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On the relationship between U.S. crude oil and natural gas for economic resilience prospects

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Abstract

As a fossil-fuel dependent economy, the United States of America relies heavily on crude oil and natural gas. Fluctuations in crude oil and natural gas prices can have a profound impact on the U.S. economy and society. Such a vulnerability to energy commodities threatens the resilience of the U.S. economy, and therefore, U.S. economic development and growth. To be able to ensure energy security and affordable energy sources, the (joint) dynamics of energy commodity prices should be captured. This article examines the complex dynamic relationship between U.S. crude oil and natural gas prices, as strategic energy sources. While crude oil prices lead natural gas prices, the relationship exhibits regime shifts that depend on technological, economic, and geopolitical factors. In each regime, we model the relationship between crude oil and natural gas price changes by using a linear Kalman filter with stochastic regression coefficients and heteroskedastic errors in the measurement equation. The random parameters and volatility illustrate the uncertainty in energy costs and prices and handle local nonlinearity. Crude oil and natural gas prices decouple for two regimes of about four and five years (i.e., short-term decoupling), while they couple for four subsequent regimes (i.e., long-term coupling for about seventeen years). The results unveil changes in the competition between oil price makers and takers and the impact of technological improvements, including the shale gas revolution and renewable energy. They also provide insights into possible short- and/or long-term hedging strategies between crude oil and natural gas, both at the producer and the end-user level. In this context, the findings are useful for the interplay between the U.S. economy, and both crude oil and natural gas. They provide an opportunity for policymakers to act and strengthen the economic resilience by sustaining energy supply and security, and mitigating energy price risk.

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(Source: Gupta et al., 2013)

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Notes

  1. Petroleum and natural gas account for 72% of the total U.S. energy consumption in 2022.

  2. The price of each commodity must be forecast separately.

  3. At least from 2007 to 2016, just before the start of U.S. LNG exports.

  4. The domestic factors consist of supply and demand, regulation, the natural gas liquids markets and recent LNG exports.

  5. Refer to https://www.eia.gov/ for more information.

  6. The Anderson–Darling test is a goodness-of-fit test that allows to test for the hypothesis that a sample random variable follows a theoretical probability distribution. It has several advantages since it is highly sensitive to the tails of the distribution and less sensitive to outliers, it is more flexible because the critical values depend on the theoretical distribution being tested (Anderson and Darling, 1954; D'Agostino & Stephens, 1986).

  7. The Phillips-Perron unit root test is robust to serial correlation and heteroskedasticity in the error term of the associated regression equation. In addition, unlike the Augmented Dickey-Füller (ADF) test, it does not need to specify the number of lags in the regression (Phillips and Perron, 1988).

  8. Incidentally, the natural gas price spike, shown in Fig. 4 on February 25, 2003 was investigated by the Federal Energy Regulatory Commission (FERC) in collaboration with the Commodity Futures Trading Commission (CFTC). However, no market manipulation was evidenced (see FERC Report, 2003). At the time, the physical natural gas market offered a very weak supply that was unable to meet the large level of demand in the U.S. in the short term. Extreme weather conditions (i.e., cold front and winter weather) both supported a sharp increase in the natural gas demand and provoked a disruption of natural gas production due to well freeze-offs. The resulting limited ability of the natural gas industry to respond to a sudden increase in demand, coupled with a low level of storage inventories worsened the supply and demand imbalances, triggering therefore a huge natural gas price spike. Similarly, a natural gas price spike also occurs on February 17, 2021, following a winter storm that triggered an increase in energy demand coupled with a disruption in natural gas production because of well freeze-offs. Finally, the negative crude oil price on April 20, 2020 results from a demand shock as a combination of the covid-19 lockdown and related economic slowdown (i.e., oversupply).

  9. We ran the same tests on the ratio of log-prices, and its adapted version in the regression of crude oil prices on natural gas prices in the presence of cointegration. We found the same number of breaks and very close break dates (when they were not similar).

