Table 5 Results of standard (OLS, Ordinary Least Square, and quantile) regressions as well as global (SAR: Spatial Autoregressive model; SDE: Spatial Error model; SDM: Spatial Durbin model) and quantile spatial models run with both contiguity and linear distance spatial weighting matrices; population growth rate (% annual) in 1971–1981 as dependent variable; population growth rate (1961–1971), demographic density (1961), population size (1961), elevation, proximity to the sea coast and a dummy indicating municipalities that act as provincial head town as predictors (*\(p< 0.05\); **\(0.001<\) \(p< 0.05\); ***\(p< 0.001\)).
From: Unraveling population trends in Italy (1921–2021) with spatial econometrics
Predictor | OLS | Quantile regression | VIF | ||||
|---|---|---|---|---|---|---|---|
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | − 0.424 | − 0.074 | 0.324 | 2.358 | ||
(0.009) | (0.007)*** | (0.007)*** | (0.010)*** | (0.131)*** | |||
Pop.growth | 0.0452 | 0.502 | 0.610 | 0.727 | 1.361 | 1.24 | |
(0.010)*** | (0.014)*** | (0.014) | (0.018)*** | (0.194)*** | |||
Dem.density | 0.123 | 0.038 | 0.048 | 0.077 | 0.281 | 1.82 | |
(0.012)*** | (0.010)*** | (0.010)*** | (0.013)*** | (0.136)*** | |||
Pop.size | 0.008 | 0.128 | 0.077 | − 0.029 | − 0.735 | 1.65 | |
(0.011) | (0.008)*** | (0.008)*** | (0.010)*** | (0.094)*** | |||
Elevation | − 0.115 | − 0.089 | − 0.069 | − 0.078 | − 0.253 | 1.45 | |
(0.010)*** | (0.007)*** | (0.008)*** | (0.010)*** | (0.148)* | |||
Sea prox. | 0.040 | 0.003 | 0.015 | 0.035 | 0.074 | 1.15 | |
(0.009)*** | (0.007) | (0.007)** | (0.010)*** | (0.113) | |||
Cap.city | − 0.066 | − 0.083 | − 0.082 | − 0.055 | \(\approx 0.000\) | 1.14 | |
(0.009)*** | (0.011)*** | (0.010)*** | (0.008)*** | (0.117) | |||
Lag. Pop.growth | |||||||
Lag.Dem.density | |||||||
Lag.Pop.size | |||||||
Lag.Elevation | |||||||
Lag.Sea prox. | |||||||
Lag.Cap.city | |||||||
Breusch-Pagan | 2703.5*** | ||||||
Durbin-Watson | 1.88*** | ||||||
Slope equality | 52.7*** | ||||||
Moran’s I(z) | |||||||
W spatial matrix | |||||||
Adjusted-\(R^2\) | 0.326 | 0.258 | 0.254 | 0.245 | 0.309 | ||
AIC | 19,799 | 16,600 | 16,412 | 19,228 | 37,343 | ||
Predictor | Contiguity-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | \(\approx 0.000\) | \(\approx 0.000\) | − 0.418 | − 0.074 | 0.319 | 2.146 |
(0.009) | (0.009) | (0.009) | (0.008)*** | (0.007)*** | (0.011)*** | (0.169)*** | |
Pop.growth | 0.045 | 0.451 | 0.451 | 0.502 | 0.605 | 0.727 | 1.366 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.023)*** | (0.024)*** | (0.253)*** | (0.115)*** | |
Dem.density | 0.119 | 0.122 | 0.106 | 0.035 | 0.047 | 0.072 | 0.279 |
(0.012)*** | (0.012)*** | (0.013)*** | (0.012)*** | (0.012)*** | (0.011)*** | (0.118)** | |
Pop.size | 0.008 | 0.005 | \(\approx 0.000\) | 0.129 | 0.076 | − 0.031 | − 0.765 |
