# Revisiting the Spatial Autoregressive Exponential Model for Counts and Other Nonnegative Variables, with Application to the Knowledge Production Function

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The SAR-E Regression

#### 2.1. Model Specification and Partial Effects

**W**is a spatial weighting matrix, and $\mathsf{\beta}$ and $\rho $ are unknown coefficients to be estimated. Observe that, according to Equation (1), the conditional mean of one location $i$, ${\mathsf{\mu}}_{i}$, is determined as a function of the characteristics of location $i$ through the observed values for the explanatory variables and of a weighted average of the conditional mean of neighboring locations.

#### 2.2. Estimation

- Run a PPML regression of $y$ on $X,WX$ and ${W}^{2}X$ and calculate the predicted values $\widehat{y}$.
- Run a PPML regression of $y$ on $W\mathrm{log}(\widehat{y})$ and $X$.

## 3. Simulation Study

#### 3.1. Simulation Design

#### 3.2. Monte Carlo Results

_{1}and X

_{2}. This result is in agreement with [26], who found biased and inconsistent estimators when spatial dependence was not taken into consideration. In addition, it is interesting to note that the distortion of results is more significant for values of $\rho $ near 1, which is in line with the results of [22].

_{1}shows spatial dependence instead of being i.i.d. Therefore, X

_{1}was simulated according to the following spatial autoregressive process:

_{1}was i.i.d. (ignoring that it is spatially autocorrelated). The results obtained for 1000 replications, considering the spatial weighting matrix W1, are included in Table A7 for bias and in Table A8 for RMSE, while Table A9 and Table A10 show, respectively, the bias and RMSE when the spatial weighted matrix is W2. Results show that ignoring spatial autocorrelation in the explanatory variable leads to noticeably higher bias and RMSE in the estimation of all parameters, especially in the estimation of the spatial autocorrelation coefficient. Both estimators show similar performance in estimating the coefficient of X

_{1}, whether the spatial matrix is based on the nearest neighbor criterion (W1) or the inverse distance (W2). The new estimator introduced, SAR-PPML 1stStep-ML, shows better performance than SAR-PPML 1stStep-OLS for the coefficient of X

_{2}when the spatial matrix is W1. The improvement in performance of SAR-PPML 1stStep-ML over the SAR-PPML 1stStep-OLS is especially visible in the estimation of $\rho $ for both spatial weighting matrices.

## 4. Empirical Application

#### 4.1. Data and Variables

#### 4.2. Exploratory Spatial Analysis

#### 4.3. Estimation Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

β1-SAR-Poisson 1stStep-ML—W1 | β1-SAR-Poisson 1stStep-OLS—W1 | β1-Aspatial Poisson ML—W1 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0028 | −0.0011 | −0.0007 | 0.0000 | 0.0000 | −0.0011 | −0.0010 | −0.0006 | −0.0003 | −0.0003 | 0.0003 | −0.0004 | −0.0005 | 0.0000 | 0.0000 |

0.2 | −0.0013 | 0.0000 | −0.0007 | 0.0001 | −0.0004 | −0.0047 | −0.0041 | −0.0044 | 0.0040 | −0.0036 | 0.0311 | 0.0312 | 0.0312 | 0.0310 | 0.0308 |

0.4 | −0.0006 | 0.0000 | −0.0006 | 0.0002 | 0.0002 | −0.0032 | −0.0047 | −0.0046 | −0.0044 | −0.0046 | 0.0891 | 0.0868 | 0.0872 | 0.0872 | 0.0870 |

0.6 | 0.0003 | 0.0007 | 0.0015 | 0.0012 | 0.0015 | −0.0013 | −0.0019 | −0.0020 | −0.0022 | −0.0020 | 0.2021 | 0.1979 | 0.1988 | 0.1969 | 0.1971 |

0.8 | 0.0049 | 0.0046 | 0.0040 | 0.0032 | 0.0043 | 0.0021 | 0.0014 | 0.0017 | 0.0005 | 0.0004 | 0.4957 | 0.4846 | 0.4886 | 0.4771 | 0.4861 |

β2-SAR-Poisson 1stStep-ML—W1 | β2-SAR-Poisson 1stStep-OLS—W1 | β2-Aspatial Poisson ML—W1 | |||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0062 | −0.0028 | 0.0026 | 0.0004 | 0.0003 | −0.0105 | −0.0021 | −0.0010 | −0.0015 | −0.0015 | −0.0064 | −0.0004 | −0.0003 | −0.0004 | −0.0007 |

0.2 | −0.0016 | 0.0012 | 0.0014 | −0.0004 | 0.0003 | −0.0132 | −0.0118 | −0.0156 | −0.0138 | −0.0130 | 0.1018 | 0.1035 | 0.1028 | 0.1027 | 0.1036 |

0.4 | 0.0024 | 0.0009 | 0.0031 | −0.0036 | 0.0043 | −0.0143 | −0.0182 | −0.0188 | −0.0191 | −0.0179 | 0.2737 | 0.2768 | 0.2760 | 0.2789 | 0.2799 |

0.6 | 0.0017 | 0.0023 | 0.0040 | −0.0001 | 0.0027 | −0.0052 | −0.0051 | −0.0097 | −0.0094 | −0.0098 | 0.6201 | 0.6327 | 0.6391 | 0.6363 | 0.6383 |

0.8 | −0.0015 | −0.0022 | −0.0058 | 0.0077 | −0.0078 | 0.0072 | 0.0066 | 0.0065 | 0.0017 | 0.0017 | 1.7530 | 1.7635 | 1.7964 | 1.8182 | 1.8423 |

Rho-SAR-Poisson 1stStep-ML—W1 | Rho-SAR-Poisson 1stStep-OLS—W1 | ||||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | |||||

0.0 | 0.0042 | 0.0015 | 0.0005 | −0.0012 | −0.0008 | 0.0037 | 0.0018 | −0.0003 | 0.0018 | 0.0014 | |||||

0.2 | 0.0012 | −0.0025 | 0.0006 | −0.0003 | 0.0003 | 0.0079 | 0.0105 | 0.0149 | 0.0131 | 0.0119 | |||||

