**2. The Improved Method of Calculating Direct Rebound E**ff**ect**

Improving energy efficiency will decrease energy consumption with the same level of energy service. For instance, if the rate of electricity use and cooling area decrease, residents will use less electricity cooling the same area. However, improving efficiency means a decrease in real power price, which will incentivize residents to use more electricity in turn. Calculating the direct rebound effect is to calculate the gap between the expected savings and the actual savings. Direct rebound effect is then defined as: Direct rebound = (expected savings − actual savings)/expected savings. The traditional calculation method of the direct rebound effect controls no other variables, so some studies recommend that the price elasticity could be an ideal proxy indicator of direct rebound effect with other variables controlled [39]. The definition and identification of direct rebound effect can be found in Appendix A.

The calculation method above only analyzes the energy consumed by the same consumers (consumers in the local region) and does not consider other consumers (consumers in the adjacent region). It implies the assumption that energy consumption in different regions is independent. However, if there is a spatial "convergence effect" in energy consumption, whereby the increased energy efficiency in a local region will have an influence on the energy consumption not only in the local region, but also in the adjacent regions, leading to a "spatial feedback effect". For the same reason, the improvement of energy efficiency in the adjacent region also induces a direct rebound effect in the local region. This paper views this as the spatial spillover effect of the direct rebound effect. It is

impossible to distinguish whether the additional energy consumption in the local region is caused by the energy efficiency improvement in the local region or in the adjacent region without considering spatial spillover effect.

Based on spatial spillover effect, this paper improves the calculating model of direct rebound effect. The spatial lag of electricity consumption is introduced into the model, and the spatial lag model (SLM) controlling other variables is:

$$y\_t = \lambda N y\_t + X\_t \beta + c + u\_t \tag{1}$$

where both the explained variable and the explanatory variable are logarithmized. *yt* is the urban residents' electricity consumption of *n* regions in year *t*. *W* is the space weight matrix, and *Wyt* is the spatial lag of *yt*. λ measures the effect of spatial lag *Wyt* on *yt*, reflecting spatial dependence. *Xt* is the explanatory variables matrix of *n* regions in year *t*. β is the coefficient of the explanatory variable. *c* is the individual effect of *n* regions. According to the individual effect, the model can be divided into fixed effect model and random effect model.

If the spatial correlation of urban residents' electricity consumption is not considered, Equation (1) is reduced to a standard static panel model.

Rewrite Equation (1) as a reduced form:

$$\mathbf{y}\_{l} = \left(I - \lambda \mathbf{W}\right)^{-1} \left(X\_{l}\boldsymbol{\beta} + \mathbf{c} + \mathbf{u}\_{l}\right) \tag{2}$$

$$E(y\_t | \mathbf{X}\_t, \mathcal{W}) = \left(I - \lambda \mathcal{W}\right)^{-1} (\mathbf{X}\_t \boldsymbol{\beta})\tag{3}$$

Equation (3) shows that measuring the direct rebound effect should consider the spatial feedback effect. According to the research on direct and indirect effects in spatial econometric models by LeSage and Pace [40], the calculation of the direct rebound effect in the space lag fixed effect model is:

$$\text{RE} = -\frac{1}{nT} \sum\_{t=1}^{T} \sum\_{i=1}^{n} \frac{\partial E(\hat{\mathcal{G}}\_{it} | \mathbf{X}\_{t\prime} \mathcal{W})}{\partial \ln P\_i} \tag{4}$$

where *y*ˆ*<sup>t</sup>* = *yt* − (*I* − λ*W*) −1 *c*. The direct rebound effect calculated here is the average value of the direct rebound effect of *n* regions, so it can be called the average direct rebound effect (abbreviated as RE).

The calculation of the space spillover effect of direct rebound effect is defined as:

$$\text{SRE} = -\frac{1}{nT(n-1)}\sum\_{t=1}^{T}\sum\_{i=1}^{n}\sum\_{j=1, j\neq i}^{n}\frac{\partial E(\hat{\rho}\_{it}|\mathbf{X}\_{t\prime}\mathcal{W})}{\partial \ln P\_{j}}\tag{5}$$

Equation (5) also calculates the average spatial spillover effects of *n* regions, so it can be called the average spatial spillover effect (abbreviated as SRE).

The spatial lag model only considers the endogenous interaction effects, ignoring the spatial correlation of unobservable random impacts. In the case of multiple spatial interactions, a more appropriate method is to use the spatial autoregressive model with spatial autoregressive disturbances (SARAR) model. Since urban residents' electricity consumption may have both the interaction of endogenous interaction and error terms, the SARAR model is introduced to measure the direct rebound effect and its spatial spillover effect:

$$y\_t = \lambda Wy\_t + X\_t\beta + \varepsilon + \varepsilon\_t, \varepsilon\_t = \rho W\varepsilon\_t + q\_t \tag{6}$$

where ρ is the coefficient of spatial lag *W*ε*t*. The calculation of the average direct rebound effect and the average spatial spillover effect of the SARAR model is consistent with the SLM model.

The process of using spatial econometric models is as follows: before establishing a spatial econometric model, it is necessary to test the spatial autocorrelation and heterogeneity in the data, using two types of spatial autocorrelation test: local autocorrelation and global autocorrelation. Then multiple models are set up. For nested models, a likelihood ratio (LR) test can be used to choose the best model. Due to endogeneity problems, the ordinary least squares (OLS) estimators are inconsistent. This paper uses the maximum likelihood estimator method (MLE) to get the consistent estimator. For panel data, individual effects need to be tested, so the Hausman test is used to select the appropriate model between the fixed effect and random effect models. And we use the method proposed by Lee and Yu [41] to estimate the panel spatial econometric model. Firstly, the individual effects are eliminated, then the maximum likelihood estimator method is performed.
