## 3 Results
### Estimation of Spatial Measurement Models  
#### 3.2.1 Ordinary Least Squares Regression and Selection Tests for Spatial Models  
Ordinary Least Squares regression analysis was employed to relate average annual NPP with various climatic factors. The R-squared value (>0.5) indicated a relatively good fit of the models (Table 2). As the use of OLS regression (Table 2) alone to estimate the effects of explanatory variables on NPP may create certain biases, further tests will be conducted applying spatial models.  
According to the above [PERSON]'s I index calculation results, there is an obvious positive spatial spillover effect of the ecological space NPP in Beijing-Tianjin-Hebei, and considering the effect of the interaction term, the SDM model, SLM model, and SEM model are chosen to analyze the effect of different climatic factors on NPP. However, it is necessary to choose the appropriate measurement model according to the test and determination rules, and the test results are shown in Table 3.  
First, the Lagrange Multiplier (LM) and Robust LM tests (Table 3) were combined to determine the form of spatial correlation (whether it exists in the error term, SEM, or the lag term, SLM). Second, the Wald test and Likelihood Ratio (LR) test were used to determine whether the Spatial Durbin Model (SDM) could be simplified into the Spatial Error Model (SEM) and Spatial Lag Model (SLM). If the original hypothesis is rejected at the same time, then SDM is the best-fit model; if the results show the rejection of the original hypothesis of the Wald test and the Robust LM lag value is significant, then the SDM model cannot be optimized as an SLM model; if the results show the rejection of the original hypothesis of the LR test and the Robust LM lag value is significant, then the SDM model cannot be optimized as an SEM model. According to the test results in the table, all the statistics in the LM test were significant, indicating that both the SLM and SEM models were applicable. Subsequently, the Wald test and LR test comparing the SEM model and the SLM model rejected the original hypothesis and passed the \(1\%\) significance test, so the SDM model could not be optimized for the SLM model and the SEM model. In the end, the SDM was used for regression analysis.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **ols\_ind** & **ols\_time** & **ols\_spatiotemporal** & **ols\_random** \\ \hline
**Tem** & \(-0.207\) * & \(-0.020\) & \(-0.436\) *** & \(-0.214\) *** \\
**Pre** & \(-0.097\) & \(1.168\) *** & \(-0.875\) *** & \(1.525\) *** \\
**Evp** & \(-1.338\) *** & \(0.625\) ** & \(1.928\) *** & \(-1.199\) *** \\
**Win** & \(-0.836\) *** & \(-0.832\) *** & \(-1.138\) *** & \(-0.641\) *** \\
**Ssd** & \(2.382\) *** & \(-0.518\) * & \(0.515\) & \(0.869\) *** \\
**Rhu** & \(-0.096\) *** & \(-0.006\) & \(-0.042\) * & \(-0.034\) *** \\
**Gst** & \(0.396\) *** & \(-0.127\) *** & \(0.078\) & \(0.161\) *** \\ \_cons** & \(2.144\) *** & \(1.051\) *** & \(-0.044\) & \(1.424\) *** \\
**R\({}^{2}\)** & \(0.516\) & \(0.521\) & \(0.564\) & \(0.477\) \\
**N** & \(1524.000\) & \(1524.000\) & \(1524.000\) & \(1524.000\) \\ \hline \end{tabular}  
*, **, and ***, respectively, indicate the significance levels at \(10\%\), \(5\%\), and \(1\%\).  
\end{table}
Table 2: OLS regression model test results.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & \_cons** & \(p\) & **Methods** & \_cons** & \(p\) \\ \hline
**LM–spatial lag** & \(97.890\) & \(0.000\) & Wald–spatial lag & \(254.78\) & \(0.000\) \\
**Robust LM–spatial lag** & \(13.702\) & \(0.000\) & LR–spatial lag & \(289.62\) & \(0.000\) \\ \hline
**LM–spatial error** & \(321.628\) & \(0.000\) & Wald–spatial error & \(138.98\) & \(0.000\) \\
**Robust LM–spatial error** & \(237.441\) & \(0.000\) & LR–spatial error & \(380.27\) & \(0.000\) \\ \hline \end{tabular}
\end{table}
Table 3: LM, LR, and Hausman testing.