**3. Methodology**

#### *3.1. Production Function*

Many countries are interested in energy-environment-growth nexus, which has gradually become a worldwide problem [56]. In this context, this paper draws on the research results of other scholars and regards energy consumption as a production factor within the nexus [57]. Moreover, innovation input is rarely included in the nexus, which reflects a country's science and technology level. In summary, the augmented Cobb-Douglas production function is as follows:

$$
\mathfrak{geg} = Ak^{a1}e^{a2} \tag{1}
$$

where *geg* is the green economic growth, which is calculated by the PCA method and consists of twelve indicators as shown in Table 1 [58,59]. *A* is the total factor productivity, *k* is the capital per capita, and *e* is the proportion of renewable energy consumption.

After logging, *i* denotes the country and *t* denotes the time period as follows:

$$
\log \mathbf{g}\_{it} = \mathbf{a}\_{it} + \mathbf{a}\_1 \mathbf{k}\_{it} + \mathbf{a}\_2 \mathbf{e}\_{it} \tag{2}
$$

Assuming that green economic growth depends on innovation input it becomes:

$$a\_{it} = a\_0 + a\_4 i m o\_{it} + \varepsilon\_{1,it} \tag{3}$$

Combining Equations (3) and (4), we can get:

$$
\log \text{g}\_{it} = \alpha \mathbf{o}\_{i} + \alpha\_{1} k\_{it} + \alpha\_{2} e\_{it} + \alpha \mathbf{j} \text{in}o\_{it} + \varepsilon\_{1,it} \tag{4}
$$

Because capital, renewable energy consumption, and innovation are conducive to green economic growth, they are expected to have a positive impact on green economic growth, indicating that *α*1, *α*2, and *α*<sup>3</sup> should be positive. Since pollution is not significant in the estimation and obeys the standard production function theory, we do not introduce pollution as an explanatory variable.

#### *3.2. Energy Consumption Function*

Referring to previous literature on the energy-environment-growth nexus [60,61], innovation input and climate change are included in the energy consumption function as follows:

$$\varepsilon\_{it} = \beta\_0 + \beta\_1 \text{geg}\_{it} + \beta\_2 ind\_{it} + \beta\_3 imo\_{it} + \beta\_4 stemp\_{it} + \beta\_5 wtemp\_{it} + \varepsilon\_{2,it} \tag{5}$$

where *e* is the proportion of renewable energy consumption, *geg* is green economic growth, *ind* is industrialization, *ino* is innovation input, *stemp* is the average temperature of three months in summer and wtemp is the average temperature of three months in winter, *ε<sup>2</sup>* is the error term.