0.4 | −0.0036 | −0.0014 | −0.0012 | −0.0023 | −0.0027 | 0.0169 | 0.0234 | 0.0246 | 0.0247 | 0.0249 | |||||

0.6 | −0.0002 | −0.0004 | −0.0018 | −0.0016 | −0.0010 | 0.0160 | 0.0181 | 0.0204 | 0.0208 | 0.0209 | |||||

0.8 | 0.0047 | 0.0039 | 0.0048 | 0.0050 | 0.0050 | 0.0018 | 0.0034 | 0.0042 | 0.0061 | 0.0082 |

β1-SAR-Poisson 1stStep-ML—W1 | β1-SAR-Poisson 1stStep-OLS—W1 | β1-Aspatial Poisson ML—W1 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.0242 | 0.0154 | 0.0102 | 0.0083 | 0.0069 | 0.0270 | 0.0163 | 0.0117 | 0.0090 | 0.0077 | 0.0171 | 0.0174 | 0.0122 | 0.0099 | 0.0084 |

0.2 | 0.0251 | 0.0152 | 0.0107 | 0.0087 | 0.0071 | 0.0271 | 0.0165 | 0.0119 | 0.0092 | 0.0083 | 0.0424 | 0.0353 | 0.0330 | 0.0321 | 0.0316 |

0.4 | 0.0223 | 0.0137 | 0.0102 | 0.0077 | 0.0070 | 0.0228 | 0.0141 | 0.0106 | 0.0089 | 0.0080 | 0.0946 | 0.0893 | 0.0885 | 0.0880 | 0.0877 |

0.6 | 0.0176 | 0.0106 | 0.0080 | 0.0065 | 0.0061 | 0.0181 | 0.0113 | 0.0079 | 0.0065 | 0.0056 | 0.2122 | 0.2027 | 0.2017 | 0.1989 | 0.1987 |

0.8 | 0.0161 | 0.0134 | 0.0124 | 0.0113 | 0.0106 | 0.0174 | 0.0155 | 0.0123 | 0.0120 | 0.0118 | 0.5380 | 0.5063 | 0.5029 | 0.4887 | 0.4954 |

β2-SAR-Poisson 1stStep-ML—W1 | β2-SAR-Poisson 1stStep-OLS—W1 | β2-Aspatial Poisson ML—W1 | |||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.1034 | 0.0649 | 0.0454 | 0.0357 | 0.0314 | 0.1085 | 0.0648 | 0.0495 | 0.0382 | 0.0330 | 0.0716 | 0.0752 | 0.0564 | 0.0449 | 0.0382 |

0.2 | 0.0912 | 0.0567 | 0.0401 | 0.0333 | 0.0287 | 0.0999 | 0.0631 | 0.0437 | 0.0357 | 0.0316 | 0.1588 | 0.1263 | 0.1132 | 0.1089 | 0.1089 |

0.4 | 0.0811 | 0.0486 | 0.0353 | 0.0300 | 0.0261 | 0.0849 | 0.0531 | 0.0411 | 0.0354 | 0.0309 | 0.2967 | 0.2854 | 0.2800 | 0.2819 | 0.2821 |

0.6 | 0.0606 | 0.0395 | 0.0305 | 0.0277 | 0.0238 | 0.0643 | 0.0403 | 0.0292 | 0.0242 | 0.0215 | 0.6483 | 0.6454 | 0.6465 | 0.6423 | 0.6432 |

0.8 | 0.0602 | 0.0548 | 0.0558 | 0.0502 | 0.0476 | 0.0589 | 0.0487 | 0.0427 | 0.0404 | 0.0428 | 1.9053 | 1.8426 | 1.8515 | 1.8604 | 1.8784 |

Rho-SAR-Poisson 1stStep-ML—W1 | Rho-SAR-Poisson 1stStep-OLS—W1 | ||||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | |||||

0.0 | 0.1112 | 0.0676 | 0.0474 | 0.0383 | 0.0320 | 0.1191 | 0.0734 | 0.0532 | 0.0398 | 0.0357 | |||||

0.2 | 0.0792 | 0.0483 | 0.0330 | 0.0276 | 0.0239 | 0.0873 | 0.0542 | 0.0395 | 0.0318 | 0.0279 | |||||

0.4 | 0.0512 | 0.0289 | 0.0214 | 0.0173 | 0.0149 | 0.0560 | 0.0390 | 0.0332 | 0.0307 | 0.0294 | |||||

0.6 | 0.0237 | 0.0148 | 0.0122 | 0.0110 | 0.0097 | 0.0311 | 0.0244 | 0.0237 | 0.0230 | 0.0224 | |||||

0.8 | 0.0142 | 0.0118 | 0.0106 | 0.0097 | 0.0094 | 0.0152 | 0.0146 | 0.0131 | 0.0137 | 0.0162 |

β1-SAR-Poisson 1stStep-ML—W2 | β1-SAR-Poisson 1stStep-OLS—W2 | β1-Aspatial Poisson ML—W2 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0006 | −0.0010 | 0.0001 | −0.0001 | 0.0003 | −0.0001 | −0.0010 | 0.0001 | −0.0002 | 0.0002 | −0.0008 | −0.0004 | 0.0001 | 0.0000 | 0.0001 |

0.2 | −0.0017 | −0.0002 | −0.0001 | −0.0005 | −0.0004 | −0.0019 | −0.0007 | −0.0004 | −0.0006 | −0.0005 | 0.0290 | 0.0291 | 0.0285 | 0.0285 | 0.0286 |

0.4 | 0.0003 | 0.0002 | −0.0006 | 0.0003 | −0.0002 | −0.0007 | −0.0002 | −0.0008 | 0.0001 | −0.0003 | 0.0769 | 0.0761 | 0.0754 | 0.0750 | 0.0740 |

0.6 | 0.0012 | −0.0005 | −0.0003 | 0.0001 | −0.0003 | 0.0008 | −0.0004 | −0.0002 | 0.0002 | −0.0001 | 0.1711 | 0.1662 | 0.1609 | 0.1609 | 0.1592 |

