with a long tradition in urban development and other regions with a relatively more recent development path. Positive (but weak) coefficients for both linear and quadratic terms were found for African (linear coefficient: 0.698; quadratic coefficient: 0.200) agglomerations. Positive linear coefficients with negative quadratic coefficients were found for North American, Latin American, Middle Eastern, and Asian/Oceanian agglomerations. Compared with evidence for Europe, Russia, and China, these empirical findings may delineate these macro-regions as affected by a more recent urban development and possibly characterized by demographically younger population structures.

^{2}= 764.8, p < 0.0001) leads to the rejection of the hypothesis of residuals’ homoscedasticity for the nine equations. The values above the main diagonal in Table 5 show that the residuals of the different equations are significantly correlated. These results confirm the initial hypothesis that the variability of population size in a region depends on what happens in the remaining areas, justifying the use of Structural Equation Modelling.

## 4. Discussion

^{2}jointly (Carlucci et al. 2020). Considering the TL assumption (linear specification), regression slopes were always positive and significant, highlighting that an increase in the average population size implies greater heterogeneity between the metropolises of each region in terms of population size (e.g., Cohen et al. 2013b). Taylor et al. (1978) estimated a TL slope for the USA equal to 2.04 ± 0.01. Similar results were reported by Cohen (2013) in the case of Norway. In our case, considering only urban sub-populations, the TL slope was found to be a bit lower. This can be justified with the assumption that we are studying more homogeneous sub-populations (metropolitan agglomerations > 300,000 inhabitants) than the total population by country. All these results are, however, congruent with those achieved in earlier studies confirming the ability of TL to describe multifaceted spatial distributions of human populations (Newman 2005).

^{2}, linear specifications are satisfactory for all regions except for Europe and Russia, since the gain in the adjusted R

^{2}when using quadratic specifications instead of the linear specification is <1% for all regions, >4% for Russia, and >8% for Europe. Moreover, the <1% gain for the seven macro-regions can be compensated by the fact that the quadratic model is less parsimonious than the linear model. However, the results of the quadratic models can be considered for all regions in order to provide additional indications as far as the spatial distribution of urban population is concerned (Gavalas et al. 2014). Quadratic models for European and Russian metropolises had a negative linear coefficient and a positive quadratic coefficient, representing a convex curve, similar to what was observed for Chinese metropolises (Cohen 2014). By contrast, the straight line for the remaining regions showed a much lower slope than that of Europe, evidencing an estimation structure with both positive linear and quadratic coefficients. This indicates that the influence of changing population size on population size variability grows more for Europe than for other regions, as the average levels increase (Egidi et al. 2020).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Scatterplots illustrating the position of world macro-regions based on tipping points for both mean (x) and variance (F(x)) of the quadratic model (see results in Table 2 and Table 4) by econometric specification (plot (

**a**): results from the standard quadratic form; plot (

**b**): results from the 3SLS estimation). Oceania includes other Asian countries except China (source: our elaboration on United Nations Population Division data).

Geographical Area | Parameter | Estimated Coefficient | Standard Error | p-Value |
---|---|---|---|---|

