# Journal of Energy Markets

**ISSN:**

1756-3607 (print)

1756-3615 (online)

**Editor-in-chief:** Derek W. Bunn

# The convenience yield implied in the European natural gas markets: the impact of storage and weather

####
Need to know

- Determination of the convenience yield implied in the European natural gas markets via an option-based approach
- Investigation of driving factors and according dynamics.
- EGARCH model to explain the convenience yield via storage and weather
- The release of natural gas storage levels generates considerably volatility

####
Abstract

**ABSTRACT**

This paper determines the convenience yield implied in the European natural gas markets and investigates driving factors and according dynamics. For this, we approximate the convenience yield via an option-based approach, in which the convenience yield is determined as the difference between two average floating-strike Asian options written on the spot and futures contracts. In a second step, we fit an exponential generalized autoregressive conditional heteroscedasticity (EGARCH) model to explain the convenience yield via storage and weather as well as other key driving factors. The empirical analysis reveals distinct results for the impact of storage, which indicates that the release of natural gas storage levels generates considerable volatility. Further, we illustrate that the mixed evidence and the absence of a clear pattern to explain the convenience yield's characteristics underline the transitory state of European natural gas markets.

####
Introduction

The literature dealing with the question of which factors drive prices of storable commodities is exhaustive, but nowhere near as many papers deal with the question of which factors affect the volatility of the convenience yield implied in these markets.

In this paper, we study the short-term price dynamics of the convenience yield implied in the European natural gas futures market. More particularly, we examine how the volatility and mean are influenced by storage, weather variables and market factors. Mu (2007), for example, shows that weather affects about 50% of the US natural gas demand. He concludes that weather has to be the most important factor that drives short-term natural gas demand.^{1}^{1}Mu (2007) draws this conclusion from the finding that industrial demand for natural gas does not vary to a great extent in the short-run; thus, weather (and to a lesser extent storage) is the most important factor in explaining the rationale behind short-term gas demand variations. In addition, he argues that in an efficient commodity market, where demand is highly volatile, storage information or storage changes may crucially affect the market’s demand and supply. This effect is closely linked to the fact that unexpected changes in the demand for natural gas create severe uncertainty about the future state of the market and, thus, affect the market price of risk (see, for example, Cartea and Williams 2008). It is also analyzed and assessed by Linn and Zhu (2004), who highlighted that the release of natural gas storage information generates considerably higher volatility levels compared with other periods. Based on these findings for the US natural gas market, we apply a similar concept to that outlined in Mu (2007) in order to analyze the factors affecting the convenience yield implied in the European natural gas markets.

Upfront, the question regarding the appropriate method for determining the convenience yield has to be answered.^{2}^{2}For an exhaustive discussion including the technical derivation, we refer the interested reader to Dockner et al (2015). For this, the properties of the underlying commodity have to be analyzed. In general, the determination of the convenience yield is a rather old concept and traces back to Working (1949). In its basic form, it deals with the benefits of owning a commodity in inventory instead of purchasing it every time it is needed, but it can also be seen as the stochastic dividend paid on the spot price of a certain commodity. In general, these benefits are not directly observable; thus, they are often hard to determine. Hence, in addition to the “Theory of Storage”, several different concepts dealing with the convenience yield approximation came up that control for certain shortcomings and restrictions in the markets.

The oldest, and denoted as “traditional”, approach to approximating the convenience yield links spot and futures prices as follows. Investors are indifferent as to whether they

- (1)
buy the commodity at the spot price and hold it by paying the storage costs and receiving the convenience yield, or

- (2)
enter a long position in a futures contract and invest in a risk-free bond.

This no-arbitrage situation is in line with the cost-of-carry approach and can be used to calculate the implied convenience yield. Pindyck (1993, 2001) provides a detailed overview on the application of this approach. As mentioned in Hochradl and Rammerstorfer (2012), the disadvantage of this approach becomes apparent as soon as, for example, market constraints hinder the implementation of arbitrage-exploiting trading strategies. In this case, a more direct way to estimate the convenience yield would be preferable. Here, models that apply option-pricing techniques, as, for example, those mentioned in Heinkel et al (1990), Milonas and Thomadakis (1997), Heaney (2002), Hochradl and Rammerstorfer (2012) and more recently Dockner et al (2015), come into play.