(0.011) | (0.011) | (0.012) | (0.007)*** | (0.008)*** | (0.011)* | (0.066)*** | |
Elevation | − 0.114 | − 0.118 | − 0.142 | − 0.088 | − 0.069 | − 0.079 | − 0.192 |
(0.010)*** | (0.011)*** | (0.012)*** | (0.007)*** | (0.008)*** | (0.011)*** | (0.174) | |
Sea prox. | 0.041 | 0.040 | 0.034 | 0.004 | 0.014 | 0.036 | 0.072 |
(0.009)*** | (0.009)*** | (0.010)*** | (0.007) | (0.007)* | (0.011)** | (0.149) | |
Cap.city | − 0.065 | − 0.065 | − 0.063 | − 0.083 | − 0.080 | − 0.054 | 0.012 |
(0.009)*** | (0.009)*** | (0.009)*** | (0.009)*** | (0.009)*** | (0.009)*** | (0.028) | |
Lag. Pop.growth | 0.028 | ||||||
(0.021) | |||||||
Lag.Dem.density | 0.039 | ||||||
(0.023)* | |||||||
Lag.Pop.size | 0.106 | ||||||
(0.022)*** | |||||||
Lag.Elevation | 0.109 | ||||||
(0.019)*** | |||||||
Lag.Sea prox. | − 0.026 | ||||||
(0.018) | |||||||
Lag.Cap.city | − 0.040 | ||||||
(0.022)* | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.069*** | ||||||
W spatial matrix | ns | ns | ns | ns | |||
Adjusted-\(R^2\) | 0.331 | 0.330 | 0.335 | ||||
AIC | 19,784 | 19,763 | 19,761 | ||||
Predictor | Distance-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | 0.016 | 0.002 | − 0.423 | − 0.075 | 0.326 | 2.36023 |
(0.009) | (0.037) | (0.009) | (0.007)*** | (0.007)*** | (0.012)*** | (0.108)*** | |
Pop.growth | 0.452 | 0.461 | 0.455 | 0.501 | 0.609 | 0.734 | 1.361 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.024)*** | (0.023)*** | (0.023)*** | (0.107)*** | |
Dem.density | 0.123 | 0.153 | 0.163 | 0.036 | 0.049 | 0.073 | 0.249 |
(0.012)*** | (0.013)*** | (0.013)*** | (0.012)*** | (0.012)*** | (0.011)*** | (0.098)*** | |
Pop.size | 0.008 | − 0.036 | − 0.046 | 0.128 | 0.075 | − 0.027 | − 0.763 |
(0.012) | (0.013)*** | (0.013)*** | (0.008)*** | (0.008)*** | (0.012)** | (0.071)*** | |
Elevation | − 0.115 | − 0.116 | − 0.126 | − 0.087 | − 0.071 | − 0.085 | − 0.279 |
(0.011)*** | (0.012)*** | (0.012)*** | (0.008)*** | (0.007)*** | (0.011)*** | (0.143)* | |
Sea prox. | 0.040 | 0.040 | 0.038 | 0.006 | 0.013 | 0.031 | 0.066 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.007) | (0.006)** | (0.011)*** | (0.145) | |
Cap.city | − 0.066 | − 0.059 | − 0.060 | − 0.083 | − 0.082 | − 0.055 | − 0.002 |
(0.009)*** | (0.009)*** | (0.009)*** | (0.008)*** | (0.008)*** | (0.009)*** | (0.027) | |
Lag. Pop.growth | 0.453 | ||||||
(0.140)*** | |||||||
Lag.Dem.density | − 0.518 | ||||||
(0.060)*** | |||||||
Lag.Pop.size | 0.743 | ||||||
(0.102)*** | |||||||
Lag.Elevation | 0.227 | ||||||
(0.046)*** | |||||||
Lag.Sea prox. | − 0.244 | ||||||
(0.041)*** | |||||||
Lag.Cap.city | − 1.588 | ||||||
(0.454)*** | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.008*** | ||||||
W spatial matrix | ns | ns | ns | ns | |||
Adjusted-\(R^2\) | 0.329 | 0.335 | 0.340 | ||||
AIC | 19,801 | 19,676 | 19,676 | ||||
