Table 8 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 2001–2011 as dependent variable; population growth rate (1991–2001), demographic density (1991), population size (1991), 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 | \(\approx 0.000\) | − 0.407 | − 0.077 | 0.315 | 2.093 | ||
(0.008) | (0.008)*** | (0.007)*** | (0.010)*** | (0.102)*** | |||
Pop.growth | 0.614 | 0.576 | 0.635 | 0.669 | 0.862 | 1.17 | |
(0.008)*** | (0.008)*** | (0.008)*** | (0.012)*** | (0.101)*** | |||
Dem.density | 0.111 | 0.066 | 0.102 | 0.135 | 0.412 | 2.20 | |
(0.011)*** | (0.009)*** | (0.009)*** | (0.014)*** | (0.101)*** | |||
Pop.size | − 0.054 | 0.073 | − 0.017 | − 0.114 | − 0.733 | 2.15 | |
(0.011)*** | (0.008)*** | (0.009)* | (0.012)*** | (0.097)*** | |||
Elevation | − 0.125 | − 0.086 | − 0.084 | − 0.128 | − 0.162 | 1.45 | |
(0.009)*** | (0.007)*** | (0.008)*** | (0.012)*** | (0.046)*** | |||
Sea prox. | − 0.046 | − 0.054 | − 0.041 | − 0.041 | 0.030 | 1.20 | |
(0.008)*** | (0.006)*** | (0.007)*** | (0.010)*** | (0.055) | |||
Cap.city | − 0.001 | − 0.010 | − 0.001 | 0.014 | 0.133 | 1.12 | |
(0.008) | (0.004)** | (0.005) | (0.007)* | (0.070)* | |||
Lag. Pop.growth | |||||||
Lag.Dem.density | |||||||
Lag.Pop.size | |||||||
Lag.Elevation | |||||||
Lag.Sea prox. | |||||||
Lag.Cap.city | |||||||
Breusch-Pagan | 2262.1*** | ||||||
Durbin-Watson | 1.88*** | ||||||
Slope equality | 47.9*** | ||||||
Moran’s I(z) | |||||||
W spatial matrix | |||||||
Adjusted-\(R^2\) | 0.475 | 0.335 | 0.326 | 0.308 | 0.290 | ||
AIC | 24,803 | 22,731 | 22,607 | 25,025 | 40,706 | ||
Predictor | Contiguity-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | \(\approx 0.000\) | \(\approx 0.000\) | − 0.371 | − 0.071 | 0.278 | 1.916 |
(0.003) | (0.009) | (0.007) | (0.008)*** | (0.007)*** | (0.011)*** | (0.155)*** | |
Pop.growth | 0.598 | 0.605 | 0.584 | 0.549 | 0.607 | 0.641 | 0.855 |
(0.008)*** | (0.008)*** | (0.008)*** | (0.014)*** | (0.011)*** | (0.016)*** | (0.080)*** | |
Dem.density | 0.101 | 0.109 | 0.094 | 0.060 | 0.093 | 0.116 | 0.346 |
(0.011)*** | (0.012)*** | (0.012)*** | (0.011)*** | (0.010)*** | (0.016)*** | (0.129)*** | |
Pop.size | − 0.042 | − 0.047 | − 0.022 | 0.083 | − 0.003 | − 0.095 | − 0.710 |
(0.011)*** | (0.011)*** | (0.012)* | (0.009)*** | (0.010) | (0.014)*** | (0.100)*** | |
Elevation | − 0.113 | − 0.125 | − 0.118 | − 0.069 | − 0.076 | − 0.112 | − 0.147 |
(0.009)*** | (0.010)*** | (0.011)*** | (0.009)*** | (0.010)*** | (0.011)*** | (0.068)** | |
Sea prox. | − 0.028 | − 0.037 | − 0.009 | − 0.035 | − 0.022 | − 0.016 | 0.085 |
(0.008)*** | (0.008)*** | (0.009) | (0.007)*** | (0.007)*** | (0.009)* | (0.073) | |
Cap.city | − 0.004 | − 0.004 | − 0.008 | − 0.016 | − 0.005 | 0.009 | 0.118 |
(0.008) | (0.008) | (0.008) | (0.006)** | (0.006) | (0.006) | (0.039)*** | |
Lag. Pop.growth | 0.076 | ||||||
(0.021)*** | |||||||
Lag.Dem.density | − 0.003 | ||||||
(0.022) | |||||||
Lag.Pop.size | − 0.058 | ||||||
(0.022)*** | |||||||
Lag.Elevation | 0.023 | ||||||
(0.017) | |||||||
Lag.Sea prox. | − 0.088 | ||||||
(0.016)*** | |||||||
Lag.Cap.city | 0.034 | ||||||
(0.019)* | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.163*** | ||||||
W spatial matrix | *** | *** | *** | ns | |||
Adjusted-\(R^2\) | 0.483 | 0.478 | 0.488 | ||||
AIC | 17,681 | 17,604 | 17,600 | ||||
Predictor | Distance-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
Quantile regression | |||||||
SAR | SDE | SDM | \(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | |
Intercept | − 0.002 | − 0.023 | 0.005 | − 0.393 | − 0.083 | 0.288 | 2.110 |
(0.007) | (0.067) | (0.008) | (0.007)*** | (0.006)*** | (0.010)*** | (0.095)*** | |
Pop.growth | 0.58 | 0.574 | 0.571 | 0.520 | 0.584 | 0.618 | 0.862 |
(0.008)*** | (0.008)*** | (0.008)*** | (0.015)*** | (0.010)*** | (0.012)*** | (0.080)*** | |
Dem.density | 0.102 | 0.105 | 0.110 | 0.060 | 0.089 | 0.141 | 0.390 |
(0.011)*** | (0.013)*** | (0.013)*** | (0.012)*** | (0.011)*** | (0.012)*** | (0.131)** | |
Pop.size | − 0.020 | − 0.020 | − 0.023 | 0.113 | 0.013 | − 0.095 | − 0.718 |
(0.011)*** | (0.012) | (0.013)* | (0.009)*** | (0.010) | (0.012)*** | (0.102)*** | |
Elevation | − 0.087 | − 0.108 | − 0.121 | − 0.040 | − 0.055 | − 0.085 | − 0.176 |
(0.010)*** | (0.010)*** | (0.011)*** | (0.008)*** | (0.010)*** | (0.012)*** | (0.080)** | |
Sea prox. | 0.004 | \(\approx 0.000\) | − 0.001 | − 0.007 | 0.013 | 0.016 | 0.016 |
(0.009) | (0.009) | (0.009) | (0.008) | (0.008) | (0.010) | (0.076) | |
Cap.city | − 0.011 | − 0.014 | − 0.014 | − 0.029 | − 0.009 | 0.006 | 0.128 |
(0.008) | (0.010)*** | (0.008)* | (0.005)*** | (0.008) | (0.007) | (0.041)*** | |
Lag. Pop.growth | 0.529 | ||||||
(0.186)*** | |||||||
Lag.Dem.density | − 0.204 | ||||||
(0.061)*** | |||||||
Lag.Pop.size | 0.006 | ||||||
(0.084) | |||||||
Lag.Elevation | 0.281 | ||||||
(0.042)*** | |||||||
Lag.Sea prox. | − 0.072 | ||||||
(0.050) | |||||||
Lag.Cap.city | 0.716 | ||||||
(0.456) | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.115*** | ||||||
W spatial matrix | *** | *** | *** | ns | |||
Adjusted-\(R^2\) | 0.489 | 0.492 | 0.494 | ||||
AIC | 24,040 | 23,906 | 23,932 | ||||