  10. OPEC stands for the Organization of the Petroleum Exporting Countries.

  11. An AO affects a single point in a series whereas an IO supports a gradual shift in the mean of the series. Sometimes outliers can generate a temporary change such that one observation is extreme and the following observations gradually reduce the magnitude of the outlier’s deviation. In other cases, outliers generate a shift in the level of the series.

  12. We have tested for a local linear trend, but the slope turns out to be insignificant. We also specify a general cycle component, which can be expanded to account for seasonal patterns (i.e., periodic patterns with integer periods).

  13. When a2 is zero, the heteroscedastic variance reduces to an ARCH(1) dynamics.

  14. The duration/period of the cycle is \(2\pi /\lambda _{c})\).

  15. The linear regression uses Heteroskedasticity and Autocorrelation Consistent covariance (HAC) estimates and pre-whitening of residuals with automatic lag selection (Hannan-Quinn criterion). HAC estimates are robust to heteroskedasticity (i.e., robust standard errors; Newey and West, 1987). The pre-whitening methodology reduces data autocorrelation and hence estimation bias (Andrews and Monahan, 1992; Newey and West, 1994).

  16. According to the EIA, the spare capacity is the “volume of production that can be brought on within 30 days and sustained for at least 90 days”.

  17. OPEC members are expected to compensate for non-OPEC production insufficiencies due to their limited production capacity. This scenario is known as the “call on OPEC” (refer to EIA website).

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Acknowledgements

We thank two anonymous referees for their questions and comments. We also thank the participants of the 67th EWGCFM (Rome, May 2023) and Hélyette Geman for their interesting questions and remarks. The usual disclaimer applies.

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Appendix

Appendix

1.1 Current natural gas and lagged crude oil price changes

To examine the relationship between crude oil and natural gas price changes, we consider the cross correlations between their price changes up to 5 leads/lags in each regime. Unreported results show that the cross-correlation between DGast and DOilt-1 is the strongest (i.e., stronger relationship between current natural gas price changes and previous crude oil price changes at lag 1). We confirm this finding by running linear regressions of DGast against DOilt-1 in each regime, using HAC estimates (see Table 

Table 10 Linear regressions of natural gas price changes against crude oil price changes

10).

Indeed, the absolute value of slope coefficient b and the R2 of regressions are larger when the independent variable is the previous crude oil price changes. Therefore, the strongest relationship should be studied when examining the link between DGast and DOilt-1 in each regime.

Furthermore, a brief comparison of the cross-correlation between natural gas and crude oil price changes at lags 0 and 1 supports the previous link (see Table 

Table 11 Cross-correlations between natural gas and crude oil price changes

11).

1.2 Testing for structural changes

Assume that there is a maximum number of N possible breaks with corresponding break dates (t1 <  < ti <  < tN), and that all the time series have a size of T. Let us set t0 = 1 as the first observation and tN+1 = T the last observation of the sample (i.e., the start and end dates of the sample). The N possible breaks determine (N + 1) regimes that occur at time intervals I1 = [t0, t1] for regime 1, and Ii+1 =]ti,ti+1] for any regime (i + 1) with i \(\in\) {2,…,N}. The test examines the break dates by considering the following regression with regime-specific coefficients (i.e., pure structural change model):

$$ DGas_{t} = \alpha_{i} + \beta_{i} DOil_{t - 1} + \varepsilon_{i,t} $$

where i is the regime, t \(\in\) Ii for i \(\in\) {1,…,N + 1}, \({\alpha }_{i}\) is the regime-specific constant, \({\beta }_{i}\) is the regime-specific slope, and \({\varepsilon }_{i,t}\) is the regime-specific residual that follows a Gaussian probability distribution with a zero mean and a constant standard deviation \({\sigma }_{i}\).