Based on these studies, this paper contributes to the existing literature in several ways. First, we give a detailed overview of the different methods of calculating the convenience yield and highlight the pros and cons of these different approaches. This leads to a clear guide to identifying the correct model for a certain market. Second, we provide insights into the dynamics of the convenience yield and the most important driving factors that allow traders to specify their expectations and market forecasts. Third, we highlight that the impact of innovations on the convenience yield’s volatility is different for positive and negative shocks, so a model that allows for asymmetries has to be applied. Finally, we shed light on the degree of efficiency in Europe and the appropriate model to determine the convenience yield in a transitioning market.^{3}^{3}All natural gas trading places in Europe are rather new, as the wholesale market for natural gas has a much shorter history here than, for example, in the United States. In Europe, the market opening traces back to the EU Directive 98/30/EC, implemented in 1998; so, the European wholesale market for natural gas can still be considered as a developing market and, in addition, has to deal with restricted storage access and pipeline capacities.

Therefore, the remainder of this paper is as follows. In Section 2, the different approaches for the convenience yield approximation are given. Section 3 describes the hubs and data with respect to the time series properties. Section 4 covers exponential generalized autoregressive conditional heteroscedasticity (EGARCH) modeling, which gives insights into the factors influencing the variance and mean of the convenience yield implied in the European gas markets. Section 5 gives the results to the empirical estimation and provides further explanations. Finally, Section 6 summarizes and concludes.

## 2 Review of the different approaches of convenience yield modeling: a guide through the literature

In the following, we offer insights into the theoretical background of the four different approaches to modeling the convenience yield. We first recall the traditional approach to approximating the convenience yield. The second approach is based on the well-known approximation provided in Heaney (2002), in which, under the assumption of perfect foresight, the investor is able to determine the convenience yield as the difference between two floating-strike lookback put options, written on the underlying commodity spot or futures contracts. For the third approximation, we refer to the convenience yield model given in Hochradl and Rammerstorfer (2012), in which the idea of Heaney (2002) is extended by relaxing the assumption of an investor’s perfect foresight. Further, we describe the approach provided in Dockner et al (2015), in which the option-based models are extended and allow the commodity price series to be mean reverting.

According to the Theory of Storage, which is exhaustively discussed in Kaldor (1939), Working (1948, 1949), Brennan (1958), Telser (1958) and Weymar (1966), the arbitrage-free futures price ${F}_{t,T}$ is equal to the cost of buying and storing the underlying minus the benefits of owning it. As storage cost data is often not available, it is hardly possible to estimate the convenience yield via the traditional approach; besides, zero costs are expected. Hence, often an indirect convenience yield test is processed, such as, for example, that given in Fama and French (1987) or Neumann et al (2008). However, some situations require an approximation of the convenience yield in a more direct way. One way of doing this is given by the option-based approaches.

In contrast to the traditional approach, the approximations given in Heaney (2002), Hochradl and Rammerstorfer (2012) and Dockner et al (2015) are based on option-pricing techniques. In Heaney (2002), it is assumed that the underlying commodity follows a geometric Brownian motion (GBM). Heaney (2002) assumes that the investor has perfect foresight, is able to sell the asset at its highest price during the time period $[t,T]$ and invests the proceeds at the risk-free rate. A rational investor would only forego this opportunity to sell the asset at its highest price if the ownership of the asset provided additional benefits. This is equivalent to the payoff of a floating-strike lookback put option, written on the spot product. Following no-arbitrage pricing techniques, the time $t$ value of this option is given by the discounted risk-neutral expected value of this payoff. For this, Levy (1997) provides a closed-form solution. For an investor who owns a futures contract (as opposed to owning the spot product), a similar line of reasoning leads to the discounted risk-neutral expected time $t$ value of the option. Following Heaney (2002), the benefits of holding the physical commodity that are not obtained by buying a futures contract, ie, the convenience yield, is the difference between the two lookback put options written on the spot and futures contracts.

Hochradl and Rammerstorfer (2012) go a step further and neglect Heaney’s assumption that investors have perfect foresight. They assume instead that investors can only trade at average prices.^{4}^{4}In line with Heaney (2002), Hochradl and Rammerstorfer (2012) assume that the underlying commodity follows a GBM. Their result is equivalent to the payoff structure of a geometric average floating-strike Asian put option. Following the same line of reasoning as in Heaney (2002), the convenience yield is given as the difference between a floating-strike option written on the spot and futures products. However, if commodity prices are mean reverting, the convenience yield approximations derived by Heaney (2002) or Hochradl and Rammerstorfer (2012) cannot be applied. In this context, Dockner et al (2015) control for this property, as they introduce arithmetic mean floating-strike Asian put options.

