Table 3 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 1951–1961 as dependent variable; population growth rate (1936–1951), demographic density (1936), population size (1936), 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 | \(\approx 0.000\) | -0.427 | -0.047 | 0.339 | 2.163 | 1.07 | |
(0.009) | (0.009)*** | (0.007)*** | (0.009)*** | (0.141)*** | |||
Pop.growth | 0.378 | 0.491 | 0.579 | 0.664 | 1.184 | ||
(0.010)*** | (0.015)*** | (0.013)*** | (0.016)*** | (0.194)*** | |||
Dem.density | 0.202 | 0.167 | 0.159 | 0.155 | 0.387 | 1.41 | |
(0.011)*** | (0.011)*** | (0.008)*** | (0.011)*** | (0.120)*** | |||
Pop.size | − 0.035 | \(\approx 0.000\) | − 0.036 | − 0.833 | − 0.312 | 1.47 | |
(0.011)** | (0.010) | (0.008)*** | (0.010)*** | (0.107)*** | |||
Elevation | 0.029 | 0.086 | 0.055 | 0.029 | 0.063 | 1.45 | |
(0.011)* | (0.010)*** | (0.009)*** | (0.010)*** | (0.114) | |||
Sea prox. | 0.101 | 0.067 | 0.062 | 0.075 | 0.198 | 1.20 | |
(0.010)*** | (0.009)*** | (0.007)*** | (0.010)*** | (0.186) | |||
Cap.city | 0.068 | 0.076 | 0.064 | 0.070 | 0.002 | 1.14 | |
(0.010)*** | (0.008)*** | (0.007)*** | (0.005)*** | (0.061) | |||
Lag. Pop.growth | |||||||
Lag.Dem.density | |||||||
Lag.Pop.size | |||||||
Lag.Elevation | |||||||
Lag.Sea prox. | |||||||
Lag.Cap.city | |||||||
Breusch-Pagan | 1760.0*** | ||||||
Durbin-Watson | 1.64*** | ||||||
Slope equality | 24.9*** | ||||||
Moran’s I(z) | |||||||
W spatial matrix | |||||||
Adjusted-\(R^2\) | 0.225 | 0.178 | 0.220 | 0.239 | 0.227 | ||
AIC | 20,928 | 17,225 | 16,649 | 18,979 | 38,6670 | ||
Predictor | Contiguity-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | \(\approx 0.000\) | \(\approx 0.000\) | − 0.410 | − 0.047 | 0.348 | 2.104 |
(0.009) | (0.012) | (0.009) | (0.012)*** | (0.006)*** | (0.012)*** | (0.263)*** | |
Pop.growth | 0.362 | 0.361 | 0.354 | 0.478 | 0.577 | 0.669 | 1.168 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.028)*** | (0.017)*** | (0.016)*** | (0.197)*** | |
Dem.density | 0.192 | 0.196 | 0.180 | 0.163 | 0.157 | 0.159 | 0.400 |
(0.011)*** | (0.011)*** | (0.012)*** | (0.014)*** | (0.010)*** | (0.012)*** | (0.089)*** | |
Pop.size | − 0.024 | − 0.007 | 0.019 | 0.006 | − 0.036 | − 0.085 | − 0.286 |
(0.011)** | (0.012) | (0.012) | (0.011) | (0.009)*** | (0.009)*** | (0.104)*** | |
Elevation | 0.024 | 0.022 | 0.004 | 0.085 | 0.055 | 0.033 | 0.093 |
(0.011)** | (0.012)* | (0.013) | (0.012)*** | (0.010)*** | (0.016)** | (0.155) | |
Sea prox. | 0.098 | 0.107 | 0.111 | 0.065 | 0.062 | 0.076 | 0.227 |
(0.010)*** | (0.010)*** | (0.011)*** | (0.011)*** | (0.007)*** | (0.011)*** | (0.159) | |
Cap.city | 0.068 | 0.062 | 0.058 | 0.075 | 0.065 | 0.069 | − 0.018 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.008)*** | (0.008)*** | (0.007)*** | (0.046) | |
Lag. Pop.growth | 0.080 | ||||||
(0.020)*** | |||||||
Lag.Dem.density | − 0.013 | ||||||
(0.021) | |||||||
Lag.Pop.size | − 0.191 | ||||||
(0.021)*** | |||||||
Lag.Elevation | 0.041 | ||||||
(0.021)* | |||||||
Lag.Sea prox. | − 0.060 | ||||||
(0.019)*** | |||||||
Lag.Cap.city | 0.043 | ||||||
(0.023)* | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.109*** | ||||||
W spatial matrix | ns | ns | ns | ns | |||
Adjusted-\(R^2\) | 0.242 | 0.244 | 0.257 | ||||
AIC | 20,800 | 20,669 | 20,674 | ||||
Predictor | Distance-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | 0.007 | 0.202 | 0.005 | − 0.355 | − 0.498 | 0.304 | 2.319 |
(0.009) | (0.325) | (0.010) | (0.008)*** | (0.008)*** | (0.009)*** | (0.173)*** | |
Pop.growth | 0.334 | 0.346 | 0.337 | 0.424 | 0.514 | 0.618 | 1.247 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.020)*** | (0.018)*** | (0.015)*** | (0.161)*** | |
Dem.density | 0.210 | 0.209 | 0.208 | 0.151 | 0.166 | 0.163 | 0.347 |
(0.011)*** | (0.012)*** | (0.012)*** | (0.012)*** | (0.011)*** | (0.009)*** | (0.086)*** | |
Pop.size | 0.013 | 0.070 | 0.079 | 0.093 | 0.017 | − 0.057 | − 0.345 |
(0.011) | (0.012)*** | (0.013)*** | (0.011)*** | (0.010)* | (0.011)*** | (0.105)*** | |
Elevation | 0.027 | 0.029 | 0.007 | 0.105 | 0.058 | 0.020 | 0.085 |
(0.011)** | (0.013)*** | (0.013) | (0.010)*** | (0.013)*** | (0.018) | (0.164) | |
Sea prox. | 0.097 | 0.130 | 0.126 | 0.081 | 0.062 | 0.073 | 0.209 |
(0.010)*** | (0.011)*** | (0.011)*** | (0.009)*** | (0.008)*** | (0.013) | (0.151) | |
Cap.city | 0.060 | 0.040 | 0.039 | 0.059 | 0.064 | 0.065 | 0.033 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.008)*** | (0.007)*** | (0.007)*** | (0.051) | |
Lag. Pop.growth | − 0.137 (0.057)** | ||||||
Lag.Dem.density | − 0.225 (0.050)*** | ||||||
Lag.Pop.size | − 0.210 (0.077)*** | ||||||
Lag.Elevation | 0.248 (0.039)*** | ||||||
Lag.Sea prox. | (0.041)*** − 0.264 | ||||||
Lag.Cap.city | 0.916 (0.569) | ||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.053*** | ||||||
W spatial matrix | *** | *** | *** | *** | |||
Adjusted-\(R^2\) | 0.237 | 0.284 | 0.292 | ||||
AIC | 20,532 | 20,147 | 20,231 | ||||
