*3.4. The Estimation Method*

As shown in Figure 1, based on the theoretical framework, the system estimation is applied to study the nexus of energy-environment-growth.

**Figure 1.** Conceptual framework.

The generalized method of moments (GMM) was first proposed by Hansen and has become one of the most popular measurement methods. Arellano and Bond [63] proposed a first difference GMM (diff-GMM) estimation method. However, Blundell and Bond [64] have found the first-order diff-GMM estimation method is vulnerable to the influence of weak instrumental variables and gets biased estimation results. To overcome the influence of weak instrumental variables, Arellano and Bover [65] and Blundell and Bond [64] proposed another more effective method system GMM (sys-GMM). With the energy-environment-growth nexus, Saidi and Hammami [66] and Sekrafi and Sghaier [67] used diff-GMM in their studies, while Bhattacharya et al. [68] used sys-GMM in the interrelationship of energy-environment-growth. The main advantage of these methods over other methods is that they rely on internal instruments for estimation. However, in the case of a reverse causal relationship, external instruments are the best. However, finding external tools is a difficult task, which varies across units and periods. Fortunately, Farhadi et al. [69] concluded that the internal tools used are different, and sys-GMM is the best choice to control the endogenous nature of explanatory variables.

$$y\_{it} = x\_{it}\beta + qy\_{i, t-1} + \varepsilon\_i + \varepsilon\_{it} \tag{8}$$

where *t* denotes time, and *i* denotes the cross-section units (countries). It appears that the error terms consist of the fixed individual effects *ci* and the idiosyncratic shocks *εit*. The properties of fixed individual effects and idiosyncratic shocks are attributed as

$$E(\mathbf{c}\_i) = E(\mathbf{c}\_{it}) = E(\mathbf{c}\_i \mathbf{c}\_{it}) = 0 \tag{9}$$

By taking the difference to eliminate the individual effects *ci* from Equation (8) resulting in:

$$
\Delta y\_{it} = (\Delta x)\_{it}\beta + \varphi(\Delta y\_{i, t-1}) + \Delta \varepsilon\_{it} \tag{10}
$$

where Δ denotes the first difference operator.

Since Roodman [70] indicated that the diff-GMM and sys-GMM estimator is suitable for data sets with large groups and few periods, the current energy-environment-growth studies do not always follow this rule. However, if groups are too small, the test of cluster robust standard error and sequence correlation becomes inaccurate. Another problem is that the quantity of instruments is quadratic in the periods, which may lead to overfitting the equation because there are too many instruments compared to the sample capacity. To overcome the problem, the quantity of instruments is expected to be less than the groups. To achieve this goal, we can limit the lag of the instruments and collapse the instrument matrix. Table 2 provides the definition and source of the variable. Descriptive statistics are shown in Table 3, which is divided into OECD and non-OECD groups.


**Table 2.** The definition and source of variables.


**Table 3.** Descriptive statistics.