Africa | $log\alpha $ | 1.305 | 0.025 | <0.001 |

β | 1.707 | 0.010 | <0.001 | |

Adjusted R-square | 0.998 | |||

AIC | −293.0 | |||

Other Asia and Oceania | $log\alpha $ | 2.683 | 0.079 | <0.001 |

β | 1.397 | 0.028 | <0.001 | |

Adjusted R-square | 0.975 | |||

AIC | −200.6 | |||

China | $log\alpha $ | 1.951 | 0.058 | <0.001 |

β | 1.495 | 0.023 | <0.001 | |

Adjusted R-square | 0.983 | |||

AIC | −188.7 | |||

Europe | $log\alpha $ | 2.023 | 0.158 | <0.001 |

β | 1.389 | 0.055 | <0.001 | |

Adjusted R-square | 0.908 | |||

AIC | −300.7 | |||

India | $log\alpha $ | 1.167 | 0.012 | <0.001 |

β | 1.832 | 0.004 | <0.001 | |

Adjusted R-square | 0.999 | |||

AIC | −422.5 | |||

Latin America | $log\alpha $ | 1.775 | 0.031 | <0.001 |

β | 1.615 | 0.011 | <0.001 | |

Adjusted R-square | 0.997 | |||

AIC | −305.1 | |||

Middle East | $log\alpha $ | 0.990 | 0.045 | <0.001 |

β | 1.805 | 0.017 | <0.001 | |

Adjusted R-square | 0.994 | |||

AIC | −209.9 | |||

North America | $log\alpha $ | 3.235 | 0.039 | <0.001 |

β | 1.098 | 0.013 | <0.001 | |

Adjusted R-square | 0.990 | |||

AIC | −373.0 | |||

Russia | $log\alpha $ | 1.837 | 0.123 | <0.001 |

β | 1.462 | 0.044 | <0.001 | |

Adjusted R-square | 0.944 | |||

AIC | −217.4 |

Geographical Area | Parameter | Estimated Coefficient | Standard Error | p-Value |
---|---|---|---|---|

Africa | $log\alpha $ | 2.552 | 0.137 | <0.001 |

β | 0.698 | 0.110 | <0.001 | |

$\gamma $ | 0.200 | 0.022 | <0.001 | |

Adjusted R-square | 0.999 | |||

AIC | −346.7 | |||

Other Asia and Oceania | $log\alpha $ | −5.483 | 0.209 | <0.001 |

β | 7.251 | 0.150 | <0.001 | |

$\gamma $ | −1.041 | 0.027 | <0.001 | |

Adjusted R-square | 0.999 | |||

AIC | −411.9 | |||

China | $log\alpha $ | 5.618 | 0.360 | <0.001 |

β | −1.377 | 0.281 | <0.001 | |

$\gamma $ | 0.554 | 0.054 | <0.001 | |

Adjusted R-square | 0.994 | |||

AIC | −251.4 | |||

Europe | $log\alpha $ | 58.078 | 2.146 | <0.001 |

β | −37.775 | 1.485 | <0.001 | |

$\gamma $ | 6.838 | 0.257 | <0.001 | |

Adjusted R-square | 0.975 | |||

AIC | −386.8 | |||

India | $log\alpha $ | 1.950 | 0.106 | <0.001 |

β | 1.246 | 0.079 | <0.001 | |

$\gamma $ | 0.108 | 0.015 | <0.001 | |

Adjusted R-square | 0.999 | |||

AIC | −461.9 | |||

Latin America | $log\alpha $ | −0.806 | 0.179 | <0.001 |

β | 3.514 | 0.132 | <0.001 | |

$\gamma $ | −0.346 | 0.024 | <0.001 | |

Adjusted R-square | 0.999 | |||

AIC | −399.5 | |||

Middle East | $log\alpha $ | −1.618 | 0.163 | <0.001 |

β | 3.936 | 0.133 | <0.001 | |

$\gamma $ | −0.427 | 0.026 | <0.001 | |

Adjusted R-square | 0.999 | |||

AIC | −315.5 | |||

North America | $log\alpha $ | −0.987 | 0.680 | 0.152 |

β | 4.005 | 0.468 | <0.001 | |

$\gamma $ | −0.499 | 0.080 | <0.001 | |

Adjusted R-square | 0.994 | |||

AIC | −402.5 | |||

Russia | $log\alpha $ | 21.689 | 2.104 | <0.001 |

β | −13.131 | 1.545 | <0.001 | |

$\gamma $ | 2.674 | 0.283 | <0.001 | |

Adjusted R-square | 0.976 | |||

AIC | −273.6 |

**Table 3.**Heteroscedasticity White test and Autocorrelation Ljung–Box test for quadratic TL estimated model, by geographical area.

Geographical Area | Heteroscedasticity | Autocorrelation of Order 1 | Autocorrelation of Order 2 | Autocorrelation of Order 3 |
---|---|---|---|---|

Africa | p-value = P(χ^{2}_{(4)} > 13.2) = 0.01 | p-value = P(F_{(1,62)} > 697.1) = 1.98 × 10^{−35} | p-value = P(F_{(2,61)} > 444.6) = 4.27 × 10^{−37} | p-value = P(F_{(3,60)} > 312.1) = 1.50 × 10^{−36} |