Overall, each of these approaches to approximating the convenience yield has several advantages and shortcomings, which are summarized in Table 1.

Asian, | Asian, | |||
---|---|---|---|---|

geometric | arithmetic | |||

Feature | Traditional | Lookback | mean | mean |

Underlying GBM | — | x | x | — |

Underlying mean reverting | — | — | — | x |

Direct approach | — | x | x | x |

Maximum level | — | x | — | — |

Storage data necessary | x | — | — | — |

Ex post analysis | x | x | x | x |

Ex ante analysis | — | x | x | x |

Not every approach perfectly suits each product. In general, the traditional approach is widely accepted among practitioners and researchers, while the option-based approaches are more important for trading and hedging purposes. The main shortcoming of the lookback option-based approach as mentioned in Hochradl and Rammerstorfer (2012) is the unrealistic assumption that investors have perfect foresight. However, the model developed by Hochradl and Rammerstorfer (2012) is not exempt from criticism either, as it restricts the underlying price process to follow a GBM, which may not be the case for several commodities.^{5}^{5}Bessembinder et al (1995), for example, analyze the term structure of commodity futures prices for the period 1982–91. They find that mean reversion is present for agricultural commodities and crude oil. In a more recent study, Pindyck (2001) undermines this finding with respect to energy commodities, although he cannot confirm this for natural gas. However, Geman (2007) states that the evidence is mixed when energy commodities are considered, and it strongly depends on the considered horizon. This criticism can be overcome by allowing the prices to mean revert, which is addressed in Dockner et al (2015). In general, the choice of a certain model depends on the statistical properties of the underlying commodity. In the following empirical analysis, we approximate the convenience yield implied in the European natural gas market. For the considered market, the traditional approach is not useful, as storage access is restricted and cost data is not readily available. Further, Heaney’s perfect foresight approach does not provide realistic insights. The empirical analysis of the underlying properties deduced from the basic augmented Dickey–Fuller (ADF) test argues in favor of a GBM, such that we prescind from the approach developed in Dockner et al (2015) and, thus, decide in favor of the geometric floating-strike Asian option-based approach.

## 3 Prices, convenience yield and data series properties

In Europe, natural gas is traded at three major trading hubs:^{6}^{6}Market opening traces back to the EU Directive 98/30/EC, implemented in 1998, so the European wholesale market for natural gas can still be considered as a developing market. the National Balancing Point in the United Kingdom (NBP, since 1996), the Zeebrugge hub in Belgium (ZEE, since 2000) and the Title Transfer Facility in the Netherlands (TTF, since 2003). NBP has been connected to ZEE through the Interconnector pipeline since October 1998, and to TTF through the Bacton-Balgzand pipeline since December 2006.^{7}^{7}The general idea of establishing transmission pipelines between NBP and the two European mainland gas trading hubs (ZEE and TTF) is to limit the potential price differential across hubs. According to Hochradl and Rammerstorfer (2012), these price differences should be limited to a no-arbitrage band, which depends on the transmission capacity as well as on the costs of gas transportation. Due to the importance of these hubs for the European market, the data from NBP, ZEE and TTF forms the basis for the following analysis of the convenience yield in the European natural gas market.

NBP is operated by the Transmission System Operator (TSO) National Grid, which covers all entry and exit points in mainland Britain. It is the most liquid trading point in Europe and constitutes the delivery point for certain contracts, traded at the Intercontinental Exchange (ICE). As mentioned above, ZEE is connected to NBP via the Interconnector. It is operated by Hubator, a 100% subsidiary of the Belgian TSO Fluxys, and its gas futures contracts are traded at the ICE Endex. The Dutch TSO Gas Transport Services operates TTF, in which natural gas is traded within the Dutch network. TTF gas futures are also traded at the ICE.

