*2.2. Methodology*

Based on the two-stage method proposed by Kilian [6], first we identified the structural shocks by estimating the SVAR model and decomposing the structural shocks. Second, we constructed a regression model to investigate how the structural shocks affected US macroeconomic aggregates such as real GDP growth and CPI inflation. This methodology is widely used in investigations related to oil and natural gas price shocks. For instance, Sim and Zhou [16] applied a structural VAR model and a quantile regression model to examine the relationship between oil prices and US stock returns. Yoshizaki and Hamori [17] utilized the same model to analyze the reaction of economic activity, inflation, and exchange rates to oil prices shocks. Ahmed and Wadud [18] also used a structural VAR model to identify the role of oil price shocks on the Malaysian economy and monetary responses.

#### 2.2.1. The SVAR Model

Consider an SVAR model based on a monthly data set for crude oil, *ot* = (<sup>Δ</sup>*oprodt*,*reat*,*rpot*), where <sup>Δ</sup>*oprodt* represents the percentage change in global crude oil production, *reat* represents an index of real economic activity, and *rpot* represents the real price of crude oil. In addition, consider an SVAR model for natural gas, *gt* = (<sup>Δ</sup>*gprodt*,*reat*,*rpgt*), where <sup>Δ</sup>*gprodt* represents the percentage change in global natural gas production, *reat* represents an index of real economic activity, and *rpgt* represents the real price of natural gas.

Thus, the representation of the SVAR model of crude oil is:

$$A\_0 \sigma\_t = \alpha + \sum\_{i}^{24} A\_i \sigma\_{t-i} + \varepsilon\_t \tag{1}$$

where ε*t* represents the vector of mutually and serially uncorrelated structural innovations. Although the lag length indicated by Akaike's information criterion was 14, we decided to use 24, as also used by Kilian, because we used monthly data series in the model. Using 24 lags avoids the dynamic misspecification problem [6]. When the error terms are related, they have a common component that cannot be recognized by any particular variable, thus, we performed an adjustment to make the

error terms orthogonal by structural decomposition. We assumed that *A*−<sup>1</sup> 0 has a recursive structure; therefore, we can decompose the errors *et* according to *et* = *A*−<sup>1</sup> 0 ε*t*:

$$\mathbf{e}\_{t} = \begin{pmatrix} e\_{t}^{\text{Δprod}} \\ e\_{t}^{\text{par}} \\ e\_{t}^{\text{pro}} \end{pmatrix} = \begin{pmatrix} a\_{11} & 0 & 0 \\ a\_{21} & a\_{22} & 0 \\ a\_{31} & a\_{32} & a\_{33} \end{pmatrix} \begin{pmatrix} \varepsilon\_{t}^{\text{oil supply shock}} \\ \varepsilon\_{t}^{\text{ag}} \varepsilon\_{t}^{\text{ag}} \end{pmatrix} \tag{2}$$

Based on Equation (2), we have three structural shocks to be identified, namely an oil supply shock, an aggregate demand shock, and an oil-specific demand shock. As noted by Kilian [6], an oil supply shock is designed to capture unexpected innovations to international oil output. An aggregate demand shock, which is driven by the global business cycle, is designed to identify the unexpected innovations to global economic activity. Finally, an oil-specific demand shock, or precautionary demand shock, which is driven by increasing uncertainty in the oil market, is designed to identify the exogenous shifts in precautionary demand.

It must be noted that, as Equation (2) shows, in the first row of *A*−<sup>1</sup> 0 , the restriction *a*12 = *a*13 = 0 implies that innovations to global oil production can only be explained by the oil supply shocks; in the middle row, the restriction *a*23 = 0 indicates that innovations to worldwide economic activity can be explained by oil supply and aggregate demand shocks rather than an oil-specific demand shock. The restriction of the last row shows that all of the three shocks could have e ffects on the real price of crude oil.

Therefore, we can obtain the structural residuals by estimating from Equations (1) and (2). Here we can plot the structural residuals to observe the changing composition of the structural shocks over time. Then, we can check the dynamic response pattern of the endogenous variables to various structural shocks by imposing a one-standard-deviation structural shock.

Similarly, the representation of the SVAR model of natural gas is:

$$A\_0 \mathbf{g}\_t = \mathfrak{G} + \sum\_{i}^{24} A\_i \mathbf{g}\_{t-i} + \mathfrak{w}\_t \tag{3}$$

We can also decompose the errors *wt* according to *wt* = *A*−<sup>1</sup> 0ω*t* to identify the structural shocks:

$$
\mathbf{w}\_t = \begin{pmatrix} \alpha\_t^{\text{Loprad}} \\ \alpha\_t^{\text{ran}} \\ \alpha\_t^{\text{ppo}} \end{pmatrix} = \begin{bmatrix} a\_{11} & 0 & 0 \\ a\_{21} & a\_{22} & 0 \\ a\_{31} & a\_{32} & a\_{33} \end{bmatrix} \begin{pmatrix} \alpha\_t^{\text{gas supply shock}} \\ \alpha\_t^{\text{aggrange demand shock}} \\ \alpha\_t^{\text{gas}-specific demand shock} \end{pmatrix} \tag{4}$$

#### 2.2.2. Regression Model

The purpose of this step is to investigate how the structural shocks estimated from the structural VAR model in Section 2.2.1 influence the US macroeconomic aggregates, such as CPI inflation ( <sup>π</sup>*t*) or real GDP growth ( Δ*yt*). The e ffects are identified by the following regressions:

$$
\Delta y\_t = \alpha\_j + \sum\_{i=0}^{12} \phi\_{ji} \mathcal{L}\_{jt-i} + u\_{jt} \tag{5}
$$

$$
\pi\_{\rm tr} = \delta\_{\rm j} + \sum\_{i=0}^{12} \psi\_{j\rm i} \hat{\mathbb{C}}\_{\rm jt-i} + \upsilon\_{j\rm t} \tag{6}
$$

$$\mathcal{L}\_{jt}^{o} = \frac{1}{3} \sum\_{i=1}^{3} \varepsilon\_{j,t,i\prime} \text{ } j = 1,2,3\tag{7}$$

$$\hat{\mathcal{L}}\_{j\text{t}}^{\mathcal{S}} = \frac{1}{3} \sum\_{i=1}^{3} \hat{\omega}\_{j,t,i\text{t}} \text{ j} = 1,2,3\tag{8}$$

In this regression model, *ujt* and *vjt* refer to errors; φ*ji* and ψ*ji* represent the impulse response coefficients, respectively; <sup>ε</sup><sup>ˆ</sup>*j*,*t*,*<sup>i</sup>* denotes the estimated residual for the *jth* structural shock in the *ith* month of the *tth* quarter in the crude oil data; and ωˆ *j*,*t*,*i* denotes the estimated residual for the *jth* structural shock in the *ith* month of the *tth* quarter in the natural gas data. Furthermore, we determined the number of lags to be 12 quarters as also used by Kilian [6].
