Table 9 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 2011–2021 as dependent variable; population growth rate (2001–2011), demographic density (2001), population size (2001), 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 | \(\approx 0.000\) | − 0.135 | − 0.052 | 0.041 | 1.398 | ||
(0.011) | (0.001)*** | (0.001)*** | (0.002)*** | (0.149)*** | |||
Pop.growth | 0.108 | 0.111 | 0.121 | 0.130 | 0.027 | 1.21 | |
(0.012)*** | (0.002)*** | (0.002)*** | (0.002)*** | (0.080) | |||
Dem.density | − 0.007 | 0.021 | 0.026 | 0.030 | − 0.069 | 2.29 | |
(0.016) | (0.002)*** | (0.002)*** | (0.002)*** | (0.126) | |||
Pop.size | − 0.051 | 0.050 | 0.020 | − 0.010 | − 0.460 | 2.23 | |
(0.016)** | (0.002)*** | (0.002)*** | (0.003)*** | (0.148)*** | |||
Elevation | − 0.009 | − 0.003 | 0.001 | 0.006 | − 0.035 | 1.48 | |
(0.013) | (0.001)* | (0.001) | (0.002)*** | (0.114) | |||
Sea prox. | 0.006 | − 0.002 | 0.002 | 0.007 | − 0.020 | 1.20 | |
(0.012) | (0.001) | (0.001) | (0.001)*** | (0.090) | |||
Cap.city | 0.023 | − 0.002 | 0.003 | 0.009 | 0.155 | 1.11 | |
(0.011)* | (0.000)** | (0.001) | (0.002)*** | (0.092)* | |||
Lag. Pop.growth | |||||||
Lag.Dem.density | |||||||
Lag.Pop.size | |||||||
Lag.Elevation | |||||||
Lag.Sea prox. | |||||||
Lag.Cap.city | |||||||
Breusch-Pagan | 144.9*** | ||||||
Durbin-Watson | 1.98 | ||||||
Slope equality | 136.5*** | ||||||
Moran’s I(z) | |||||||
W spatial matrix | |||||||
Adjusted-\(R^2\) | 0.011 | 0.226 | 0.198 | 0.129 | 0.083 | ||
AIC | 44,275 | 20,402 | 20,681 | 25,805 | 61,907 | ||
Predictor | Contiguity-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
Quantile regression | |||||||
SAR | SDE | SDM | \(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | |
Intercept | \(\approx 0.000\) | \(\approx 0.000\) | \(\approx 0.000\) | − 0.120 | − 0.046 | 0.039 | 1.394 |
(0.011) | (0.011) | (0.011) | (0.003)*** | (0.002)*** | (0.002)*** | (0.213)*** | |
Pop.growth | 0.107 | 0.107 | 0.096 | 0.108 | 0.117 | 0.123 | 0.025 |
(0.012)*** | (0.012)*** | (0.012)*** | (0.002)*** | (0.002)*** | (0.003)*** | (0.066) | |
Dem.density | − 0.007 | − 0.007 | − 0.009 | 0.018 | 0.022 | 0.025 | − 0.074 |
(0.016) | (0.016) | (0.018) | (0.003)*** | (0.002)*** | (0.003)*** | (0.052) | |
Pop.size | − 0.005 | − 0.050 | − 0.038 | 0.053 | 0.024 | − 0.006 | − 0.451 |
(0.016)*** | (0.016)*** | (0.017)** | (0.002)*** | (0.002)*** | (0.003)* | (0.086)*** | |
Elevation | − 0.009 | − 0.009 | \(\approx 0.000\) | \(\approx 0.000\) | 0.002 | 0.007 | − 0.027 |
(0.013) | (0.013) | (0.015) | (0.001) | (0.001) | (0.002)*** | (0.052) | |
Sea prox. | 0.007 | 0.007 | 0.023 | \(\approx 0.000\) | 0.004 | 0.014 | − 0.019 |
(0.012) | (0.012) | (0.013)* | (0.001) | (0.001)*** | (0.002)*** | (0.030) | |
Cap.city | 0.023 | 0.023 | 0.020 | − 0.001 | 0.003 | 0.007 | 0.153 |
(0.011)** | (0.011)** | (0.011)* | (0.001) | (0.001)*** | (0.001)*** | (0.033)*** | |
Lag. Pop.growth | 0.032 | ||||||
(0.024) | |||||||
Lag.Dem.density | − 0.012 | ||||||
(0.030) | |||||||
Lag.Pop.size | − 0.037 | ||||||
(0.031) | |||||||
Lag.Elevation | − 0.025 | ||||||
(0.024) | |||||||
Lag.Sea prox. | − 0.051 | ||||||
(0.023)** | |||||||
Lag.Cap.city | − 0.005 | ||||||
(0.026) | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.016*** | ||||||
W spatial matrix | *** | *** | *** | ns | |||
Adjusted-\(R^2\) | 0.012 | 0.011 | 0.013 | ||||
AIC | 44,274 | 44,273 | 44,272 | ||||
Predictor | Distance-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | 0.002 | 0.007 | − 0.004 | − 0.126 | − 0.050 | 0.037 | 1.226 |
(0.011) | (0.025) | (0.011) | (0.001)*** | (0.001)*** | (0.002)*** | (0.145)*** | |
Pop.growth | 0.096 | 0.094 | 0.090 | 0.102 | 0.109 | 0.117 | − 0.016 |
(0.012)*** | (0.012)*** | (0.012)*** | (0.003)*** | (0.003)*** | (0.003)*** | (0.044) | |
Dem.density | − 0.014 | − 0.017 | − 0.021 | 0.017 | 0.020 | 0.023 | − 0.070 |
(0.016) | (0.018) | (0.018) | (0.002)*** | (0.002)*** | (0.002)*** | (0.033)** | |
Pop.size | − 0.036 | − 0.034 | − 0.028 | 0.060 | 0.031 | 0.003 | − 0.357 |
(0.016) | (0.017)* | (0.018) | (0.002)*** | (0.002)*** | (0.002) | (0.079)*** | |
Elevation | − 0.001 | − 0.001 | \(\approx 0.000\) | 0.003 | 0.005 | 0.013 | − 0.035 |
(0.013) | (0.014) | (0.015) | (0.001)** | (0.001)*** | (0.002)*** | (0.036) | |
Sea prox. | 0.021 | 0.023 | 0.030 | 0.005 | 0.009 | 0.020 | 0.021 |
(0.012)* | (0.013)* | (0.013)** | (0.001)*** | (0.002)*** | (0.002)*** | (0.020) | |
Cap.city | 0.020 | 0.020 | 0.018 | − 0.004 | \(\approx 0.000\) | 0.006 | 0.089 |
(0.011)* | (0.011)* | (0.011) | (0.001)*** | (0.001) | (0.002)*** | (0.026)*** | |
Lag. Pop.growth | 0.184 | ||||||
(0.103) | |||||||
Lag.Dem.density | − 0.014 | ||||||
(0.090) | |||||||
Lag.Pop.size | 0.039 | ||||||
(0.115) | |||||||
Lag.Elevation | 0.019 | ||||||
(0.057) | |||||||
Lag.Sea prox. | − 0.016 | ||||||
(0.064) | |||||||
Lag.Cap.city | − 0.509 | ||||||
(0.598) | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.007*** | ||||||
W spatial matrix | *** | *** | *** | *** | |||
Adjusted-\(R^2\) | 0.013 | 0.013 | 0.014 | ||||
AIC | 44,261 | 44,265 | 44,266 | ||||
