*4.2. Spatial Correlation Test*

Before applying the spatial econometric model, it is necessary to analyze the local and global spatial correlation to test whether there is spatial dependence in the urban residents' electricity consumption between regions. First, the local correlation types are analyzed. Although the Moran scatter plot can infer spatial correlation to some extent, the Moran scatter plot cannot determine whether the local correlation type is statistically significant. So the local indicators of spatial association (LISA) map is used to analyze the local spatial autocorrelation. Figures 1 and 2 show the LISA maps of China's urban residents' electricity consumption in 2006 and 2016, respectively.

**Figure 1.** Spatial aggregation of urban residents' electricity consumption in 2006.

**Figure 2.** Spatial aggregation of urban residents' electricity consumption in 2016.

The LISA maps show that there are four types of spatial agglomeration in China's urban residents' electricity consumption, and there is little change in local correlation patterns over time. From the perspective of aggregation effect, various types of spatial aggregation reflect the spatial heterogeneity of urban residents' electricity consumption. In terms of time, although the provinces with low-low aggregation and low-high aggregation have a small increase, it does not show a significant leap (for example, high-high to low-low), indicating that the spatial aggregation in urban residents' electricity consumption is stable.

The local spatial aggregation in urban residents' electricity consumption indicates that the spatial dependence cannot be ignored when the direct rebound effect is examined. Table 3 lists the test results of Moran's *I* index of urban residents' electricity consumption, in order to judge the global correlation in urban residents' electricity consumption.


**Table 3.** Spatial autocorrelation test.

Table 3 shows that there is a significant spatial autocorrelation in urban residents' electricity consumption, and the spatial correlation is positive, indicating that the urban residents' electricity consumption mainly reflects convergence effect.

#### *4.3. Analysis of Estimation Results of SARAR and SLM Models*

Table 4 displays the estimation results of the SLM and the SARAR model and the robust test results. Hausman test results of the SLM and the SARAR model reject the null hypothesis at the 1% level, meaning that the individual fixed effect model is superior to the individual random effect model. The SARAR fixed effect model has a larger log likelihood value than the SLE fixed effect model. The statistic of the LR test for the SLM model and the SARAR model is 7.448, rejecting the null hypothesis at the 1% level, showing that the SARAR fixed effect model is better than the SLM fixed effect model. Then the direct rebound effect for residents' electricity consumption and its spatial spillover effect are calculated based on SARAR model estimation results.


**Table 4.** Estimation results of SLM and SARAR model and robust test.

Note: The number in parentheses is the level of significance. \*\*\*, \*\*, and \* indicate significance levels at 1%, 5%, and 10%, respectively.

All the variable coefficients in the SARAR fixed effect models are significant. However, the absolute value of all variable coefficients in the SARAR fixed effect model is lower than that in static panel fixed effect model, indicating that ignoring the spatial correlation will overestimate the influence of these variables on electricity consumption.

This is because residents' electricity consumption in the local region is affected not only by power price, population and per capita income in the local region, but also by the positive impact of the spatial lag of residents' electricity consumption. The static panel model classifies the positive impact of spatial lag on residents' electricity consumption into other explanatory variables. So, the contribution of these explanatory variables is exaggerated.

#### *4.4. Analysis of RE and SRE*

In the SARAR fixed effect model, due to the existence of spatial lag, the spatial feedback effect should be considered to measure the direct rebound effect. Combined with Equation (4), the average direct rebound effect is 37.00%, indicating that improving the electricity efficiency does induce a direct rebound effect. However, the direct rebound effect for urban residents' electricity consumption is much lower than 100%, and is lower than that of the static panel model. This means that the direct rebound effect value is reduced after considering the spatial correlation. Increasing the efficiency of electricity consumption will ultimately reduce the urban residents' electricity consumption. 37% of the expected savings are offset, and 63% of the expected targets can be achieved actually. So, improving the efficiency plays an important role in reducing the urban residents' electricity consumption. Table 4 also shows that in addition to the decline in power price, the growth of population, per capita income and degree day value will also increase the urban residents' electricity consumption, especially when the inter-regional urban residents' electricity consumption has a mutual pulling effect. When the government measures the restraining effect of electricity efficiency on residents' electricity consumption, the factors above should be controlled. Otherwise, the direct rebound effect for residents' electricity consumption will be overestimated, and the inhibition effect of improving efficiency on electricity conservation will be underestimated.

The spatial spillover effect of direct rebound effect for urban residents' electricity consumption can be calculated and tested by using Equation (5). The test results confirm that the direct rebound effect for urban residents' electricity consumption has a significant spatial spillover effect at 1% level, and the spatial spillover effect is 13.30%. That is to say, per 1% decrease in power price due to the increased efficiency in adjacent areas will increase the urban residents' electricity consumption in the local region by 0.133%. Adding RE and SRE together, the total electricity consumption induced by the increased efficiency is 50.30%. The proportion of RE is 73.56%, and the proportion of SRE is 26.44%.

The calculating results above show that if the spatial dependence in urban residents' electricity consumption is not considered, the direct rebound effect and its spatial spillover effect will be confused. Due to the spatial spillover effect, the realization of energy-saving targets in local area depends on the implementation effect of energy efficiency in surrounding areas. Moreover, due to the low spatial spillover effect, direct rebound effect induced by efficiency improvement in the local region is still the main reason affecting the implementation effect of energy efficiency policies in the local region.

#### *4.5. Robust Test*

In addition to the two-part decomposition method adopted above, some studies also adopt a three-part decomposition method. Then the three-part decomposition method is used for a robustness test, shown in the last column of Table 4. The results of the robustness test are consistent with the empirical results above, indicating that the direct rebound effect measurement value for urban residents' electricity consumption is not sensitive to the price decomposition methods.
