Modeling the Natural Gas Spot-futures Markets as a Regime Switching Vector Error Correction Model

Abstract

While using the jump diffusion system controlled by the Markov-switching technique, this study designs a generalized dynamic conditional correlation (GDCC) time-varying transition probabilities (TVTP) Markov-switching vector error correction model (MRS-VECM) in which the parameters of the error correction term and the spot-futures correlations switches according the phase of volatility regime to reexamine the dynamics of the relationship between the natural gas spot and futures prices. The data used in this study is unleaded gasoline traded on the New York Mercantile Exchange, covering the period November 2, 1994 to June 2, 2009, resulting in 3,642 daily observations. Our empirical results are consistent with the following notions. First, the gasoline market reacts more quickly to the spot-futures price discrepancy when a high variance regime is encountered. Second, the spot-futures correlations vary considerably across various variance states, and the correlation for the high variance state is significantly lower than that of the low variance state. Third, the state-varying hedge ratios by our GDCC-TVTP-MRS-VECM could provide a more efficient futures hedging strategy when compared to conventional hedging models, such as the naïve, VECM, constant conditional correlation multivariate generalized ARCH (CCC-MVGARCH) and dynamic conditional correlation multivariate generalized ARCH (DCC-MVGARCH) models. Finally, the lagged spot-futures price deviation functions as an indicator for the variance-turning process.

Keywords:

• dynamic conditional correlation
• Markov-switching
• natural gas
• volatility

Notes

¹For convenience, we assume p = q in Eqs. (2) and (3).

²In theory, when spot prices differ from futures prices, arbitrage trading will be triggered: simultaneous short-selling of spot positions and purchasing of futures positions when the mispricing term, z _t−1, is negative, the opposite being the case when z _t−1 is positive. Clearly, this price adjustment process implied by such arbitrage behaviors causes spot and futures prices to move in opposite directions, and thus reduces the degree of the co-movements between them. More specifically, the ω parameter in Eq. (9) should have negative sign.

³If spot and futures prices differ, arbitrage trading between the spot and futures markets will be triggered regardless of whether the price deviation is positive or negative. Consequently, this study adopts the absolute value of spot-futures price deviation as an indicator of market variances.

⁵The naïve hedging strategy is to set the hedge ratio as unity.

⁶The MVGARCH model specification is well documented in the literature. Therefore, this study omits any discussion of the details and refers readers to Bollerslev (1990), Chan et al. (1991) and Engle (2002).

⁴Please refer to Li (2009) for the related discussions.

⁷The estimation procedure for the MRS models is well documented in the literature. Therefore, this study omits any discussion of estimation and refers readers to Hamilton (1989).

⁸Even with this simple structure involving one lagged ARCH component, there are 24 parameters that require estimation. A more general structure with a higher-order ARCH term could increase the number of parameters to be estimated. Furthermore, similarly to Hamilton and Susmel (1994), this study also found that the higher-order ARCH parameter estimates do not differ significantly from 0 after filtering out the variance-switching process. To save space, this study does not report the results of the higher lag order setting.

⁹The algorithm of Boyden, Fletcher, Goldfarb, and Shanno can effectively yield the maximum value of the non-linear likelihood functions, as demonstrated by Luenberger (1984).

¹⁰This study uses four-order lag lengths for the augmented Dickey-Fuller (ADF) test. Additionally, the conclusion from the unit root and co-integration tests is robust for the setting with various lag length numbers.

* denotes the significance at the 1% level. The empirical results indicate that both series of futures and spot prices are nonstationary in all cases. However, the logarithmic first difference of stock price including futures and spot is stationary. Additionally, the cointegration test indicates that the EC term, namely z_t , of the futures and spot prices is stationary in all cases. Consequently, the cointegration relationship between the futures and spot process holds. The data source is consistent with Table 1.

* and ** denote the significance at the 5% and 1%, respectively. # denotes the estimate is significantly greater than unity at the 1% significance level. The data source is consistent Table 1.

¹¹Notably, with the constraint of g ^f ₁ = g ^f ₂ = 1 and g ^s ₁ = g ^s ₂ = 1, one would encounter an unidentified problem under null hypothesis when testing the model. Specifically, the conventional LR statistic under this condition no longer follows the standard χ² distribution. Hansen (1992; 1996) proposes a bound test that addresses these problems, but its computational difficulty has limited its applicability. See Hansen (1992; 1996) for a detailed explanation of these problems. The Hansen test is not reported in this study, but the present results show that the 99% confidence levels of estimates g ^f ₂ and g ^s ₂ do not overlap with unity, the value of g ^f ₁ and g ^s ₁, and thus the two measures of volatility differ significantly from one another.

¹²Please refer to Li (2007; 2008a) for the related discussions.

¹³To test the null hypothesis of identical intercept involved in the sopt-futures correlations, our GDCC-TVTP-MRS-VECM is first estimated on the basis of a two-state intercept and L(H _A ), representing the log likelihood function. The model is then estimated assuming the existence of a single constant intercept (π₁ = π₂ = π), which allows for the subsequent derivation of the log likelihood function of the restricted model, L(H ₀). Finally, this function is used to carry out a likelihood ratio test, LR = −2[L(H ₀)– L(H_A )]. In terms of the null hypothesis, this test displays a χ² distribution with 3 (= 4 – 1) degrees of freedom. Further, the null hypothesis of π₁ = π₂₂ = π, a single intercept, is rejected at the 1% level.

¹⁴Please refer to Li (2008b; 2010) for the related discussions.

¹⁵The naïve hedging strategy is to set the hedge ratio as unity.

¹⁶The MVGARCH model specification is well documented in the literature. Therefore, this study omits any discussion of the details and refers readers to Bollerslev (1990) and Chan et al. (1991).

¹⁷Since the estimation procedure for the MRS models is well documented in the literature, this study omits related discussion of estimation and refers readers to Hamilton (1989). However, the raw data and the computer programs and codes used in this study are available upon request.

Notes: * denotes the minimum value in the column.