0.8 | 0.0025 | −0.0015 | −0.0016 | −0.0017 | −0.0015 | 0.0003 | 0.0002 | 0.0000 | 0.0000 | 0.0001 | 0.4010 | 0.3873 | 0.3743 | 0.3630 | 0.3577 |

β2-SAR-Poisson 1stStep-ML—W2 | β2-SAR-Poisson 1stStep-OLS—W2 | β2- Aspatial Poisson ML—W2 | |||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0012 | 0.0010 | −0.0027 | 0.0005 | −0.0004 | −0.0005 | 0.0010 | −0.0029 | −0.0009 | −0.0005 | −0.0785 | −0.0014 | −0.0011 | 0.0004 | −0.0005 |

0.2 | −0.0038 | −0.0021 | −0.0011 | −0.0003 | 0.0001 | −0.0058 | −0.0035 | −0.0019 | −0.0003 | −0.0003 | −0.0520 | 0.0991 | 0.0998 | 0.1022 | 0.1009 |

0.4 | 0.0031 | 0.0020 | −0.0003 | −0.0003 | 0.0008 | −0.0006 | −0.0002 | −0.0012 | −0.0003 | 0.0002 | −0.0219 | 0.2671 | 0.2663 | 0.2682 | 0.2670 |

0.6 | 0.0045 | 0.0018 | 0.0013 | −0.0007 | 0.0005 | −0.0007 | 0.0010 | 0.0015 | 0.0009 | 0.0008 | 0.0143 | 0.5905 | 0.6000 | 0.6034 | 0.6027 |

0.8 | 0.0397 | 0.0172 | 0.0109 | −0.0069 | 0.0081 | 0.0015 | 0.0006 | −0.0001 | 0.0000 | 0.0004 | 0.0364 | 1.6308 | 1.6558 | 1.6638 | 1.6922 |

Rho-SAR-Poisson 1stStep-ML—W2 | Rho-SAR-Poisson 1stStep-OLS—W2 | ||||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | |||||

0.0 | −0.0152 | −0.0020 | 0.0001 | −0.0010 | −0.0017 | −0.0082 | 0.0007 | 0.0018 | 0.0010 | −0.0006 | |||||

0.2 | −0.0033 | 0.0006 | 0.0007 | 0.0009 | 0.0015 | −0.0122 | −0.0108 | −0.0115 | −0.0121 | −0.0119 | |||||

0.4 | −0.0038 | −0.0009 | 0.0024 | 0.0002 | 0.0007 | −0.0064 | −0.0030 | 0.0000 | −0.0026 | −0.0016 | |||||

0.6 | 0.0124 | 0.0072 | 0.0040 | 0.0021 | 0.0026 | 0.0126 | 0.0144 | 0.0152 | 0.0152 | 0.0154 | |||||

0.8 | 0.0501 | 0.0341 | 0.0236 | 0.0166 | 0.0143 | 0.0047 | 0.0044 | 0.0045 | 0.0045 | 0.0043 |

β1-SAR-Poisson 1stStep-ML—W2 | β1-SAR-Poisson 1stStep-OLS—W2 | β1-Aspatial Poisson ML—W2 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.0387 | 0.0232 | 0.0165 | 0.0142 | 0.0114 | 0.0402 | 0.0236 | 0.0167 | 0.0143 | 0.0114 | 0.0276 | 0.0178 | 0.0120 | 0.0095 | 0.0082 |

0.2 | 0.0363 | 0.0207 | 0.0151 | 0.0121 | 0.0110 | 0.0360 | 0.0207 | 0.0151 | 0.0121 | 0.0110 | 0.0386 | 0.0327 | 0.0300 | 0.0296 | 0.0294 |

0.4 | 0.0304 | 0.0187 | 0.0136 | 0.0103 | 0.0088 | 0.0300 | 0.0186 | 0.0136 | 0.0103 | 0.0088 | 0.0820 | 0.0781 | 0.0763 | 0.0757 | 0.0745 |

0.6 | 0.0241 | 0.0134 | 0.0093 | 0.0076 | 0.0066 | 0.0233 | 0.0134 | 0.0093 | 0.0076 | 0.0065 | 0.1784 | 0.1694 | 0.1625 | 0.1621 | 0.1603 |

0.8 | 0.0305 | 0.0156 | 0.0123 | 0.0102 | 0.0092 | 0.0100 | 0.0056 | 0.0038 | 0.0031 | 0.0026 | 0.4256 | 0.4019 | 0.3816 | 0.3692 | 0.3626 |

β2-SAR-Poisson 1stStep-ML—W2 | β2-SAR-Poisson 1stStep-OLS—W2 | β2-Aspatial Poisson ML—W2 | |||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.1487 | 0.0914 | 0.0669 | 0.0541 | 0.0488 | 0.1520 | 0.0924 | 0.0674 | 0.0543 | 0.0489 | 0.1224 | 0.0797 | 0.0551 | 0.0436 | 0.0394 |

0.2 | 0.1368 | 0.0843 | 0.0624 | 0.0495 | 0.0421 | 0.1376 | 0.0845 | 0.0625 | 0.0496 | 0.0422 | 0.1512 | 0.1238 | 0.1109 | 0.1095 | 0.1075 |

0.4 | 0.1182 | 0.0758 | 0.0515 | 0.0412 | 0.0362 | 0.1186 | 0.0761 | 0.0515 | 0.0411 | 0.0361 | 0.2866 | 0.2732 | 0.2714 | 0.2691 | 0.2709 |

0.6 | 0.0902 | 0.0527 | 0.0387 | 0.0302 | 0.0274 | 0.0894 | 0.0529 | 0.0385 | 0.0300 | 0.0273 | 0.6054 | 0.6064 | 0.6061 | 0.6050 | 0.6043 |

0.8 | 0.1310 | 0.0613 | 0.0402 | 0.0270 | 0.0277 | 0.0376 | 0.0216 | 0.0156 | 0.0126 | 0.0105 | 1.6870 | 1.6898 | 1.6781 | 1.7039 | 1.7125 |

Rho-SAR-Poisson 1stStep-ML—W2 | Rho-SAR-Poisson 1stStep-OLS—W2 | ||||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | |||||

