Table 1 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 1931–1936 as dependent variable; population growth rate (1921–1931), demographic density (1921), population size (1921), 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 | \(\approx 0.000\) | -0.192 | 0.005 | 0.189 | 1.485 | ||
(0.010) | (0.005)*** | (0.003) | (0.004)*** | (0.836)* | |||
Pop.growth | 0.079 | 0.350 | 0.401 | 0.388 | 0.347 | 1.02 | |
(0.011)*** | (0.031)*** | (0.031)*** | (0.036)*** | (0.392) | |||
Dem.density | − 0.129 | − 0.011 | − 0.028 | − 0.039 | − 0.120 | 1.32 | |
(0.012)*** | (0.003)*** | (0.004)*** | (0.004)*** | (0.055)** | |||
Pop.size | − 0.004 | 0.046 | 0.033 | 0.015 | − 0.107 | 1.38 | |
(0.012) | (0.004)*** | (0.004)*** | (0.005) | (0.048)** | |||
Elevation | − 0.142 | − 0.045 | − 0.016 | − 0.005 | − 0.042 | 1.40 | |
(0.012)*** | (0.003)*** | (0.004)*** | (0.005) | (0.075) | |||
Sea prox. | 0.045 | 0.028 | 0.036 | 0.043 | 0.106 | 1.18 | |
(0.011)*** | (0.003)*** | (0.004)*** | (0.004)*** | (0.115) | |||
Cap.city | 0.091 | − 0.001 | 0.006 | 0.021 | 5.387 | 1.13 | |
(0.011)*** | (0.003) | (0.004) | (0.004)*** | (0.654) | |||
Lag. Pop.growth | |||||||
Lag.Dem.density | |||||||
Lag.Pop.size | |||||||
Lag.Elevation | |||||||
Lag.Sea prox. | |||||||
Lag.Cap.city | |||||||
Breusch-Pagan | 5204.2*** | ||||||
Durbin-Watson | 1.89** | ||||||
Slope equality | 10.8*** | ||||||
Moran’s I(z) | |||||||
W spatial matrix | |||||||
Adjusted-\(R^2\) | 0.047 | 0.098 | 0.104 | 0.100 | 0.088 | ||
AIC | 22,604 | 8,684 | 6,722 | 8,425 | 33,576 | ||
Predictor | Contiguity-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | \(\approx 0.000\) | \(\approx 0.000\) | − 0.176 | 0.006 | 0.152 | 4.710 |
(0.010) | (0.011) | (0.010) | (0.015)*** | (0.004) | (0.007)*** | (0.657)*** | |
Pop.growth | 0.078 | 0.077 | 0.075 | 0.344 | 0.380 | 0.350 | 0.178 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.079)*** | (0.043)*** | (0.028)*** | (0.227) | |
Dem.density | − 0.127 | − 0.130 | − 0.132 | − 0.010 | − 0.020 | − 0.032 | − 0.048 |
(0.012)*** | (0.012)*** | (0.013)*** | (0.006) | (0.005)*** | (0.006)*** | (0.519) | |
Pop.size | − 0.007 | − 0.008 | − 0.020 | 0.042 | 0.027 | \(\approx 0.000\) | − 1.666 |
(0.012) | (0.012) | (0.013) | (0.006)*** | (0.006)*** | (0.006) | (0.528)** | |
Elevation | − 0.143 | − 0.151 | − 0.200 | − 0.046 | − 0.019 | − 0.015 | − 0.691 |
(0.012)*** | (0.013)*** | (0.014)*** | (0.006)*** | (0.005)*** | (0.004)*** | (0.770) | |
Sea prox. | 0.042 | 0.038 | 0.002 | 0.026 | 0.028 | 0.033 | 0.121 |
(0.011)*** | (0.011)*** | (0.012) | (0.006)*** | (0.006)*** | (0.005)*** | (1.067) | |
