*3.2. VAR Model*

The coupling coordination model can conduct an intuitive evaluation of the coupling coordination of the two systems from a comparative static point of view, and the VAR model is an econometric model used to estimate the dynamic relationship of joint endogenous variables. The model is established according to the statistical characteristics of the data without setting any constraints in advance. Each variable in the system is regarded as endogenous, and the lag term of all variables is included in the constructed function model. The VAR model is mainly used to analyse the response of interconnected time-series systems under the dynamic impact of system variables. The analysis of the model is mainly to observe the impulse response function and variance decomposition of the system. The former refers to the system's response to a random impact of one of the variables and how long this response will last. The latter is an important method to judge the dynamic correlation between economic series variables. In essence, it decomposes the prediction mean square error of the system into the contribution of the shocks of various variables in the system. Through the Granger causality test, we can analyse the causal effect of variables in time. This paper used the VAR model to analyse the dynamic action of the atmospheric environment system and the industrial system in Taiyuan. The specific model is constructed as follows:

$$\begin{cases} y\_{1t} = \mathfrak{a}\_{10} + \gamma\_{11} y\_{1, t-1} + \dots + \gamma\_{1p} y\_{1, t-p} + \beta\_{11} y\_{2, t-1} + \dots + \beta\_{1p} \beta\_{2, t-p} + \varepsilon\_{1t} \\ y\_{2t} = \mathfrak{a}\_{20} + \gamma\_{21} y\_{1, t-1} + \dots + \gamma\_{2p} y\_{1, t-p} + \beta\_{21} y\_{2, t-1} + \dots + \beta\_{2p} \beta\_{2, t-p} + \varepsilon\_{2t} \end{cases} \tag{10}$$

where *y*<sup>1</sup> and *y*<sup>2</sup> are the comprehensive indexes of the atmospheric environment system and the industrial structure system, respectively. *p* represents the lag order, *t* represents the time, *γ* and *β* represent the regression coefficient, *α* represents the intercept term, and *ε* represents the residual term.