0.0 | 0.2196 | 0.1283 | 0.0934 | 0.0803 | 0.0687 | −0.0082 | 0.0007 | 0.0018 | 0.0010 | −0.0006 | |||||

0.2 | 0.1419 | 0.0860 | 0.0636 | 0.0484 | 0.0420 | −0.0122 | −0.0108 | −0.0115 | −0.0121 | −0.0119 | |||||

0.4 | 0.0814 | 0.0513 | 0.0360 | 0.0287 | 0.0245 | −0.0064 | −0.0030 | 0.0000 | −0.0026 | −0.0016 | |||||

0.6 | 0.0571 | 0.0377 | 0.0264 | 0.0187 | 0.0157 | 0.0126 | 0.0144 | 0.0152 | 0.0152 | 0.0154 | |||||

0.8 | 0.0840 | 0.0493 | 0.0290 | 0.0276 | 0.0245 | 0.0047 | 0.0044 | 0.0045 | 0.0045 | 0.0043 |

β1-SAR-Poisson 1stStep-ML—W3 | β1-SAR-Poisson 1stStep-OLS—W3 | β1-Aspatial Poisson ML—W3 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0013 | −0.0009 | −0.0006 | 0.0000 | 0.0000 | −0.0015 | −0.0004 | −0.0002 | −0.0007 | 0.0002 | −0.0008 | −0.0004 | −0.0004 | 0.0000 | 0.0001 |

0.2 | −0.0002 | −0.0003 | 0.0001 | −0.0003 | 0.0002 | −0.0068 | −0.0057 | −0.0049 | −0.0051 | −0.0049 | 0.0334 | 0.0320 | 0.0328 | 0.0327 | 0.0327 |

0.4 | 0.0018 | 0.0031 | 0.0030 | 0.0033 | 0.0031 | −0.0074 | −0.0067 | −0.0061 | −0.0066 | −0.0059 | 0.0945 | 0.0955 | 0.0961 | 0.0950 | 0.0954 |

0.6 | 0.0053 | 0.0055 | 0.0049 | 0.0045 | 0.0076 | −0.0043 | −0.0036 | −0.0040 | −0.0047 | −0.0041 | 0.2162 | 0.2243 | 0.2249 | 0.2249 | 0.2227 |

0.8 | 0.0333 | 0.0256 | 0.0271 | 0.0269 | 0.0258 | 0.0024 | 0.0008 | 0.0008 | −0.0021 | −0.0002 | 0.5565 | 0.5744 | 0.5710 | 0.5781 | 0.5701 |

β2-SAR-Poisson 1stStep-ML—W3 | β2-SAR-Poisson 1stStep-OLS—W3 | β2-Aspatial Poisson ML—W3 | |||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0016 | −0.0020 | 0.0028 | −0.0006 | 0.0001 | −0.0069 | −0.0032 | −0.0038 | −0.0011 | −0.0013 | −0.0014 | −0.0011 | 0.0034 | −0.0005 | 0.0000 |

0.2 | 0.0020 | 0.0014 | 0.0053 | 0.0043 | 0.0041 | −0.0173 | −0.0166 | −0.0191 | −0.0190 | −0.0184 | 0.1017 | 0.1032 | 0.1052 | 0.1047 | 0.1046 |

0.4 | 0.0175 | 0.0206 | 0.0198 | 0.0221 | 0.0212 | −0.0227 | −0.0236 | −0.0241 | −0.0259 | −0.0267 | 0.2822 | 0.2859 | 0.2864 | 0.2869 | 0.2874 |

0.6 | 0.0382 | 0.0308 | 0.0296 | 0.0263 | 0.0248 | −0.0144 | −0.0147 | −0.0169 | −0.0191 | −0.0196 | 0.6469 | 0.6572 | 0.6612 | 0.6640 | 0.6653 |

0.8 | 0.0978 | 0.0919 | 0.0982 | 0.0490 | 0.0311 | 0.0067 | 0.0086 | −0.0031 | −0.0136 | −0.0153 | 1.8195 | 1.9117 | 1.9474 | 1.9597 | 1.9623 |

Rho-SAR-Poisson 1stStep-ML—W3 | Rho-SAR-Poisson 1stStep-OLS—W3 | ||||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | |||||

0.0 | 0.0006 | 0.0000 | 0.0002 | −0.0013 | −0.0002 | 0.0043 | 0.0006 | 0.0018 | 0.0015 | −0.0005 | |||||

0.2 | −0.0008 | 0.0015 | −0.0006 | 0.0007 | 0.0002 | 0.0193 | 0.0199 | 0.0215 | 0.0227 | 0.0219 | |||||

0.4 | −0.0050 | −0.0086 | −0.0081 | −0.0088 | −0.0090 | 0.0303 | 0.0329 | 0.0336 | 0.0358 | 0.0346 | |||||

0.6 | 0.0002 | −0.0041 | −0.0048 | −0.0059 | −0.0114 | 0.0236 | 0.0247 | 0.0275 | 0.0296 | 0.0288 | |||||

0.8 | 0.0118 | 0.0060 | 0.0003 | −0.0034 | −0.0015 | 0.0114 | 0.0112 | 0.0158 | 0.0239 | 0.0231 |

β1-SAR-Poisson 1stStep-ML—W3 | β1-SAR-Poisson 1stStep-OLS—W3 | β1-Aspatial Poisson ML—W3 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.0315 | 0.0198 | 0.0134 | 0.0109 | 0.0090 | 0.0358 | 0.0212 | 0.0148 | 0.0119 | 0.0100 | 0.0276 | 0.0178 | 0.0119 | 0.0095 | 0.0082 |

0.2 | 0.0326 | 0.0198 | 0.0133 | 0.0117 | 0.0099 | 0.0356 | 0.0212 | 0.0147 | 0.0118 | 0.0109 | 0.0427 | 0.0367 | 0.0345 | 0.0339 | 0.0336 |

0.4 | 0.0316 | 0.0197 | 0.0139 | 0.0118 | 0.0107 | 0.0324 | 0.0199 | 0.0144 | 0.0119 | 0.0106 | 0.1015 | 0.0983 | 0.0976 | 0.0959 | 0.0962 |