For all hubs, the most liquid futures are considered, ie, futures with one- to three-month maturities, apart from TTF; here, only futures with one or two months are included. The data consists of end-of-day mid-quotes provided by Platts Power Vision. Due to constraints in data availability, the evaluation period is restricted to the following ranges: October 1, 1996–December 31, 2010 for NBP; January 4, 2000–December 31, 2010 for ZEE; and April 1, 2004–December 31, 2010 for TTF.

### 3.1 Evolution of spot and futures prices

For all trading places, we see a rather similar evolution of prices and peaks.^{8}^{8}In order to see the similar evolution of TTF and the other hubs, you have to convert the Euro/MWh into pence per therm units. Markets are mainly in contango, as in general futures prices exceed spot prices. We see that, around the change of the gas year in September, the integration between spot and futures prices becomes weaker, ie, the close evolution of spot and futures prices across maturities and hubs widens. Further, the price differences across hubs decrease over time and toward the end of the sample period. The importance of the connecting pipelines can be seen directly in the periods of maintenance (using the example of the Interconnector pipeline), ie, from July 17 to 25, 2000; September 17 to 26, 2001; August 31 to September 9, 2002; September 8 to 22, 2003; September 13 to 28, 2004; July 4 to 19, 2005; June 5 to 20, 2006; September 3 to 17, 2007; August 27 to September 10, 2008; September 8 to 23, 2009; and September 7 to 24, 2010, when the pipeline was out-of-service.^{9}^{9}A similar pattern is observable for the maintenance periods of Bacton-Balgzand. In the periods of maintenance, ie, from August 31 to September 14, 2009, and from September 1 to 15, 2010, the spot price differentials widened. At these dates, the spot price differentials between NBP and ZEE widened immediately and turned back to their previous level once maintenance was completed.

### 3.2 Integration of price series

In order to decide between the mean-reverting and the GBM-option-based approaches, we apply the ADF test (see Dickey and Fuller 1979) on the basic price series.^{10}^{10}This is in line with Hochradl and Rammerstorfer (2012), who applied a similar specification to test whether European natural gas markets are integrated or not. The lag length for the ADF test is selected by the Akaike information criterion (AIC), whereas for the Phillips–Perron (PP) test we used the Parzen kernel to determine the probability density function and the Newey–West bandwidth for lag selection.

Table 2 gives the results of the ADF test in levels and first differences. The null hypothesis that a certain price series has a unit root cannot be rejected for any price series in levels. When taking first differences, the null hypothesis is rejected for all price series. Consequently, the series are not mean reverting but integrated with order one.^{11}^{11}Further, we implemented the PP test as well as the classical approach (${X}_{t+1}=\rho {X}_{t}+{\u03f5}_{t}$), as mentioned in Geman (2007), to test whether series of prices are mean reverting. Overall, we find the PP test confirms the findings of the ADF test. In addition, the classical approach proposed by Geman (2007) highlighted that $\rho $ is close or equal to 1, which indicates the existence of a unit root. Thus, the process of the considered prices follows a random walk.

NBP | ZEE | TTF | ||||
---|---|---|---|---|---|---|

Series | Levels | Differences | Levels | Differences | Levels | Differences |

Spot | $-$0.9696 | $-$15.9830*** | $-$0.7697 | $-$14.0151*** | $-$0.2871 | $-$13.4236*** |

1m | $-$0.9908 | $-$10.9592*** | $-$0.4953 | $-$35.0197*** | $-$0.2859 | $-$22.4150*** |

2m | $-$1.1197 | $-$9.5961*** | $-$0.5009 | $-$20.7815*** | $-$0.2851 | $-$14.4634*** |

3m | $-$1.3374 | $-$8.8756*** | $-$0.9077 | $-$8.0802*** | — | — |

Thus, for approximating the convenience yield, we refer to the convenience yield as the difference between two lookback put options.

### 3.3 Convenience yield approximation

In the following, we highlight the outcomes of the convenience yield approximation for the one-month yield exemplarily.

First, the signs of the estimates are nearly always positive for all hubs. Second, the convenience yield estimates for each trading location peak at almost the same yield level and tend toward zero as the maturity of the futures contracts expires (some minor exemptions). Following Dockner et al (2015), this pattern is to be expected; it corresponds to the constant volatility of the underlying price process and the forward-looking behavior implicitly present in the lookback option approach. This model further shows the reducing benefits of holding the physical commodity as time to maturity decreases.