Other Asia and Oceania | p-value = P(χ^{2}_{(4)} > 31.7) = 2.16 × 10^{−6} | p-value = P(F_{(1,62)} > 1010.2) = 4.38 × 10^{−40} | p-value = P(F_{(2,61)} > 708.6) = 5.97 × 10^{−43} | p-value = P(F_{(3,60)} > 481.2) = 6.58 × 10^{−42} |

China | p-value = P(χ^{2}_{(4)} > 34.2) = 6.69 × 10^{−7} | p-value = P(F_{(1,62)} > 191.6) = 1.25 × 10^{−20} | p-value = P(F_{(2,61)} > 96.7) = 1.22 × 10^{−19} | p-value = P(F_{(3,60)} > 64.1) = 1.06 × 10^{−18} |

Europe | p-value = P(χ^{2}_{(4)} > 13.4) = 0.009 | p-value = P(F_{(1,62)} > 744.2) = 3.05 × 10^{−36} | p-value = P(F_{(2,61)} > 416.7) = 2.69 × 10^{−36} | p-value = P(F_{(3,60)} > 291.0) = 1.07 × 10^{−35} |

India | p-value = P(χ^{2}_{(4)} > 13.7) = 0.008 | p-value = P(F_{(1,62)} > 539.7) = 2.68 × 10^{−32} | p-value = P(F_{(2,61)} > 289.9) = 7.06 × 10^{−32} | p-value = P(F_{(3,60)} > 207.7) = 1.23 × 10^{−31} |

Latin America | p-value = P(χ^{2}_{(4)} > 13.5) = 0.009 | p-value = P(F_{(1,62)} > 545.0) = 2.05 × 10^{−32} | p-value = P(F_{(2,61)} > 309.1) = 1.20 × 10^{−32} | p-value = P(F_{(3,60)} > 215.1) = 4.63 × 10^{−32} |

Middle East | p-value = P(χ^{2}_{(4)} > 20.4) = 0.0004 | p-value = P(F_{(1,62)} > 1939.9) = 1.70 × 10^{−48} | p-value = P(F_{(2,61)} > 3780.0) = 1.12 × 10^{−64} | p-value = P(F_{(3,60)} > 2482.6) = 7.47 × 10^{−63} |

North America | p-value = P(χ^{2}_{(4)} > 28.1) = 1.21 × 10^{−5} | p-value = P(F_{(1,62)} > 1392.7) = 3.40 × 10^{−44} | p-value = P(F_{(2,61)} > 2436.5) = 6.46 × 10^{−59} | p-value = P(F_{(3,60)} > 1649.8) = 1.40 × 10^{−57} |

Russia | p-value = P(χ^{2}_{(4)} > 24.2) = 7.29 × 10^{−5} | p-value = P(F_{(1,62)} > 1032.6) = 2.31 × 10^{−40} | p-value = P(F_{(2,61)} > 544.1) = 1.29 × 10^{−39} | p-value = P(F_{(3, 60)} > 369.6) = 1.25 × 10^{−38} |

Geographical Area | Parameter | Estimated Coefficient | Standard Error | p-Value |
---|---|---|---|---|

