Table 6 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 1981–1991 as dependent variable; population growth rate (1971–1981), demographic density (1971), population size (1971), 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 | \(\approx 0.000\) | − 0.344 | − 0.043 | 0.280 | 1.819 | ||
(0.009) | (0.007)*** | (0.006)*** | (0.009)*** | (0.120)*** | |||
Pop.growth | 0.528 | 0.459 | 0.521 | 0.604 | 1.092 | 1.27 | |
(0.010)*** | (0.009)*** | (0.009)*** | (0.012)*** | (0.134)*** | |||
Dem.density | 0.084 | 0.041 | 0.065 | 0.092 | 0.281 | 1.94 | |
(0.012)*** | (0.008)*** | (0.007)*** | (0.008)*** | (0.109)** | |||
Pop.size | − 0.078 | 0.047 | − 0.038 | − 0.135 | − 0.566 | 1.87 | |
(0.012)*** | (0.007)*** | (0.007)*** | (0.007)*** | (0.080)*** | |||
Elevation | − 0.027 | − 0.014 | 0.020 | 0.028 | 0.179 | 1.43 | |
(0.011)* | (0.005)*** | (0.005)*** | (0.007)*** | (0.058)*** | |||
Sea prox. | 0.010 | − 0.006 | 0.019 | 0.035 | 0.114 | 1.19 | |
(0.010) | (0.006) | (0.005)*** | (0.008)*** | (0.043)*** | |||
Cap.city | − 0.029 | − 0.051 | − 0.027 | − 0.019 | 0.062 | 1.15 | |
(0.009)** | (0.009)*** | (0.005)*** | (0.003)*** | (0.189) | |||
Lag. Pop.growth | |||||||
Lag.Dem.density | |||||||
Lag.Pop.size | |||||||
Lag.Elevation | |||||||
Lag.Sea prox. | |||||||
Lag.Cap.city | |||||||
Breusch-Pagan | 376.9*** | ||||||
Durbin-Watson | 1.94** | ||||||
Slope equality | 95.1*** | ||||||
Moran’s I(z) | |||||||
W spatial matrix | |||||||
Adjusted-\(R^2\) | 0.309 | 0.260 | 0.254 | 0.256 | 0.295 | ||
AIC | 20,006 | 14,525 | 13,873 | 16,413 | 35,423 | ||
Predictor | Contiguity-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | \(\approx 0.000\) | \(\approx 0.000\) | − 0.325 | − 0.042 | 0.266 | 1.600 |
(0.009) | (0.009) | (0.009) | (0.008)*** | (0.006)*** | (0.010)*** | (0.173)*** | |
Pop.growth | 0.526 | 0.527 | 0.523 | 0.459 | 0.519 | 0.597 | 1.093 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.015)*** | (0.014)*** | (0.019)*** | (0.113)*** | |
Dem.density | 0.078 | 0.082 | 0.064 | 0.030 | 0.062 | 0.084 | 0.231 |
(0.012)*** | (0.013)*** | (0.013)*** | (0.011)** | (0.010)*** | (0.011)*** | (0.121)* | |
Pop.size | − 0.078 | − 0.080 | − 0.080 | 0.049 | − 0.039 | − 0.138 | − 0.537 |
(0.012)*** | (0.012)*** | (0.013)*** | (0.010)*** | (0.008)*** | (0.011)*** | (0.090)*** | |
Elevation | − 0.027 | − 0.030 | − 0.052 | − 0.016 | 0.020 | 0.026 | 0.127 |
(0.011)** | (0.011)** | (0.012)*** | (0.005)*** | (0.006)*** | (0.005)*** | (0.102) | |
Sea prox. | 0.011 | 0.010 | 0.022 | − 0.006 | 0.021 | 0.037 | 0.094 |
(0.010) | (0.010) | (0.010) | (0.005) | (0.005)*** | (0.009)*** | (0.073) | |
Cap.city | − 0.029 | − 0.029 | − 0.027 | − 0.052 | − 0.028 | − 0.018 | 0.038 |
(0.009)*** | (0.009)*** | (0.009)*** | (0.007)*** | (0.005)*** | (0.003)*** | (0.043) | |
Lag. Pop.growth | 0.022 | ||||||
(0.024) | |||||||
Lag.Dem.density | 0.055 | ||||||
(0.024)** | |||||||
Lag.Pop.size | 0.051 | ||||||
(0.024)** | |||||||
Lag.Elevation | 0.090 | ||||||
(0.020)*** | |||||||
Lag.Sea prox. | 0.011 | ||||||
(0.018) | |||||||
Lag.Cap.city | − 0.024 | ||||||
(0.022) | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.057*** | ||||||
W spatial matrix | *** | *** | *** | * | |||
Adjusted-\(R^2\) | 0.311 | 0.310 | 0.313 | ||||
AIC | 19,988 | 19,976 | 19,976 | ||||
Predictor | Distance-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | − 0.004 | − 0.006 | − 0.337 | − 0.043 | 0.278 | 1.859 |
(0.009) | (0.035) | (0.009) | (0.007)*** | (0.006)*** | (0.010)*** | (0.122)*** | |
Pop.growth | 0.528 | 0.530 | 0.526 | 0.459 | 0.522 | 0.604 | 1.052 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.013)*** | (0.013)*** | (0.017)*** | (0.127)*** | |
Dem.density | 0.083 | 0.090 | 0.088 | 0.036 | 0.064 | 0.092 | 0.270 |
(0.012)*** | (0.014)*** | (0.014)*** | (0.010)*** | (0.009)*** | (0.009)*** | (0.101)*** | |
Pop.size | − 0.078 | − 0.103 | − 0.106 | 0.046 | − 0.038 | − 0.137 | − 0.508 |
(0.012)*** | (0.013)*** | (0.014)*** | (0.008)*** | (0.009)*** | (0.011)*** | (0.101)*** | |
Elevation | − 0.026 | − 0.055 | − 0.072 | − 0.012 | 0.019 | 0.028 | 0.198 |
(0.011)** | (0.012)*** | (0.013)*** | (0.005)** | (0.006)*** | (0.008)*** | (0.115)* | |
Sea prox. | 0.012 | 0.001 | − 0.003 | − 0.002 | 0.020 | 0.036 | 0.124 |
(0.010) | (0.011) | (0.011) | (0.006) | (0.005)*** | (0.008)*** | (0.072)* | |
Cap.city | − 0.026 | − 0.025 | − 0.026 | − 0.015 | − 0.027 | − 0.019 | 0.020 |
(0.009)*** | (0.009)** | (0.009)** | (0.007)*** | (0.004)*** | (0.004)*** | (0.053) | |
Lag. Pop.growth | 0.123 | ||||||
(0.150) | |||||||
Lag.Dem.density | − 0.126 | ||||||
(0.050)** | |||||||
Lag.Pop.size | 0.442 | ||||||
(0.87)*** | |||||||
Lag.Elevation | 0.335 | ||||||
(0.051)*** | |||||||
Lag.Sea prox. | − 0.189 | ||||||
(0.045)*** | |||||||
Lag.Cap.city | − 0.839 | ||||||
(0.439)* | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.003*** | ||||||
W spatial matrix | ** | ns | ns | ns | |||
Adjusted-\(R^2\) | 0.309 | 0.313 | 0.3416 | ||||
AIC | 20,008 | 19,926 | 19,928 | ||||