Cap.city | 0.092 | 0.093 | 0.094 | 0.000 | 0.008 | 0.024 | 0.648 |
(0.011)*** | (0.011)*** | (0.011)*** | (0.004) | (0.004)** | (0.006)*** | (0.246) | |
Lag. Pop.growth | 0.012 | ||||||
(0.010) | |||||||
Lag.Dem.density | 0.039 | ||||||
(0.023)* | |||||||
Lag.Pop.size | 0.080 | ||||||
(0.023)*** | |||||||
Lag.Elevation | 0.184 | ||||||
(0.023)*** | |||||||
Lag.Sea prox. | 0.124 | ||||||
(0.021)*** | |||||||
Lag.Cap.city | − 0.055 | ||||||
(0.026)** | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.0304*** | ||||||
W spatial matrix | ns | * | *** | ns | |||
Adjusted-\(R^2\) | 0.050 | 0.051 | 0.060 | ||||
AIC | 22,592 | 22,533 | 22,533 | ||||
Predictor | Distance-based spatial weights | ||||||
|---|---|---|---|---|---|---|---|
SAR | SDE | SDM | Quantile regression | ||||
\(\tau = 0.25\) | \(\tau = 0.50\) | \(\tau = 0.75\) | \(\tau = 0.99\) | ||||
Intercept | \(\approx 0.000\) | − 0.039 | − 0.008 | − 0.133 | 0.006 | 0.152 | 1.417 |
(0.010) | (0.107) | (0.011) | (0.004)*** | (0.003)* | (0.004)*** | (0.288)*** | |
Pop.growth | 0.066 | 0.065 | 0.067 | 0.216 | 0.288 | 0.319 | 0.366 |
(0.010)*** | (0.010)*** | (0.010)*** | (0.080)*** | (0.050)*** | (0.024)*** | (0.102)*** | |
Dem.density | − 0.103 | − 0.136 | − 0.136 | 0.016 | 0.007 | − 0.015 | − 0.112 |
(0.012)*** | (0.013)*** | (0.013)*** | (0.005)*** | (0.004) | (0.005)*** | (0.041)*** | |
Pop.size | − 0.043 | − 0.042 | − 0.040 | 0.012 | − 0.011 | − 0.029 | − 0.097 |
(0.012)*** | (0.014)*** | (0.014)*** | (0.005) | (0.005)*** | (0.006)*** | (0.040)** | |
Elevation | − 0.161 | − 0.213 | − 0.221 | − 0.064 | − 0.042 | − 0.041 | − 0.046 |
(0.013)*** | (0.014)*** | (0.014)*** | (0.006)*** | (0.005)*** | (0.005)*** | (0.065) | |
Sea prox. | 0.015 | − 0.003 | − 0.004 | − 0.005 | 0.012 | 0.019 | 0.105 |
(0.011) | (0.012) | (0.012) | (0.004)*** | (0.003)*** | (0.005)*** | (0.103) | |
Cap.city | 0.099 | 0.101 | 0.100 | 0.007 | 0.016 | 0.03 | 5.385 |
(0.011)*** | (0.011)*** | (0.011)*** | (0.004)** | (0.003)*** | (0.004)*** | (2.654)** | |
Lag. Pop.growth | 0.765 | ||||||
(0.196)*** | |||||||
Lag.Dem.density | 0.236 | ||||||
(0.062)*** | |||||||
Lag.Pop.size | 0.122 | ||||||
(0.087) | |||||||
Lag.Elevation | 0.433 | ||||||
(0.052)*** | |||||||
Lag.Sea prox. | − 0.055 | ||||||
(0.044) | |||||||
Lag.Cap.city | 0.040 | ||||||
(0.588) | |||||||
Breusch-Pagan | |||||||
Durbin-Watson | |||||||
Slope equality | |||||||
Moran’s I(z) | 0.0282*** | ||||||
W spatial matrix | *** | *** | ** | * | |||
Adjusted-\(R^2\) | 0.070 | 0.077 | 0.078 | ||||
AIC | 22,422 | 22,360 | 22,352 | ||||