0.6 | 0.0271 | 0.0174 | 0.0125 | 0.0103 | 0.0134 | 0.0266 | 0.0160 | 0.0119 | 0.0104 | 0.0087 | 0.2288 | 0.2300 | 0.2284 | 0.2275 | 0.2246 |

0.8 | 0.0655 | 0.0502 | 0.0472 | 0.0455 | 0.0439 | 0.0494 | 0.0361 | 0.0327 | 0.0344 | 0.0324 | 0.6065 | 0.6061 | 0.5963 | 0.5972 | 0.5870 |

β2-SAR-Poisson 1stStep-ML—W3 | β2-SAR-Poisson 1stStep-OLS—W3 | β2-Aspatial Poisson ML—W3 | |||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.1356 | 0.0855 | 0.0592 | 0.0474 | 0.0413 | 0.1408 | 0.0865 | 0.0648 | 0.0509 | 0.0433 | 0.1224 | 0.0797 | 0.0540 | 0.0436 | 0.0394 |

0.2 | 0.1226 | 0.0759 | 0.0525 | 0.0437 | 0.0398 | 0.1295 | 0.0801 | 0.0573 | 0.0458 | 0.0423 | 0.1544 | 0.1242 | 0.1146 | 0.1126 | 0.1109 |

0.4 | 0.1143 | 0.0788 | 0.0645 | 0.0619 | 0.0558 | 0.1152 | 0.0718 | 0.0521 | 0.0456 | 0.0425 | 0.3083 | 0.2969 | 0.2915 | 0.2902 | 0.2899 |

0.6 | 0.1184 | 0.0851 | 0.0721 | 0.0582 | 0.0311 | 0.0898 | 0.0571 | 0.0394 | 0.0360 | 0.0341 | 0.6891 | 0.6764 | 0.6717 | 0.6715 | 0.6705 |

0.8 | 0.2134 | 0.1867 | 0.1864 | 0.1999 | 0.1949 | 0.1701 | 0.1297 | 0.1155 | 0.1217 | 0.1164 | 2.0317 | 2.0655 | 2.1007 | 2.0681 | 2.0419 |

Rho-SAR-Poisson 1stStep-ML—W3 | Rho-SAR-Poisson 1stStep-OLS—W3 | ||||||||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 | |||||

0.0 | 0.1272 | 0.0784 | 0.0547 | 0.0436 | 0.0368 | 0.1414 | 0.0885 | 0.0621 | 0.0478 | 0.0441 | |||||

0.2 | 0.0983 | 0.0554 | 0.0380 | 0.0320 | 0.0274 | 0.1116 | 0.0647 | 0.0482 | 0.0400 | 0.0368 | |||||

0.4 | 0.0641 | 0.0377 | 0.0294 | 0.0268 | 0.0244 | 0.0717 | 0.0501 | 0.0419 | 0.0404 | 0.0382 | |||||

0.6 | 0.0440 | 0.0308 | 0.0254 | 0.0203 | 0.0248 | 0.0422 | 0.0324 | 0.0308 | 0.0322 | 0.0307 | |||||

0.8 | 0.0774 | 0.0542 | 0.0431 | 0.0417 | 0.0416 | 0.0429 | 0.036563 | 0.0342 | 0.0400 | 0.0387 |

**Table A7.**Bias: SAR-Poisson. SAR-LogLinear with W1 for 1000 replicates; X

_{1}with spatial dependence.

β1-SAR-Poisson 1stStep-ML—W1 | β1-SAR-Poisson 1stStep-OLS—W1 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0123 | −0.0068 | −0.0035 | −0.0022 | −0.0031 | −0.0077 | −0.0056 | −0.0062 | −0.0018 | −0.0028 |

0.2 | −0.0024 | −0.0021 | 0.0001 | 0.0003 | 0.0008 | 0.0044 | 0.0032 | 0.0056 | 0.0066 | 0.0068 |

0.4 | −0.0024 | −0.0016 | −0.0009 | −0.0004 | 0.0004 | 0.0077 | 0.0086 | 0.0102 | 0.0104 | 0.0113 |

0.6 | −0.0029 | −0.0009 | −0.0033 | −0.0019 | −0.0008 | 0.0086 | 0.0121 | 0.0107 | 0.0126 | 0.0136 |

0.8 | −0.0093 | −0.0164 | −0.0191 | −0.0220 | −0.0226 | 0.0097 | 0.0089 | 0.0100 | 0.0107 | 0.0097 |

β2-SAR-Poisson 1stStep-ML—W1 | β2-SAR-Poisson 1stStep-OLS—W1 | |||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0205 | −0.0086 | −0.0054 | −0.0020 | −0.0030 | −0.0187 | −0.0094 | −0.0031 | −0.0021 | −0.0038 |

0.2 | −0.0123 | −0.0024 | −0.0039 | −0.0028 | −0.0011 | −0.0364 | −0.0302 | −0.0326 | −0.0303 | −0.0289 |

0.4 | −0.0073 | 0.0012 | −0.0009 | 0.0018 | 0.0017 | −0.0583 | −0.0553 | −0.0573 | −0.0554 | −0.0564 |

0.6 | 0.0036 | 0.0033 | −0.0014 | 0.0022 | 0.0021 | −0.0694 | −0.0757 | −0.0789 | −0.0795 | −0.0788 |

0.8 | 0.0013 | 0.0045 | 0.0016 | −0.0018 | 0.0001 | −0.0532 | −0.0631 | −0.0686 | −0.0789 | −0.0778 |

Rho-SAR-Poisson 1stStep-ML—W1 | Rho-SAR-Poisson 1stStep-OLS—W1 | |||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.0264 | 0.0079 | 0.0035 | 0.0021 | 0.0045 | 0.0206 | 0.0124 | 0.0071 | 0.0029 | 0.0063 |

0.2 | −0.0012 | −0.0032 | 0.0034 | 0.0028 | 0.0001 | −0.0056 | 0.0035 | 0.0094 | 0.0066 | 0.0051 |

0.4 | 0.0073 | 0.0052 | 0.0075 | 0.0062 | 0.0050 | 0.0204 | 0.0365 | 0.0419 | 0.0428 | 0.0435 |

0.6 | 0.0173 | 0.0088 | 0.0106 | 0.0081 | 0.0060 | 0.0586 | 0.0807 | 0.0868 | 0.0893 | 0.0895 |

0.8 | 0.0400 | 0.0306 | 0.0323 | 0.0329 | 0.0314 | 0.0555 | 0.0766 | 0.0837 | 0.0926 | 0.0920 |

**Table A8.**RMSE: SAR-Poisson. SAR-LogLinear with W1 for 1000 replicates; X

_{1}with spatial dependence.

