2.1.3. Another Matrix

With the popularity of spatial econometric method, adjacency matrix and geographical distance matrix have been unable to meet the needs of economic research, and more and more new spatial weight matrices have appeared. For example, technological distance matrix reflects the impact of technological consistency between regions on spillover effects [41], institutional distance matrix reflects the impact of institutional disparity on spillover effects [42], and economic distance reflects the impact of economic disparity on spillover effects [43], etc. Among them, the economic distance matrix is used most frequently.

#### *2.2. Moran's I Test*

Spatial econometric models require that there must be significant spatial autocorrelation of the explained variables. A commonly used test is the Moran's I Test, specifically, the global Moran's I test and the local Moran's I test.

#### 2.2.1. Global Moran's I Test

The global spatial autocorrelation test tests the spatial autocorrelation degree of the whole sample, which reflects the spatial dependence of attribute values. Global Moran's I test is the most commonly used method, which can visualize the spatial aggregation characteristics of attribute values [44]. Formula (3) shows the calculation process of global Moran's I value.

$$\mathbf{I}\_{\mathbf{t}}^{\mathbf{G}} = \sum\_{\mathbf{i}=1}^{\mathbf{n}} \sum\_{\mathbf{j}=1}^{\mathbf{n}} \mathbf{W}\_{\overline{\mathbf{i}}\} \left(\mathbf{X}\_{\mathbf{i},\mathbf{t}} - \overline{\mathbf{X}\_{\mathbf{t}}}\right) \left(\mathbf{X}\_{\mathbf{j},\mathbf{t}} - \overline{\mathbf{X}\_{\mathbf{t}}}\right) \bigg/ \mathbf{S}^{2} \sum\_{\mathbf{i}=1}^{\mathbf{n}} \sum\_{\mathbf{j}=1}^{\mathbf{n}} \mathbf{W}\_{\overline{\mathbf{i}}\} \tag{3}$$

where i and j are individual labels; n is the number of samples; t is the time; Xi, Xj and X are the attribute value of i, j and average; Wij is the spatial weight matrix; S2 is the variance of the attribute value. The value of Moran's I ∈ [−1, 1]. When 0 < I G < 1, there is a positive spatial agglomeration; when I G = 0, there is no spatial agglomeration, which means the attribute value is randomly distributed; when −1 < I G < 0, there is a negative spatial agglomeration.

#### 2.2.2. Local Moran's I Test

Global spatial autocorrelation test can only test whether there is spatial autocorrelation in the whole sample, but it cannot judge the spatial autocorrelation characteristics of attribute values. We need to use the local spatial autocorrelation test to further explore the spatial autocorrelation characteristics of each attribute value (high-high agglomeration, low-low agglomeration, high-low agglomeration and low-high agglomeration). Moran scatter plot is one of the methods commonly used to test the characteristics of local regional aggregation. Formula (4) shows its calculation process.

$$\mathbf{I}\_{\mathbf{i}\_{\text{'}}}^{\mathcal{L}} = \left(\boldsymbol{\chi}\_{\text{i},\text{t}} - \overline{\boldsymbol{\chi}\_{\text{t}}}\right) \sum\_{\mathbf{j}=1}^{n} \mathbf{W}\_{\text{ij}} (\boldsymbol{\chi}\_{\text{j},\text{t}} - \overline{\boldsymbol{\chi}\_{\text{t}}}) \bigg/ \mathbf{S}^{2} \tag{4}$$

The meaning of each symbol is the same as above. In addition, when I L > 0, there is H-H or L-L agglomeration characteristics in adjacent areas; when I L < 0, there is L-H or H-L agglomeration characteristics in adjacent areas; when I L = 0, there is no local agglomeration characteristics in adjacent areas.

#### *2.3. Description of Spatial Econometric Model*

Common spatial econometric models include spatial lag model (SLM), spatial error model (SEM), and spatial Durbin model (SDM). SLM refers to the spatial lag of the explained variable, rather than the traditional time lag. The combination of the spatial lag and the explained variable is used as an explanatory variable to reflect the influence of the explained variable in other regions in the whole on the local. SEM can deal with spatial spillover effects caused by missing important variables or unobservable random shocks, that is, assuming that the disturbance term has spatial dependence. The characteristic of SDM is to add a combination of explanatory variables and spatial lags as new explanatory variables to reflect the influence of explanatory variables in other regions in the whole on the local. Formulas (5)–(7) are the general forms of SLM, SEM and SDM, respectively:

$$\mathbf{Y} = \mathbf{a}\mathbf{W}\mathbf{Y} + \mathbf{\beta}\mathbf{X} + \varepsilon \tag{5}$$

$$\mathbf{Y} = \mathbf{a}\mathbf{W}\mathbf{Y} + \beta\mathbf{X} + \delta\mathbf{W}\mathbf{X} + \varepsilon \tag{6}$$

$$\mathbf{Y} = \underset{\mathbf{x}}{\otimes} \mathbf{X} + \boldsymbol{\mu} \tag{7}$$

$$
\mu = \mathbf{M}\mu + \varepsilon \tag{6}
$$

where W and M are spatial weight matrices; α, β and δ are coefficients of corresponding variables, and ε is a random error term.
