Table 4 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 1961–1971 as dependent variable; population growth rate (1951–1961), demographic density (1951), population size (1951), 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 | \(\approx 0.000\) | − 0.329 | − 0.051 | 0.235 | 1.966 | ||
(0.008) | (0.006)*** | (0.005)*** | (0.007)*** | (0.162)*** | |||
Pop.growth | 0.547 | 0.499 | 0.567 | 0.646 | 1.363 | 1.13 | |
(0.009)*** | (0.008)*** | (0.008)*** | (0.010)*** | (0.179)*** | |||
Dem.density | 0.188 | 0.146 | 0.157 | 0.132 | 0.244 | 1.56 | |
(0.010)*** | (0.007)*** | (0.007)*** | (0.008)*** | (0.161) | |||
Pop.size | − 0.029 | 0.039 | − 0.011 | − 0.048 | − 0.496 | 1.53 | |
(0.010)*** | (0.006)*** | (0.006)* | (0.007)*** | (0.116)*** | |||
Elevation | − 0.071 | − 0.08 | − 0.073 | − 0.090 | − 0.055 | 1.46 | |
(0.010)*** | (0.004)*** | (0.005)*** | (0.007)*** | (0.109) | |||
Sea prox. | − 0.035 | − 0.033 | − 0.034 | − 0.047 | 0.054 | 1.20 | |
(0.009)*** | (0.005)*** | (0.004)*** | (0.007)*** | (0.178) | |||
Cap.city | − 0.015 | − 0.011 | − 0.011 | − 0.017 | − 0.048 | 1.14 | |
(0.009)* | (0.006)* | (0.003)*** | (0.005)*** | (0.144) | |||
Lag. Pop.growth | |||||||
Lag.Dem.density | |||||||
Lag.Pop.size | |||||||
Lag.Elevation | |||||||
Lag.Sea prox. | |||||||
Lag.Cap.city | |||||||
Breusch-Pagan | 238.2*** | ||||||
Durbin-Watson | 1.83*** | ||||||
Slope equality | 48.1*** | ||||||
Moran’s I(z) | |||||||
W spatial matrix | |||||||
Adjusted-\(R^2\) | 0.403 | 0.327 | 0.361 | 0.379 | 0.353 | ||
AIC | 18,821 | 12,164 | 11,953 | 14,735 | 37,300 | ||
Predictor | Contiguity-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | − 0.001 | − 0.001 | − 0.310 | − 0.049 | 0.230 | 1.895 |
(0.008) | (0.009) | (0.008) | (0.006)*** | (0.005)*** | (0.007)*** | (0.237)*** | |
Pop.growth | 0.539 | 0.543 | 0.534 | 0.492 | 0.563 | 0.641 | 1.322 |
(0.009)*** | (0.009)*** | (0.009)*** | (0.013)*** | (0.011)*** | (0.013)*** | (0.123)*** | |
Dem.density | 0.178 | 0.183 | 0.171 | 0.140 | 0.154 | 0.130 | 0.218 |
(0.010)*** | (0.010)*** | (0.011)*** | (0.008)*** | (0.008)*** | (0.008)*** | (0.128)* | |
Pop.size | − 0.020 | − 0.016 | 0.012 | 0.041 | − 0.008 | − 0.046 | − 0.489 |
(0.010)* | (0.010) | (0.011) | (0.006)*** | (0.006) | (0.007)*** | (0.113)*** | |
Elevation | − 0.067 | − 0.065 | − 0.049 | − 0.074 | − 0.07 | − 0.089 | − 0.115 |
(0.010)*** | (0.010)*** | (0.011)*** | (0.006)*** | (0.006)*** | (0.007)*** | (0.090) | |
Sea prox. | − 0.029 | − 0.023 | 0.002 | − 0.031 | − 0.031 | − 0.045 | 0.072 |
(0.009)*** | (0.009)** | (0.010) | (0.006)*** | (0.006)*** | (0.006)*** | (0.119) | |
Cap.city | − 0.015 | − 0.018 | − 0.021 | − 0.013 | − 0.012 | − 0.014 | − 0.033 |
(0.009)* | (0.009)** | (0.009)** | (0.007)* | (0.003)*** | (0.005)*** | (0.033) | |
Lag. Pop.growth | − 0.021 | ||||||
(0.020) | |||||||
Lag.Dem.density | 0.022 | ||||||
(0.020) | |||||||
Lag.Pop.size | − 0.119 | ||||||
(0.019)*** | |||||||
Lag.Elevation | 0.047 | ||||||
(0.018)** | |||||||
Lag.Sea prox. | − 0.106 | ||||||
(0.017)*** | |||||||
Lag.Cap.city | 0.046 | ||||||
(0.020)** | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.099*** | ||||||
W spatial matrix | *** | *** | * | ns | |||
Adjusted-\(R^2\) | 0.408 | 0.409 | 0.416 | ||||
AIC | 33,100 | 33,015 | 33,011 | ||||
Predictor | Distance-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | 0.003 | 0.044 | 0.002 | − 0.295 | − 0.053 | 0.209 | 1.951 |
(0.008) | (0.108) | (0.008) | (0.006)*** | (0.005)*** | (0.007)*** | (0.214)*** | |
Pop.growth | 0.525 | 0.531 | 0.526 | 0.479 | 0.540 | 0.621 | 1.318 |
(0.009)*** | (0.009)*** | (0.009)*** | (0.014)*** | (0.009)*** | (0.010)*** | (0.155)*** | |
Dem.density | 0.172 | 0.175 | 0.175 | 0.131 | 0.142 | 0.131 | 0.240 |
(0.010)*** | (0.011)*** | (0.011)*** | (0.007)*** | (0.008)*** | (0.009)*** | (0.142)* | |
Pop.size | 0.028 | 0.050 | 0.054 | 0.085 | 0.036 | − 0.010 | − 0.481 |
(0.010)*** | (0.011)*** | (0.012)*** | (0.007)*** | (0.008)*** | (0.008) | (0.127)*** | |
Elevation | − 0.042 | − 0.029 | − 0.036 | − 0.043 | − 0.049 | − 0.006 | − 0.084 |
(0.010)*** | (0.011)** | (0.012)*** | (0.006)*** | (0.007)*** | (0.008) | (0.117) | |
Sea prox. | 0.002 | 0.028 | 0.027 | − 0.006 | − 0.001 | − 0.025 | 0.109 |
(0.009) | (0.010)*** | (0.010)*** | (0.005) | (0.005) | (0.006)*** | (0.123) | |
Cap.city | − 0.025 | − 0.035 | − 0.035 | − 0.018 | − 0.021 | − 0.015 | − 0.038 |
(0.009)*** | (0.009)*** | (0.009)*** | (0.006)*** | (0.003)*** | (0.006) | (0.035) | |
Lag. Pop.growth | − 0.331 | ||||||
(0.065)*** | |||||||
Lag.Dem.density | − 0.137 | ||||||
(0.038)*** | |||||||
Lag.Pop.size | − 0.154 | ||||||
(0.061)** | |||||||
Lag.Elevation | 0.081 | ||||||
(0.035)** | |||||||
Lag.Sea prox. | − 0.115 | ||||||
(0.427)*** | |||||||
Lag.Cap.city | 0.742 | ||||||
(0.414)* | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.039*** | ||||||
W spatial matrix | *** | *** | *** | ns | |||
Adjusted-\(R^2\) | 0.418 | 0.429 | 0.430 | ||||
AIC | 32,948 | 32,741 | 32,796 | ||||