β1-SAR-Poisson 1stStep-ML—W1 | β1-SAR-Poisson 1stStep-OLS—W1 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.1142 | 0.0703 | 0.0477 | 0.0399 | 0.0359 | 0.1140 | 0.0702 | 0.0472 | 0.0395 | 0.0353 |

0.2 | 0.1068 | 0.0700 | 0.0477 | 0.0404 | 0.0348 | 0.1043 | 0.0663 | 0.0462 | 0.0391 | 0.0340 |

0.4 | 0.1103 | 0.0666 | 0.0462 | 0.0394 | 0.0325 | 0.1051 | 0.0636 | 0.0996 | 0.0824 | 0.0324 |

0.6 | 0.0950 | 0.0604 | 0.0408 | 0.0326 | 0.0302 | 0.0910 | 0.0586 | 0.0396 | 0.0333 | 0.0320 |

0.8 | 0.0763 | 0.0487 | 0.0408 | 0.0385 | 0.0373 | 0.0707 | 0.0462 | 0.0332 | 0.0291 | 0.0263 |

β2-SAR-Poisson 1stStep-ML—W1 | β2-SAR-Poisson 1stStep-OLS—W1 | |||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.1740 | 0.1077 | 0.0767 | 0.0637 | 0.0536 | 0.2156 | 0.1323 | 0.0948 | 0.0771 | 0.0659 |

0.2 | 0.1635 | 0.1042 | 0.0727 | 0.0617 | 0.0520 | 0.2057 | 0.1274 | 0.0920 | 0.0794 | 0.0681 |

0.4 | 0.1638 | 0.1003 | 0.0720 | 0.0573 | 0.0498 | 0.2003 | 0.1274 | 0.0449 | 0.0385 | 0.0769 |

0.6 | 0.1548 | 0.0956 | 0.0695 | 0.0586 | 0.0476 | 0.1822 | 0.1257 | 0.1042 | 0.0958 | 0.0892 |

0.8 | 0.1548 | 0.1073 | 0.1017 | 0.0969 | 0.0900 | 0.1582 | 0.1117 | 0.0959 | 0.0962 | 0.0900 |

Rho-SAR-Poisson 1stStep-ML—W1 | Rho-SAR-Poisson 1stStep-OLS—W1 | |||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.3733 | 0.2327 | 0.1647 | 0.1259 | 0.1109 | 0.4042 | 0.2554 | 0.1768 | 0.1393 | 0.1236 |

0.2 | 0.2934 | 0.1744 | 0.1175 | 0.1017 | 0.0818 | 0.3358 | 0.1986 | 0.1342 | 0.1173 | 0.0986 |

0.4 | 0.2205 | 0.1202 | 0.0830 | 0.0675 | 0.0575 | 0.2615 | 0.1477 | 0.1102 | 0.0904 | 0.0795 |

0.6 | 0.1368 | 0.0733 | 0.0510 | 0.0423 | 0.0357 | 0.1610 | 0.1142 | 0.0999 | 0.0968 | 0.0948 |

0.8 | 0.0873 | 0.0517 | 0.0451 | 0.0443 | 0.0417 | 0.1109 | 0.0927 | 0.0926 | 0.0998 | 0.0974 |

**Table A9.**Bias: SAR-Poisson. SAR-LogLinear with W2 for 1000 replicates; X

_{1}with spatial dependence.

β1-SAR-Poisson 1stStep-ML—W2 | β1-SAR-Poisson 1stStep-OLS—W2 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0123 | −0.0024 | −0.0026 | −0.0002 | −0.0015 | −0.0064 | −0.0010 | −0.0022 | 0.0016 | −0.0013 |

0.2 | −0.0040 | −0.0032 | 0.0005 | −0.0001 | 0.0007 | −0.0016 | −0.0023 | −0.0043 | 0.0002 | 0.0008 |

0.4 | −0.0042 | 0.0027 | −0.0006 | −0.0008 | 0.0006 | −0.0033 | 0.0034 | −0.0002 | −0.0006 | 0.0008 |

0.6 | −0.0016 | 0.0016 | 0.0002 | 0.0009 | 0.0005 | −0.0014 | 0.0020 | 0.0003 | 0.0009 | 0.0005 |

0.8 | 0.0001 | 0.0049 | 0.0029 | 0.0013 | 0.0005 | −0.0007 | 0.0012 | 0.0013 | −0.0005 | −0.0007 |

β2-SAR-Poisson 1stStep-ML—W2 | β2-SAR-Poisson 1stStep-OLS—W2 | |||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | −0.0205 | −0.0105 | −0.0039 | −0.0002 | −0.0039 | −0.0101 | −0.0040 | −0.0026 | 0.0002 | −0.0034 |

0.2 | −0.0122 | −0.0033 | −0.0042 | 0.0030 | −0.0031 | −0.0088 | −0.0045 | 0.0009 | 0.0025 | −0.0038 |

0.4 | −0.0128 | −0.0010 | −0.0044 | 0.0011 | −0.0026 | −0.0116 | −0.0055 | −0.0058 | −0.0005 | −0.0036 |

0.6 | −0.0010 | −0.0026 | −0.0007 | −0.0009 | 0.0013 | −0.0064 | −0.0063 | −0.0018 | −0.0018 | 0.0004 |

0.8 | −0.0030 | −0.0034 | −0.0002 | −0.0011 | −0.0009 | −0.0024 | −0.0001 | 0.0024 | 0.0014 | 0.0006 |

