As an important emerging branch of economics, spatial econometrics mainly deals with spatial autocorrelation, spatial structure analysis, and the spatial effect in the regression model of cross-sectional data and panel data. There are three kinds of basic spatial metrology models: the spatial lag model (SLM), the spatial error model (SEM) and the spatial Durbin model (SDM). Elhorst [
17] pointed out that the spatial model of the space panel relative to the cross-sectional form has more degrees of freedom, which can improve the validity of the estimation in the space measurement economy. Therefore, space measurement models are increasingly used in the field of social sciences, thereby becoming a major highlight of econometric theory.
2.1.2. Economic Weight Matrix
In this paper, the global Moran’s I index was determined using three spatial weight matrices. The most commonly used spatial weight matrices are the first-order adjacency spatial weight matrix and inverse distance spatial weight matrix with geographic characteristics [
21,
22]. Under these two spatial weight matrices, the influence intensity between adjacent cities is the same. However, in reality, cities with high economic development have a stronger influence on cities with lower economic development [
23]. Since the economic distance spatial weight matrix considers not only the geographical factors but also the economic factors, we used an economic distance spatial weight matrix to explore the impact of the spatial spillover effect of residents’ income on the per capita electricity consumption of urban residents. Note that we also established the inverse distance spatial weight matrix (henceforward noted as W_distance) and a four nearest neighbours spatial weight matrix (henceforward noted as W_k4) to test the robustness of the empirical results.
In Equation (2), is the geographic distance spatial weight matrix. In this paper, the average GDP was selected as the indicator to reflect the economic development of one city. is the average value of the GDP spanning 2005 to 2016 in city i; and denotes the average value of the GDP for 278 cities in China spanning 2005 to 2016. For consistency, the GDP indicators in different years were converted into the price GDP index in 2005.
2.1.3. Spatial Panel Regression Models
This paper employs the Impact of Population, Affluence and Technology (IPAT) model to identify the effect of the average wage on the per capita electricity consumption of urban residents. This model, however, ignores the effects of other determinants [
24]. Therefore, through further improvements to the IPAT model, some scholars have proposed the Stochastic Impacts by Regression on Population, Affluence and Technology (STIRPAT) model. By considering the factors of population, affluence, and technological progress, the model can randomly expand other important factors that affect the environment [
25]. This paper uses the STIRPAT model to test the impact of average wage on the per capita electricity consumption of urban residents. Previous studies have used the STIRPAT model to study the driving factors of energy consumption [
26]. In order to mitigate the impact of heteroscedasticity, the variables selected in this paper are logarithmic. Accordingly, the STIRPAT model is constructed as follows:
where REC is the per capita electricity consumption of urban residents; WAGE is the average wage of urban residents, denoting the degree of affluence; POP is the urban population density, representing the population factor; EI is the urban electricity intensity, reflecting the technical level; EDU stands for the educational level of urban residents; LPG represents the per capita household liquefied petroleum gas consumption in urban areas; and ε
it denotes the error term.
Considering the spatial interaction between regional production and the regional interdependence of electric utilities, it seems reasonable that the per capita electricity consumption of urban residents has a spatial correlation in China. Conversely, ignoring the potential spatial correlation of the per capita electricity consumption of urban residents may affect the validity of elasticity estimations and the accurate forecasting of the per capita electricity consumption of urban residents, thereby affecting the achievement of the goal of securing supply in China’s electricity sector. The SDM jointly considers the impacts of spatial lag of the dependent variables and the independent variables [
27]. Therefore, in this paper, we established a spatial econometric model for the per capita electricity consumption of urban residents in China based on the STIRPAT model.
Here, i and j represent different cities; Wij is the spatial weight matrix; Xit is the vector of influencing factors; LnRECit is the per capita electricity consumption of urban residents, ; is an intercept term; β is the regression coefficient for the influencing factors; is the spatial autoregressive coefficient for the per capita electricity consumption of urban residents; is the spatial regression coefficient for the influencing factors; λ denotes the spatial regression coefficient of the error term; and εit denotes the error term.
For Equations (6) and (7), if ρ ≠ 0, φ = 0, and λ = 0, then Equation (6) is the SLM, indicating that the per capita electricity consumption of urban residents in one city is affected by that in the neighbouring cities; if λ ≠ 0, ρ = 0 and φ = 0, then Equations (6) and (7) are the SEM, which indicates that the per capita electricity consumption of urban residents in one city is affected by factors other than the average wage of urban residents, urban population density, urban electricity intensity, educational level of urban residents, and the per capita household liquefied petroleum gas consumption in urban areas in neighbouring cities; if ρ ≠ 0, λ = 0 and φ ≠ 0, then Equation (6) is the SDM, which shows that the per capita electricity consumption of urban residents in one city is affected not only by the per capita electricity consumption of urban residents in the neighbouring cities, but also by the average wage of urban residents, urban population density, urban electricity intensity, educational level of urban residents, and the per capita household liquefied petroleum gas consumption in urban areas in the neighbouring cities.
The SDM jointly considers the impacts of the spatial lag of the dependent variables and the independent variables. The use of two Lagrange multiplier tests (i.e., LM_test for no spatial lag and Robust LM_test for no spatial lag; LM_test for no spatial error and Robust LM_test for no spatial error) determines whether the spatial lag effect or the spatial error effect is significant. If one LM test shows a significant effect while the other effect is not significant, this study should adopt the significant form spatial effect model. If the LM test results show that the two effects are significant or not significant simultaneously, this study should adopt the SDM, and, by the Wald or likelihood ratio (LR) test, determine whether the SDM can be simplified into the SLM or SEM. A flowchart summarizing the regression analysis is shown in
Figure 1.
The parameter estimates in the nonspatial model represent the marginal effect, whereas the coefficients in the spatial Durbin model do not. For this purpose, one should use the direct and indirect effect estimates to interpret the model [
28]. Meanwhile, one should note that the direct effects of the explanatory variables are different from their coefficient estimates. The direct effect represents the influence of a local independent variable on the local dependent variable. This measure includes feedback effects that arise as a result of impacts passing through adjacent areas (e.g., from region i to j to k) and back to the area that the change originated from (region i). The indirect effect, known as spatial spillover, represents the impact of a local independent variable on the dependent variables in the adjacent areas. The total effect is simply the sum of the direct and indirect effects. LeSage and Pace [
28] and Elhorst [
29] point out that the indirect effects of the independent variables should be used to examine whether or not spatial spillovers exist. By estimating the direct and indirect effects, the regression model can accurately reflect the marginal effect of explanatory variables. Therefore, this paper mainly observes the impact of average wage on per capita electricity consumption of urban residents through direct and indirect effects.