Table 3 Summary of gravity models

(a) Regression table

(Intercept)

1.5829***

1.6187***

1.5723***

1.5766***

1.6713***

1.8964***

1.6534***

2.0972***

2.1456***

\(D_{ij}\)

− 0.4726***

− 0.4725***

− 0.4727***

− 0.4729***

− 0.4735***

− 0.4717***

− 0.4718***

− 0.4711***

− 0.4725***

\(\text{SR}_i\)

− 0.1155***

− 0.1193***

− 0.0897***

− 0.1106***

− 0.1283***

− 0.1316***

− 0.1113***

− 0.1105***

− 0.0969***

\(\text{RB}_i\)

− 0.0717***

− 0.0551***

− 0.0695***

− 0.0603***

− 0.0616***

− 0.0670***

− 0.0910***

− 0.0747***

− 0.0866***

\(\text{RCB}_i\)

0.0086

0.0079

− 0.0035

− 0.0062

− 0.0214***

0.0047

0.0001

− 0.0020

0.0071

\(\text{FA}_i\)

0.0206***

0.0183***

0.0159***

0.0163***

0.0128***

0.0210***

0.0177***

0.0149***

0.0111***

\(\text{CR}_i\)

0.1236***

0.1099***

0.1219***

0.1689***

0.1986***

0.1280***

0.1765***

0.1138***

0.0795***

\(\text{CO}_i\)

0.2193***

0.2172***

0.2215***

0.2202***

0.2209***

0.2255***

0.2287***

0.2281***

0.2299***

\(\text{GDP}_i\)

− 0.0023*

− 0.0019*

− 0.0016

− 0.0013

− 0.0023*

− 0.0390***

− 0.0258***

− 0.0285***

− 0.0223**

\(\text{PI}_i\)

− 0.0042**

− 0.0030*

− 0.0040**

− 0.0044**

− 0.0029*

− 0.0021

− 0.0031*

− 0.0009

0.0026

\(\text{SR}_j\)

− 0.0054

− 0.0099

0.0043

0.0017

0.0158

0.0594***

0.0758***

0.0260

0.0058

\(\text{RB}_j\)

− 0.0057

− 0.0112

− 0.0291***

− 0.0351***

− 0.0459***

− 0.0430***

− 0.0548***

− 0.0473***

− 0.0599***

\(\text{RCB}_j\)

− 0.0152**

− 0.0130**

− 0.0197***

− 0.0213***

− 0.0326***

− 0.0178***

− 0.0160***

− 0.0229***

− 0.0134***

\(\text{FA}_j\)

− 0.0122***

− 0.0120***

− 0.0114***

− 0.0111***

− 0.0123***

− 0.0086***

− 0.0107***

− 0.0159***

− 0.0199***

\(\text{CR}_j\)

0.0464 **

0.0537***

0.0514***

0.0726***

0.0811***

0.0100

− 0.0210

− 0.0661***

− 0.0583***

\(\text{CO}_j\)

0.2218***

0.2205***

0.2216***

0.2223***

0.2244***

0.2304***

0.2326***

0.2327***

0.2344***

\(\text{GDP}_j\)

− 0.0076***

− 0.0080***

− 0.0085***

− 0.0082***

− 0.0086***

− 0.0139**

− 0.0062

− 0.0163**

− 0.0152*

\(\text{PI}_j\)

− 0.0105***

− 0.0096***

− 0.0091***

− 0.0101***

− 0.0095***

− 0.0139***

− 0.0139***

− 0.0091***

− 0.0089***

Adj. \(R^2\)

0.4034***

0.4029***

0.4029***

0.4032***

0.4041***

0.4026***

0.4032***

0.4019***

0.4030***

\(\epsilon ^{\text {grav}}\)

0.0078

0.0073

0.0078

0.0077

0.0076

0.0080

0.0081

0.0080

0.0082

(b) Absolute errors of estimated centralities

\(\epsilon _{C_{D}^+}\)

3.0414

5.7021

3.3355

3.5763

4.2312

3.8885

4.6273

6.5212

4.7929

\(\epsilon _{C_{D}^-}\)

4.5173

3.6933

5.1694

4.9229

4.9568

5.5349

4.1927

6.3334

5.5961

\(\epsilon _{C_{B}}\)

142.7669

183.9388

139.3517

160.2476

161.9900

147.5215

167.2923

185.0560

182.4788

\(\epsilon _{C_{C}^+}\)

2.72E−06

3.25E−06

2.46E−06

2.22E−06

1.75E−06

2.77E−06

2.99E−06

1.91E−06

2.43E−06

\(\epsilon _{C_{C}^-}\)

4.62E−06

1.97E−06

4.71E−06

4.06E−06

3.60E−06

4.59E−06

2.77E−06

2.26E−06

3.77E−06

\(\epsilon _{C_{H}}\)

1.98E−05

1.67E−05

2.06E−05

1.94E−05

1.85E−05

2.13E−05

1.81E−05

1.64E−05

2.06E−05

\(\epsilon _{C_{A}}\)

1.42E−05

1.93E−05

1.28E−05

1.38E−05

1.39E−05

1.47E−05

1.45E−05

1.22E−05

1.53E−05

\(\epsilon _{C_{P}}\)

2.83E−05

3.45E−05

2.62E−05

3.15E−05

3.18E−05

3.25E−05

2.23E−05

2.05E−05

3.00E−05