Rho-SAR-Poisson 1stStep-ML—W2 | Rho-SAR-Poisson 1stStep-OLS—W2 | |||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.0264 | −0.0183 | −0.0201 | −0.0258 | −0.0070 | −0.0024 | −0.0045 | 0.0007 | −0.0084 | 0.0027 |

0.2 | 0.0155 | −0.0226 | −0.0082 | −0.0167 | −0.0048 | −0.0681 | −0.0597 | −0.0580 | −0.0656 | −0.0571 |

0.4 | 0.0100 | −0.0100 | 0.0002 | −0.0058 | 0.0005 | −0.0904 | −0.0728 | −0.0654 | −0.0701 | −0.0653 |

0.6 | 0.0071 | 0.0018 | 0.0012 | 0.0023 | −0.0018 | −0.0497 | −0.0225 | −0.0170 | −0.0142 | −0.0152 |

0.8 | 0.0150 | 0.0136 | 0.0056 | 0.0066 | 0.0044 | 0.0135 | 0.0472 | 0.0570 | 0.0597 | 0.0620 |

**Table A10.**RMSE: SAR-Poisson. SAR-LogLinear with W2 for 1000 replicates; X

_{1}with spatial dependence.

β1-SAR-Poisson 1stStep-ML—W2 | β1-SAR-Poisson 1stStep-OLS—W2 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.1142 | 0.0709 | 0.0503 | 0.0418 | 0.0353 | 0.1117 | 0.0717 | 0.0504 | 0.0421 | 0.0354 |

0.2 | 0.1113 | 0.0656 | 0.0472 | 0.0395 | 0.0337 | 0.1116 | 0.0657 | 0.0473 | 0.0397 | 0.0337 |

0.4 | 0.1072 | 0.0668 | 0.0445 | 0.0393 | 0.0330 | 0.1063 | 0.0666 | 0.0444 | 0.0390 | 0.0330 |

0.6 | 0.0970 | 0.0594 | 0.0407 | 0.0345 | 0.0309 | 0.0964 | 0.0591 | 0.0406 | 0.0343 | 0.0308 |

0.8 | 0.0774 | 0.0457 | 0.0308 | 0.0242 | 0.0214 | 0.0770 | 0.0441 | 0.0306 | 0.0241 | 0.0213 |

β2-SAR-Poisson 1stStep-ML—W2 | β2-SAR-Poisson 1stStep-OLS—W2 | |||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.1740 | 0.1337 | 0.0986 | 0.0834 | 0.0710 | 0.2279 | 0.1466 | 0.1007 | 0.0854 | 0.0723 |

0.2 | 0.2144 | 0.1343 | 0.0949 | 0.0797 | 0.0690 | 0.2264 | 0.1397 | 0.0962 | 0.0811 | 0.0698 |

0.4 | 0.2003 | 0.1293 | 0.0908 | 0.0753 | 0.0660 | 0.2143 | 0.1322 | 0.0915 | 0.0758 | 0.0661 |

0.6 | 0.1901 | 0.1190 | 0.0861 | 0.0670 | 0.0612 | 0.1950 | 0.1185 | 0.0854 | 0.0671 | 0.0612 |

0.8 | 0.1522 | 0.0970 | 0.0624 | 0.0518 | 0.0443 | 0.1542 | 0.0901 | 0.0621 | 0.0518 | 0.0442 |

Rho-SAR-Poisson 1stStep-ML—W2 | Rho-SAR-Poisson 1stStep-OLS—W2 | |||||||||

Rho/n | 100 | 250 | 500 | 750 | 1000 | 100 | 250 | 500 | 750 | 1000 |

0.0 | 0.3733 | 0.4956 | 0.3579 | 0.3020 | 0.2538 | 0.4872 | 0.3062 | 0.2027 | 0.1741 | 0.1505 |

0.2 | 0.3981 | 0.3621 | 0.2393 | 0.2000 | 0.1718 | 0.4180 | 0.2643 | 0.1808 | 0.1573 | 0.1370 |

0.4 | 0.4164 | 0.2356 | 0.1561 | 0.1264 | 0.1106 | 0.3929 | 0.2239 | 0.1568 | 0.1347 | 0.1222 |

0.6 | 0.3435 | 0.1368 | 0.0897 | 0.0667 | 0.0619 | 0.2933 | 0.1541 | 0.1102 | 0.0852 | 0.0786 |

0.8 | 0.1594 | 0.0571 | 0.0338 | 0.0294 | 0.0250 | 0.2220 | 0.0781 | 0.0662 | 0.0643 | 0.0650 |

## Appendix B. Countries in the Sample

Pat | R&D_B | R&D_G | R&D_U | Pers_B | Pers_G | Pers_U | Educ | Pop | GDP | Mort | |
---|---|---|---|---|---|---|---|---|---|---|---|

Pat | 1.000 | ||||||||||

R&D_B | 0.717 | 1.000 | |||||||||

R&D_G | 0.301 | 0.438 | 1.000 | ||||||||

R&D_U | 0.390 | 0.533 | 0.510 | 1.000 | |||||||

Pers_B | 0.474 | 0.601 | 0.332 | 0.211 | 1.000 | ||||||

Pers_G | 0.153 | 0.215 | 0.591 | 0.117 | 0.622 | 1.000 | |||||

Pers_U | 0.164 | 0.286 | 0.329 | 0.261 | 0.746 | 0.674 | 1.000 | ||||

Educ | 0.263 | 0.450 | 0.408 | 0.440 | 0.296 | 0.253 | 0.355 | 1.000 | |||

Pop | 0.056 | 0.082 | 0.119 | −0.088 | 0.663 | 0.651 | 0.775 | 0.010 | 1.000 | ||

GDP | 0.573 | 0.639 | 0.519 | 0.642 | 0.368 | 0.148 | 0.226 | 0.559 | −0.043 | 1.000 | |

Mort | −0.285 | −0.248 | −0.176 | −0.266 | −0.141 | −0.017 | −0.099 | −0.275 | 0.082 | −0.466 | 1.000 |