In a second step, we analyze the properties and dynamic behavior of the convenience yield’s process in order to determine the appropriate empirical model. For all trading places considered (see Table 3), the calculated average convenience yield and the median exhibit positive values. We observe that excess kurtosis is positive, which corresponds to a leptokurtic convenience yield distribution. Further, the convenience yield is skewed and concentrated on the right tail of the distribution. It should also be mentioned that the convenience yield’s volatility exceeds the mean for all considered contract maturities and, therefore, indicates that the European natural gas markets are rather volatile. Again, and in line with the data described earlier, the convenience yields implied in NBP and ZEE are closely related (similar amounts and volatility), while the convenience yield implied in TTF evolves a bit more distinctively, especially at the beginning of the sample period.

Hub | Series | Nrobs | Mean | Median | Max | Min | Std Dev | Skew | Kurt | JB |
---|---|---|---|---|---|---|---|---|---|---|

NBP | ${\text{CY}}_{\text{1m}}$ | 3579 | 0.0122 | 0.0028 | 0.5858 | $-$0.1701 | 0.0471 | 4.3542 | 35.6814 | 170 585 |

${\text{CY}}_{\text{2m}}$ | 3579 | 0.0103 | 0.0029 | 0.7636 | $-$0.2562 | 0.0603 | 2.2748 | 25.0747 | 75 754 | |

${\text{CY}}_{\text{3m}}$ | 3579 | 0.0100 | 0.0044 | 0.6431 | $-$0.4954 | 0.0756 | $-$0.2732 | 13.8777 | 17 689 | |

ZEE | ${\text{CY}}_{\text{1m}}$ | 2767 | 0.0095 | 0.0021 | 0.5001 | $-$0.1463 | 0.0447 | 4.4906 | 38.0066 | 150 584 |

${\text{CY}}_{\text{2m}}$ | 2767 | 0.0137 | 0.0035 | 0.6388 | $-$0.2170 | 0.0536 | 2.7013 | 21.8821 | 44 470 | |

${\text{CY}}_{\text{3m}}$ | 2767 | 0.0097 | 0.0045 | 0.5444 | $-$0.4719 | 0.0612 | 0.1731 | 18.9531 | 29 355 | |

TTF | ${\text{CY}}_{\text{1m}}$ | 1697 | 0.0060 | 0.0009 | 0.3000 | $-$0.1228 | 0.0361 | 3.9917 | 29.5472 | 54 338 |

${\text{CY}}_{\text{2m}}$ | 1697 | 0.0046 | 0.0007 | 0.4512 | $-$0.2218 | 0.0434 | 2.6462 | 30.0394 | 53 677 |

The according autocorrelation patterns for the convenience yield series as well as the squared convenience yield series strongly indicate the existence of a time-varying volatility, which needs to be taken into account for the empirical analysis on factors driving the convenience yield (see, for example, Hochradl and Rammerstorfer 2012). The approximated convenience yields are far from being normally distributed, and they follow a mean-reverting pattern, which is also indicated by the ADF test (see Table 4).^{12}^{12}Further, we applied the PP test, which confirms the findings from the ADF test. For deeper insights into the discussion on mean-reversion testing in the context of commodity prices and convenience yield, see, for example, Geman (2005).