Africa | $log\alpha $ | 2.498 | 0.099 | <0.001 |

β | 0.745 | 0.079 | <0.001 | |

$\gamma $ | 0.190 | 0.016 | <0.001 | |

Adjusted R-square | 0.999 | |||

Other Asia and Oceania | $log\alpha $ | −5.625 | 0.165 | <0.001 |

β | 7.354 | 0.118 | <0.001 | |

$\gamma $ | −1.060 | 0.021 | <0.001 | |

Adjusted R-square | 0.999 | |||

China | $log\alpha $ | 5.475 | 0.255 | <0.001 |

β | −1.248 | 0.198 | <0.001 | |

$\gamma $ | 0.526 | 0.526 | <0.001 | |

Adjusted R-square | 0.997 | |||

Europe | $log\alpha $ | 61.716 | 2.505 | <0.001 |

β | −40.281 | 1.747 | <0.001 | |

$\gamma $ | 7.270 | 0.305 | <0.001 | |

Adjusted R-square | 0.975 | |||

India | $log\alpha $ | 1.876 | 0.071 | <0.001 |

β | 1.303 | 0.053 | <0.001 | |

$\gamma $ | 0.098 | 0.010 | <0.001 | |

Adjusted R-square | 0.999 | |||

Latin America | $log\alpha $ | −0.945 | 0.135 | <0.001 |

β | 3.615 | 0.099 | <0.001 | |

$\gamma $ | −0.364 | 0.018 | <0.001 | |

Adjusted R-square | 0.999 | |||

Middle East | $log\alpha $ | −1.798 | 0.134 | <0.001 |

β | 4.085 | 0.108 | <0.001 | |

$\gamma $ | −0.457 | 0.022 | <0.001 | |

Adjusted R-square | 0.998 | |||

North America | $log\alpha $ | −1.366 | 0.511 | <0.001 |

β | 4.278 | 0.351 | <0.001 | |

$\gamma $ | −0.548 | 0.060 | <0.001 | |

Adjusted R-square | 0.993 | |||

Russia | $log\alpha $ | 25.472 | 1.395 | <0.001 |

β | −15.903 | 1.021 | <0.001 | |

$\gamma $ | 3.182 | 0.187 | <0.001 | |

Adjusted R-square | 0.975 |

Africa | Other Asia and Oceania | China | Europe | India | Latina America | Middle East | North America | Russia | |
---|---|---|---|---|---|---|---|---|---|

Africa | 0.00028 | (0.027) | (0.555) | (0.748) | (0.843) | (−0.606) | (−0.043) | (0.854) | (0.686) |

Other Asia and Oceania | 4.63 × 10^{−6} | 0.00011 | (0.071) | (0.280) | (−0.274) | (0.410) | (0.842) | (0.308) | (0.349) |

China | 0.00032 | 2.56 × 10^{−5} | 0.0012 | (0.861) | (0.466) | (−0.625) | (−0.047) | (0.416) | (0.747) |

Europe | 0.00016 | 3.60 × 10^{−5} | 0.00038 | 0.00016 | (0.682) | (−0.479) | (0.292) | (0.765) | (0.942) |

India | 9.96 × 10^{−5} | −1.98 × 10^{−5} | 0.00011 | 6.02 × 10^{−5} | 4.95 × 10^{−5} | (−0.512) | (−0.031) | (0.778) | (0.673) |

Latin America | −0.00011 | 4.74 × 10^{−5} | −0.00025 | −6.78 × 10^{−5} | −4.06 × 10^{−5} | 0.00013 | (0.627) | (−0.298) | (−0.235) |

Middle East | 1.54 × 10^{−5} | 0.00018 | −3.53 × 10^{−5} | 7.86 × 10^{−5} | −4.63 × 10^{−5} | 0.00015 | 0.00046 | (0.400) | (0.478) |

North America | 0.00016 | 3.51 × 10^{−5} | 0.00016 | 0.00011 | 6.07 × 10^{−5} | −3.73 × 10^{−5} | 9.52 × 10^{−5} | 0.00012 | (0.778) |

Russia | 0.00034 | 0.00011 | 0.00078 | 0.00035 | 0.00014 | −7.91 × 10^{−5} | 0.00031 | 0.00026 | 0.00089 |

Geographical Area | $\tilde{\mathit{\beta}}$ |
---|---|

Africa | 1.26 |

Other Asia and Oceania | 4.22 |

China | 0.17 |

Europe | −19.14 |

India | 1.58 |

Latin America | 2.56 |

Middle East | 2.87 |

North America | 2.64 |

Russia | −6.92 |

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## Share and Cite

**MDPI and ACS Style**

Benassi, F.; Naccarato, A.; Salvati, L.
Testing Taylor’s Law in Urban Population Dynamics Worldwide with Simultaneous Equation Models. *Economies* **2023**, *11*, 56.
https://doi.org/10.3390/economies11020056

**AMA Style**

Benassi F, Naccarato A, Salvati L.
Testing Taylor’s Law in Urban Population Dynamics Worldwide with Simultaneous Equation Models. *Economies*. 2023; 11(2):56.
https://doi.org/10.3390/economies11020056

**Chicago/Turabian Style**

Benassi, Federico, Alessia Naccarato, and Luca Salvati.
2023. "Testing Taylor’s Law in Urban Population Dynamics Worldwide with Simultaneous Equation Models" *Economies* 11, no. 2: 56.
https://doi.org/10.3390/economies11020056