Bai and Perron (1998) test for the equality of \({\alpha }_{i}\) coefficients on the one hand, and \({\beta }_{i}\) coefficients across the different possible regimes on the other hand (i.e., H0: \({\alpha }_{0}\)=… = \({\alpha }_{i}\)=…=\({\alpha }_{N}\) and \({\beta }_{0}\)=… = \({\beta }_{i}\)=… = \({\beta }_{N}\)). In the presence of multiple unknown breaks, a version of the test sequentially tests for the null hypothesis of l breaks versus no break for l \(\in\) {1,…,N} (i.e., when the constancy of parameters is rejected). Such an approach consists of starting with an unknown break date and detecting whether the regression coefficients move across any set of two sub-samples that can be built from the entire sample (i.e., detecting when parameter constancy is rejected). A second break date is then added to investigate the stability of coefficients across the further sub-samples that can be built (i.e., alternatively testing for the existence of l = 2 breaks versus one break). The process is repeated until the maximum number of breaks N is reached or until no more breaks can be detected (i.e., the null is not rejected). The test is based on minimising the sum of squared residuals across the listed regimes (i.e., ordinary least squares estimation). The null is tested by using a \(\text{sup}F\) statistic, which is scaled when several regression coefficients are regime-dependent (Bai & Perron, 1998, 2003a, 2003b). If the statistic is larger than its critical value at 5% for a given number of breaks, the constancy of regression parameters across the detected regimes is rejected, and the identified number of breaks and related break dates are statistically significant.

1.3 Unobserved component model with heteroskedastic errors

First, Eqs. (2), (4), (5), (7), and (8) can be summarized in a general multivariate setting as follows (see Kim & Nelson, 1999):

$$ y_{t} = X_{t - 1}^{*} \beta_{t}^{*} $$
(14)
$$ \beta_{t}^{*} = F^{*} \beta_{t - 1}^{*} + v_{t}^{*} $$
(15)

where

$$ y_{t} = DGas_{t} ,X_{t - 1}^{*} = \left( {\begin{array}{*{20}c} 1 & {DOil_{t - 1} } & 1 \\ \end{array} \begin{array}{*{20}c} 0 & 1 \\ \end{array} } \right) $$
$$ \beta_{t}^{*} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\mu_{t} } \\ {\beta_{t} } \\ {\psi_{t} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\psi_{t}^{*} } \\ {\varepsilon_{t}^{*} } \\ \end{array} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\beta_{0t} } \\ {\beta_{1t} } \\ {\beta_{2t} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\beta_{3t} } \\ {\beta_{4t} } \\ \end{array} } \\ \end{array} } \right),v_{t}^{*} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\omega_{0t} } \\ {\omega_{1t} } \\ {\kappa_{t} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\kappa_{t}^{*} } \\ {\varepsilon_{t}^{*} } \\ \end{array} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {v_{0t} } \\ {v_{1t} } \\ {v_{2t} } \\ \end{array} } \\ {\begin{array}{*{20}c} {v_{3t} } \\ {\varepsilon_{t}^{*} } \\ \end{array} } \\ \end{array} } \right),F^{*} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & {\rho cos\left( {\lambda_{c} } \right)} \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ {\rho sin\left( {\lambda_{c} } \right)} & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} { 0} & 0 & { - \rho sin\left( {\lambda_{c} } \right)} \\ { 0} & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} {\rho cos\left( {\lambda_{c} } \right)} & 0 \\ 0 & 0 \\ \end{array} } \\ \end{array} } \right) $$

\(v_{t}^{*} \sim N\left( {0_{5} ,Q_{t}^{*} } \right)\) with \(0_{5} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ \end{array} } \right)\) and \( Q_{t}^{*} = E\left[ {\left. {v_{t}^{*} v_{t}^{*{\prime}} } \right|{\mathcal{F}}_{t - 1} } \right] = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\sigma_{0}^{2} } & 0 & 0 \\ 0 & {\sigma_{1}^{2} } & 0 \\ 0 & 0 & {\sigma_{2}^{2} } \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {0 } & {0 } & 0 \\ {0 } & {0 } & 0 \\ \end{array} } & {\begin{array}{*{20}c} {\sigma_{2}^{2} } & 0 \\ 0 & {h_{t} } \\ \end{array} } \\ \end{array} } \right)\).