Levels | |||
---|---|---|---|

ADF | ${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ |

NBP | $-$19.4530*** | $-$13.1001*** | $-$10.0505*** |

ZEE | $-$16.0772*** | $-$9.8918*** | $-$7.8640*** |

TTF | $-$10.5780*** | $-$8.6394*** | — |

### 3.4 Modeling time-varying volatility in convenience yield series: discussion on the appropriate model

Recall that the significant autocorrelation patterns in levels and squared levels for all considered trading hubs strongly indicate the existence of a time-varying volatility; thus, the implementation of an autoregressivemoving-average model, ARMA$(p,q)$, or a generalized autoregressive conditional heteroscedasticity model, GARCH$(p,q)$, is crucial in capturing the dynamic behavior of the underlying convenience yield series (see, for example, Gibson and Schwartz 1990; Schwartz 1997). In line with the existing literature on nonconstant volatility in financial data, we applied the following test procedure to determine the appropriate empirical model for the convenience yield estimates. In a first step, we implemented an ARMA$(p,q)$ ordinary least squares (OLS) regression on the convenience yield series to determine whether this model is able to capture the characteristics of the convenience yield. In a second step, we implemented a GARCH$(p,q)$ model for the considered time series and hubs to model the nonconstant volatility property of the time series.^{13}^{13}Brooks (2008) states that a GARCH(1,1) model is sufficient to capture all the volatility clustering in the data. Nevertheless, we implement GARCH$(p,q)$ models up to the values $\mathrm{p}\leqq 4$ and $\mathrm{q}\leqq 4$, as this provides enough flexibility and parsimony to determine the appropriate characteristics of the data. We choose the appropriate order of the GARCH$(p,q)$ model by comparing the maximum value of the likelihood functions, as well as applying a diagnostic test to the residuals. Mazaheri (1999) and Carpantier and Dufays (2013) expect asymmetric responses of the volatility to negative and positive convenience yield shocks, whereas positive shocks are more likely to cause higher volatility levels. Consequently, the basic GARCH$(p,q)$ representation is not able to capture the dynamic behavior of the volatility (see Brooks 2008). Following Nelson (1991), the selection of the appropriate volatility model crucially depends on the processing of innovations on news, which may be processed in a symmetric or asymmetric way. For this, Engle and Ng (1993) propose four diagnostic tests to analyze asymmetries in the volatility. In these, the squared residuals (${\widehat{\u03f5}}_{t}^{2}$) of the GARCH$(p,q)$ fit are regressed on a dummy variable that takes 1 if $$ and $0$ otherwise.^{14}^{14}Engle and Ng (1993) suggested four diagnostic tests for volatility models: the sign bias test, ${\widehat{\u03f5}}_{t}^{2}={\alpha}_{0}+{\alpha}_{1}{\mathrm{Dummy}}_{t-1}+{u}_{t}$; the negative size bias test, ${\widehat{\u03f5}}_{t}^{2}={\alpha}_{0}+{\alpha}_{1}{\mathrm{Dummy}}_{t-1}*{\u03f5}_{t-1}+{u}_{t}$; the positive size bias test, ${\widehat{\u03f5}}_{t}^{2}={\alpha}_{0}+{\alpha}_{1}(1-{\mathrm{Dummy}}_{t-1})*{\u03f5}_{t-1}+{u}_{t}$; and the joint bias test, ${\widehat{\u03f5}}_{t}^{2}={\alpha}_{0}+{\alpha}_{1}{\mathrm{Dummy}}_{t-1}+{\alpha}_{2}{\mathrm{Dummy}}_{t-1}*{\u03f5}_{t-1}+{\alpha}_{3}(1-{\mathrm{Dummy}}_{t-1})*{\u03f5}_{t-1}+{u}_{t}$. The variance model is misspecified if the included variables are significant and, thus, able to predict the squared residuals (${\widehat{\u03f5}}_{t}^{2}$). For all trading hubs considered, the test results argue in favor of asymmetric responses of the volatility to negative and positive shocks; thus, they support the implementation of an asymmetric GARCH model such as the EGARCH. The results of the above test procedure are given at length in Appendix 2 (available online).

## 4 Exponential generalized autoregressive conditional heteroscedasticity modeling: driving factors, storage and weather

The conditional variance equation of the proposed EGARCH model to overcome the asymmetry restriction in the standard GARCH setting is given by (see Nelson 1991)

$$\mathrm{log}({\sigma}_{t}^{2})=\omega +\sum _{j=1}^{q}{\beta}_{j}\mathrm{log}({\sigma}_{t-j}^{2})+\sum _{i=1}^{p}{\alpha}_{i}\left(|\frac{{\u03f5}_{t-i}}{{\sigma}_{t-i}}|-E|\frac{{\u03f5}_{t-i}}{{\sigma}_{t-i}}|\right)+\sum _{k=1}^{r}{\gamma}_{k}\frac{{\u03f5}_{t-k}}{{\sigma}_{t-k}},$$ | (4.1) |

where $\mathrm{log}({\sigma}_{t}^{2})$ is the natural logarithm of the conditional variance, $\omega $ represents the constant and $\beta $ denotes the parameter of previous lags of the natural logarithm of the conditional variance. Further, $\alpha $ represents the sizes and $\gamma $ represents the sign effect; thus, they capture the asymmetry relation between returns and volatility changes.^{15}^{15}Following Franke et al (2010) and interpreting the model according to Nelson (1991), the EGARCH model contains two parameters that are defined as the “size effect” and the “sign effect” to accommodate the asymmetric relation between returns and volatility changes. In line with Mazaheri (1999), a nonzero and significant $\gamma $ provides evidence of an asymmetric effect in the volatility.

