## 3 Materials and Methods
### Methods
#### 3.3.1 Panel Threshold Model  
The panel threshold model explains the phenomenon that when one parameter reaches a specific value, it causes another parameter to undergo a sudden shift to other forms of development. The critical value that causes the shift is called the threshold value. One or more threshold levels can be generated in the threshold effects model, thus classifying the data into two or more regimes depending on the threshold level [54]. Significant differences exist in the changes in ADR caused by NGCL in different threshold values. The development of the panel threshold model consists of three steps: hypothesis, estimation, and test. Firstly, the null hypothesis that there is no threshold is: Ho: both coefficients are the same, while constructing the LM statistic. According to [PERSON]'s [53], the calculation formula of the model is as follows.  
\[\ln res_{it}=\beta_{0}+\beta_{11}\ln rngc_{it}d(q\leq\lambda_{i})+\beta_{12} \ln rngc_{it}\times d(q>\lambda_{i})+\sum_{k=2}^{4}Z_{kit}\beta_{k}+\varepsilon _{it} \tag{1}\]  
where, \(res_{it}\) and \(rngc_{it}\) are the ADR and NGCL level of the \(i\)th city in the \(t\)th year, respectively. \(q\) and \(\lambda_{i}\) are the threshold variable and threshold value. The relationship between \(q\) and \(\lambda_{i}\) is the variable of elasticity coefficient of NGCL level affecting the change of ADR. \(d()\) is the indicator function, \(\varepsilon\) represents the error disturbance term, \(Z_{kit}\) is a set of control variables, and \(\beta_{1}\) and \(\beta_{k}\) are estimated coefficients of explanatory variables.  
Generally speaking, apart from the external factors of the economic system, the labor force quality and the agricultural operation scale are the most significant variables affecting the allocation of production factors in theory. The effect of policy intervention and the role of market in resource allocation are beneficial tools to improve the mismatch in production factors. Thus, in this study, cultivated land per labor force, NGCL level, proportion of agricultural laborers (refers to the proportion of agricultural employees in the total population), and agricultural machinery per capita were selected as alternative threshold variables. Among them, the cultivated land area per agricultural practitioner reflects the agricultural production level and the actual situation of agricultural production [54]. The proportion of agricultural employees represents the scale of regional agricultural labor force, and it somewhat reflects the level of regional agricultural labor force. Thus, the three variables above represent the quality of agricultural production factors. Furthermore, an alternative threshold variable was selected for analysis to express the threshold characteristics of the direct impact of the NGCL level on ADR.  
To eliminate individual fixed effects in panel data, the software Stata 17 is used, and hypothesized threshold variables are estimated. The variables passing the significance test are further analyzed based on a bootstrap method. The principle of this method is to obtain a self-sampling sample by simulating a sequence of dependent variables that meet the standard distribution, followed by a simulated LM statistic. The simulation process is repeated 1000 times, and the \(p\)-value in the statistical test is obtained when the LM statistic generated by the simulation is greater than the proportion of the given number of times to the total number of simulations, resulting in a meaningful threshold variable [55]. In addition, the exact number of thresholds needs to be confirmed separately. After the first threshold is estimated, the second threshold is then searched for. After two thresholds are determined, a third threshold is found, in the same way as before, until the null hypothesis cannot be rejected. The parameters of each variable in model are also estimated by ordinary least squares (OLS) and so on, combining with robustness tests.