# Urbanization and Grain Production Pattern of China: Dynamic Effect and Mediating Mechanism
## 3 Materials and Methods
### Methodology
#### 3.1.3 Dynamic Spatial Panel Econometric Model  
The change of GPP is a dynamic and cumulative process, and there is also an obvious path dependence and inertia effect of regional grain production, which is not only related to the current period grain production but also influenced by the previous period's grain production, so the econometric model setting needs to consider the previous period's GPP. In addition, although the agglomeration of grain production between regions is constrained by geographical and natural conditions, in socioeconomic terms, the increasingly frequent spatial correlation of production factors such as labor force transfer and cross-area operation of mechanization services makes it possible for grain production in one region to influence grain production in neighboring regions through the demonstration and spillover effects of factor flows [38]. It is necessary to test the impact of urbanization on GPP under the perspective of spatial correlation. So, the dynamic spatial panel econometric model is used for empirical testing, which can not only effectively deal with the endogenous problem caused by other variables besides the dependent variable time lag term and spatial lag term but also can significantly reduce the bias of the spatial lag coefficient, which can effectively compensate for the defects of the basic spatial econometric model [39]. The basic models of spatial econometrics mainly include the Spatial Lag Model (SLM) and Spatial Error Model (SEM), where the SLM examines the spatial spillover effect arising from the spatial dependence of dependent variable, and the SEM examines the spillover effects caused by the shocks of the error terms in neighboring regions to the region [40]. Considering the above, the dynamic spatial econometric models set in this paper are as follows:  
\[\text{SLM}:\ \ln GPP_{it}=\alpha+\theta\ln GPP_{it-1}+\rho W\ln GPP_{it}+\beta URBAN _{it}+\sigma{\sum}control+\mu_{i}+\omega_{t}+\varepsilon_{it} \tag{9}\]\[\text{SEM}:\ \left\{ \begin{array}{l}\ln GPP_{tt}=\alpha+\theta\ln GPP_{tt-1}+ \betaURBAN_{it}+\sigma\sum control+\mu_{i}+\omega_{t}+\varepsilon_{it}\\ \varepsilon_{it}=\lambda WE_{jit}+\varphi_{it}\end{array} \right. \tag{10}\]  
Equations (9) and (10) are the spatial lag form and spatial error form of the dynamic spatial panel model, respectively. \(i\) is province, \(t\) is year, _Ingrain_ is the explained variable, which represents the grain production pattern (GPP), expressed as the natural logarithm of the percentage of grain production concentration index (GPCI), _Ingrain\({}_{i\text{-}1}\)_ is the time lag term, _WlnGPP\({}_{it}\)_ is the spatial lag term, and \(W\) is the spatial weight matrix. URBAN is the comprehensive urbanization level measured based on the entropy method, which is the core explanatory variable of this paper; \(\alpha\) is a constant term; _Control_ represents a collection of control variables; \(\mu_{i}\) is the individual fixed effect; \(\omega_{t}\) is the time fixed effect; \(\varepsilon_{it}\), \(\varphi_{it}\) are the random error term of white noise, and the others are the variable parameters to be estimated; and \(\rho\) and \(\lambda\) are the coefficients of the spatial lag term and spatial error term, respectively. The parameter estimation methods of dynamic spatial models mainly include the generalized method of moments (GMM), unconditional maximum likelihood estimation (ML), and bias-corrected LSDV method, and the GMM also includes Diff-GMM and Sys-GMM [41]. It should be considered that the dynamic spatial panel model includes both the time-lagged term and the spatial-lagged term of the dependent variable, which may cause estimation bias and endogeneity. The system generalized method of moments (Sys-GMM) can select appropriate instrumental variables from the time trends of the variables, avoiding the inappropriate selection of external instrumental variables [42], thus increasing the efficiency of estimation, and has a smaller finite sample bias and a more prominent advantage of parameter estimation.