Let’s now introduce some conditional expectations and variances and other necessary computations, where the symbosl \(\boldsymbol{^{\prime}}\) denotes the transposition operator:

  • The Expected value of \({\beta }_{t}^{*}\) conditional on the information set available up to (t-1) writes as \({\beta }_{\left.t\right|t-1}^{*}={F}^{*}{\beta }_{\left.t-1\right|t-1}^{*}\)

  • The Covariance matrix of \({\beta }_{t}^{*}\) conditional on the information set available up to (t-1) writes as \({P}_{\left.t\right|t-1}^{*}={F}^{*}{P}_{\left.t-1\right|t-1}^{*}{F}^{*{\prime}}+{Q}_{t}^{*}\)

  • The Prediction error writes as \({\eta }_{\left.t\right|t-1}^{*}={y}_{t}-{{X}_{t-1}^{*}\beta }_{\left.t\right|t-1}^{*}\)

  • The Conditional variance of the prediction error writes as \({f}_{\left.t\right|t-1}^{*}={X}_{t-1}^{*}{P}_{\left.t-1\right|t-1}^{*}{X}_{t-1}^{*{\prime}}\)

  • The Expected value of \({\beta }_{t}^{*}\) conditional on the information set available up to time (t) writes as \({\beta }_{\left.t\right|t}^{*}={\beta }_{\left.t\right|t-1}^{*}+{P}_{\left.t\right|t-1}^{*}{X}_{t-1}^{*{\prime}}{f}_{\left.t\right|t-1}^{*-1}{\eta }_{\left.t\right|t-1}^{*}\)

  • The Covariance matrix of \({\beta }_{t}^{*}\) conditional on the information set available up to (t) writes as \({P}_{\left.t\right|t}^{*}={P}_{\left.t\right|t-1}^{*}-{P}_{\left.t\right|t-1}^{*}{X}_{t-1}^{*{\prime}}{f}_{\left.t\right|t-1}^{*-1}{X}_{t-1}^{*}{P}_{\left.t\right|t-1}^{*}\)

The log-likelihood function (ln L) to be maximized then writes:

$$ \ln L = - \frac{1}{2}\mathop \sum \limits_{t = 1}^{T} ln\left( {2\pi f_{\left. t \right|t - 1}^{*} } \right) - \frac{1}{2}\mathop \sum \limits_{t = 1}^{T} \eta_{\left. t \right|t - 1}^{*{\prime}} f_{\left. t \right|t - 1}^{* - 1} \eta_{\left. t \right|t - 1}^{*} $$
(16)

where T is the sample size.

For regimes 2 and 6, Eqs. (3), (4), (5), (7), and (8) can be summarized in a general multivariate setting as follows (i.e., endogenous trend and periodic components, and an additional exogenous component):

$$ y_{t} = X_{t - 1}^{*} \beta_{t}^{*} + d\,\, Dummy_{t} $$
(17)
$$ \beta_{t}^{*} = F^{*} \beta_{t - 1}^{*} + v_{t}^{*} $$
(18)

where all notations and definitions remain the same as above except for the prediction error which writes now as \({\eta }_{\left.t\right|t-1}^{*}={y}_{t}-{{X}_{t-1}^{*}\beta }_{\left.t\right|t-1}^{*}-\) \({d}\,\,{Dummy}_{t}\). The log-likelihood function (ln L) to be maximized is still expressed as in Eq. (16). Note that, when a cyclical component is included in the measurement Eqs. (2) and (3), the corresponding period estimate is 6 months.

1.4 Summary of the data analysis and econometric study

The Table 12 below summarises the various tests and econometric models carried out on the data sample. As the data exhibit structural breaks and therefore regime shifts, the econometric model is estimated in each regime.

Table 12 Summary of the data analysis steps and econometric models

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Gatfaoui, H. On the relationship between U.S. crude oil and natural gas for economic resilience prospects. Ann Oper Res (2024). https://doi.org/10.1007/s10479-024-06411-9

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