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Variables | Abbrev. | Unit | Expected Outcome |
---|---|---|---|

The number of patents registered (dependent variable) | Pat | Unit per million inhabitants | − |

Intramural Expenditure on R&D by private business | R&D_B | Euros per inhabitant | + |

Intramural Expenditure on R&D by the government | R&D_G | Euros per inhabitant | Ambiguous |

Intramural Expenditure on R&D by universities | R&D_U | Euros per inhabitant | Ambiguous |

Total R&D personnel and researchers in private business (no. of full-time workers) | Pers_B | - | + |

Total R&D personnel and researchers in the government (no. of full-time workers) | Pers_G | - | Ambiguous |

Total R&D personnel and researchers in universities (no. of full-time workers) | Pers_U | - | Ambiguous |

% Population aged 25–64 with Bachelor’s degree | Educ | Percentage | + |

Population | Pop | Number of inhabitants | − |

GDP per capita | GDP | Thousand euros per capita | + |

Tuberculosis mortality | Mort | Rate per 100,000 inhabitants | − |

Variables | N | Mean | Std Dev | Min | Max |
---|---|---|---|---|---|

Pat | 234 | 89.171 | 106.045 | 0.000 | 590 |

R&D_B | 234 | 318.248 | 382.444 | 0.000 | 2441.700 |

R&D_G | 234 | 59.819 | 87.684 | 0.000 | 480.600 |

R&D_U | 234 | 135.917 | 152.388 | 0.000 | 891.700 |

Pers_B | 234 | 5744.342 | 9291.554 | 0.000 | 97,982.000 |

Pers_G | 234 | 1467.979 | 2628.740 | 0.000 | 17,934.000 |

Pers_U | 234 | 3294.923 | 3558.347 | 0.000 | 34,836.000 |

Educ | 234 | 27.334 | 8.700 | 11.200 | 50.100 |

Pop | 234 | 1,982,780.5 | 1,563,839.6 | 126,620.0 | 11,898,502.0 |

GDP | 234 | 26.922 | 13.874 | 3.561 | 84.047 |

Mort | 234 | 1.009 | 1.375 | 0.100 | 8.800 |

SAR-PPML 1stStep-ML | SAR-PPML 1stStep-OLS | |||
---|---|---|---|---|

Variable | Coefficients | Bootstrap SE | Coefficients | Bootstrap SE |

$\rho $ | 6.81 × 10^{−1} *** | 0.06838 | 6.19 × 10^{−1} *** | 0.07821 |

R&D_B | 8.91 × 10^{−4} *** | 0.00034 | 9.18 × 10^{−4} *** | 0.00035 |

R&D_G | −2.15 × 10^{−3} * | 0.00130 | −2.01 × 10^{−3} | 0.00157 |

R&D_U | −3.21 × 10^{−4} | 0.00079 | −5.85 × 10^{−4} | 0.00083 |

Pers_B | −1.33 × 10^{−5} | 0.00003 | −1.59 × 10^{−5} | 0.00003 |

Pers_G | 2.75 × 10^{−5} | 0.00006 | 3.43 × 10^{−5} | 0.00006 |

Pers_U | 5.07 × 10^{−5} | 0.00005 | 1.86 × 10^{−5} | 0.00005 |

Educ | 2.58 × 10^{−4} | 0.01165 | 7.55 × 10^{−5} | 0.01279 |

Pop | −3.21 × 10^{−9} | 9.62 × 10^{−8} | 6.10 × 10^{−8} | 1.14 × 10^{−7} |

GDP | 3.81 × 10^{−2} *** | 0.01003 | 4.42 × 10^{−2} *** | 0.01127 |

Mort | −1.95 × 10^{−1} ** | 0.09624 | −6.72 × 10^{−2} | 0.09071 |

Log Likelihood | −6557.154 | −7704.048 | ||

N | 234 | 234 |

SAR-PPML 1stStep-ML | SAR-PPML 1stStep-OLS | |||||
---|---|---|---|---|---|---|

Variable | Direct | ASpill-in | ASpill-out | Direct | ASpill-in | ASpill-out |

R&D_B | 0.0934 | 0.1743 | 0.1683 | 0.0878 | 0.1126 | 0.1093 |

R&D_G | −0.2250 | −0.4200 | −0.4054 | −0.1921 | −0.2464 | −0.2391 |

R&D_U | −0.0336 | −0.0628 | −0.0606 | −0.0560 | −0.0718 | −0.0697 |

Pers_B | −0.0014 | −0.0026 | −0.0025 | −0.0015 | −0.0020 | −0.0019 |

Pers_G | 0.0029 | 0.0054 | 0.0052 | 0.0033 | 0.0042 | 0.0041 |

Pers_U | 0.0053 | 0.0099 | 0.0096 | 0.0018 | 0.0023 | 0.0022 |

Educ | 0.0270 | 0.0504 | 0.0487 | 0.7221 | 0.9258 | 0.8984 |

Pop | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

GDP | 3.9933 | 7.4536 | 7.1953 | 4.2288 | 5.4222 | 5.2617 |

Mort | −20.4060 | −38.0886 | −36.7686 | −6.4270 | −8.2409 | −7.9969 |

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## Share and Cite

**MDPI and ACS Style**

Proença, I.; Glórias, L.
Revisiting the Spatial Autoregressive Exponential Model for Counts and Other Nonnegative Variables, with Application to the Knowledge Production Function. *Sustainability* **2021**, *13*, 2843.
https://doi.org/10.3390/su13052843

**AMA Style**

Proença I, Glórias L.
Revisiting the Spatial Autoregressive Exponential Model for Counts and Other Nonnegative Variables, with Application to the Knowledge Production Function. *Sustainability*. 2021; 13(5):2843.
https://doi.org/10.3390/su13052843

**Chicago/Turabian Style**

Proença, Isabel, and Ludgero Glórias.
2021. "Revisiting the Spatial Autoregressive Exponential Model for Counts and Other Nonnegative Variables, with Application to the Knowledge Production Function" *Sustainability* 13, no. 5: 2843.
https://doi.org/10.3390/su13052843