## 3 Variables Description and Research Method
### Research Method  
The use of agricultural film varies across regions and times, and artificial grouping would introduce bias. The threshold value of agricultural film usage was determined by automatic identification of the data, and then the nonlinear relationship between agricultural film and agricultural development was analyzed. This paper, which drew on the fixed-effect panel threshold model of [PERSON] [61], determined the threshold value of agricultural film by automatically identifying the data and then analyzing the impact of agricultural film usage, pesticide usage and fertilizer input on agricultural development. To avoid missing important explanatory variables, control variables such as land input, water input, agricultural machinery total power and agricultural financial input were added during the research process. The single threshold model was  
\[ \begin{cases}y_{it}=\alpha_{i}+\beta^{{}^{\prime}}_{1}\mathbf{x_{it}}+u_{it}, \ \ q_{it}\leq\gamma\\ y_{it}=\alpha_{i}+\beta^{{}^{\prime}}_{2}\mathbf{x_{it}}+u_{it},\ \ q_{it}>\gamma \end{cases}  \tag{1}\]  
where \(i\) represented the region (\(1\leq i\leq n\)), \(t\) represented the time (\(1\leq t\leq T\)), \(y_{it}\) was the explained variable, \(\ \mathbf{x_{it}}\) was the explanatory variable, \(\beta_{1}\ \&\ \beta_{2}\) were the parameters to be estimated, \(q_{it}\) was the threshold variable, \(\gamma\) was the threshold value to be estimated, \(\alpha_{it}\) was the individual effect and \(u_{it}\) was the random error term. Using the indicative function \(1(\cdot)\), Formula (1) could be more concisely expressed as  
\[y_{it}=\alpha_{i}+\beta^{{}^{\prime}}_{1}\mathbf{x_{it}}\cdot 1(q_{it}\leq\gamma )+\beta^{{}^{\prime}}_{2}\mathbf{x_{it}}\cdot 1(q_{it}>\gamma)+u_{it} \tag{2}\]  
Two tests were needed for the threshold variable \(q_{it}\). The first was to test for the presence of a threshold effect. The second was to test whether or not the threshold estimate is equal to its true value. To test for the presence of a threshold effect, the null hypothesis was \(H_{0}:\mathbf{\beta_{1}}=\mathbf{\beta_{2}}\). The unconstrained sum of squared residuals was \(SSR(\hat{\gamma})\). The sum of squared residuals using \(H_{0}:\mathbf{\beta_{1}}=\mathbf{\beta_{2}}\) as the constraint is \(SSR^{*}\). The larger \(SSR^{*}-SSR(\hat{\gamma})\) was, the more inclined it was to reject \(H_{0}:\mathbf{\beta_{1}}=\mathbf{\beta_{2}}\). This indicates that there was a threshold effect. According to the likelihood ratio proposed by [PERSON], it could be judged that  
\[LR\equiv[SSR^{*}-SSR(\hat{\gamma})]/\phi^{2} \tag{3}\]  
where \(\phi^{2}\equiv\frac{SSR(\hat{\gamma})}{n(T-1)}\) was the consistent estimator of the variance of the random error term. To test whether the threshold estimate was equal to the true value, the null hypothesis was \(H_{0}:\gamma=\gamma_{0}\). The likelihood ratio test statistic was  
\[LR(\gamma)\equiv[SSR(\gamma)-SSR(\hat{\gamma})]/\hat{\phi}^{2} \tag{4}\]  
Although the distribution of \(LR(\gamma)\) was nonstandard, its cumulative distribution function was \((1-e^{\frac{-\gamma}{2}})^{2}\) and its critical value could be calculated on this basis. If \(LR(\gamma_{0})\) exceeded the critical value, then the null hypothesis \(H_{0}:\gamma=\gamma_{0}\) was rejected, and the threshold estimate was not equal to its true value. Similarly, according to Model (2), a panel threshold regression model of \"multiple threshold values\" could be established as shown below  
\[y_{it}=\alpha_{i}+\beta^{{}^{\prime}}_{1}\mathbf{x_{it}}\cdot 1(q_{it}\leq \gamma_{1})+\beta^{{}^{\prime}}_{2}\mathbf{x_{it}}\cdot 1(\gamma_{1}<q_{it}\leq \gamma_{2})\cdot\ldots\ldots\beta^{{}^{\prime}}_{n}\mathbf{x_{it}}\cdot 1(q_{it}> \gamma_{n})+u_{it} \tag{5}\]  
Because the values of all variables in the model were relatively large, the variables were processed by taking the logarithm. The advantage of taking the logarithm was that the nature and correlation of the data would not be changed after the logarithm was taken, but the scale of the variables would be compressed.