In this section, we determine the appropriate mean and variance equations of the EGARCH model, as well as possible driving factors of the convenience yield. For this, we distinguish storage and weather variables, as well as other driving factors, which allow us to deduce the impact of the different influencing factors on the mean and variance of the convenience yield.

Therefore, we estimate the following EGARCH model by employing the maximum likelihood procedure based on the Gaussian distribution:

$$\begin{array}{cc}\hfill {y}_{t}& ={\alpha}_{0}+{\alpha}_{1}{W}_{t}^{\mathrm{CDD}}+{\alpha}_{2}{W}_{t}^{\mathrm{HDD}}+{\alpha}_{3}{\mathrm{CRET}}_{t}+{\alpha}_{4}{R}_{m,t}\hfill \\ & +{\alpha}_{5}{\mathrm{SPRET}}_{t}+{\alpha}_{6}{\mathrm{CRB}}_{t}+{\alpha}_{7}{\mathrm{CCDum}}_{t}+{\epsilon}_{t},\hfill \end{array}$$ | ||

$${\epsilon}_{t}\mid {Q}_{t-1}\sim (0,{\sigma}_{t}),$$ |

to

$\mathrm{log}({\sigma}_{t}^{2})$ | $=\omega +{\displaystyle \sum _{j=1}^{q}}{\beta}_{j}\mathrm{log}({\sigma}_{t-j}^{2})+{\displaystyle \sum _{i=1}^{p}}{\alpha}_{i}\left(|{\displaystyle \frac{{\u03f5}_{t-i}}{{\sigma}_{t-i}}}|-E|{\displaystyle \frac{{\u03f5}_{t-i}}{{\sigma}_{t-i}}}|\right)+{\displaystyle \sum _{k=1}^{r}}{\gamma}_{k}{\displaystyle \frac{{\u03f5}_{t-k}}{{\sigma}_{t-k}}}$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}+{\delta}_{1}{W}_{t}^{\mathrm{CDD}}+{\delta}_{2}{W}_{t}^{{\mathrm{CDD}}^{2}}+{\delta}_{3}{W}_{t}^{\mathrm{HDD}}+{\delta}_{4}{W}_{t}^{{\mathrm{HDD}}^{2}}+{\delta}_{5}{\mathrm{SADum}}_{t}$ | ||||

$\mathrm{\hspace{1em}\hspace{1em}}+{\delta}_{6}{\mathrm{AUTDum}}_{t}+{\delta}_{7}{\mathrm{WINDum}}_{t},$ | (4.2) |

where ${y}_{t}$ denotes the convenience yield series, ${\epsilon}_{t}$ represents the error term, ${Q}_{t-1}$ is the information set available at time $(t-1)$ and $\mathrm{log}({\sigma}_{t}^{2})$ is the natural logarithm of the conditional variance of the error term resulting from the mean equation.^{16}^{16}We use two different models: one without storage level information that incorporates the whole sample period, and one that analyzes further the impact of storage levels on the implied convenience yield. For the latter, the variables in brackets replace the storage dummy variable in the variance equation: $\mathrm{log}({\sigma}_{t}^{2})=\omega +{\displaystyle \sum _{j=1}^{q}}{\beta}_{j}\mathrm{log}({\sigma}_{t-j}^{2})+{\displaystyle \sum _{i=i}^{p}}{\alpha}_{i}\left(|{\displaystyle \frac{{\u03f5}_{t-i}}{{\sigma}_{t-i}}}|-E|{\displaystyle \frac{{\u03f5}_{t-i}}{{\sigma}_{t-i}}}|\right)+{\displaystyle \sum _{k=1}^{r}}{\gamma}_{k}{\displaystyle \frac{{\u03f5}_{t-k}}{{\sigma}_{t-k}}}+{\delta}_{1}{W}_{t}^{\mathrm{CDD}}+{\delta}_{2}{W}_{t}^{{\mathrm{CDD}}^{2}}+{\delta}_{3}{W}_{t}^{\mathrm{HDD}}+{\delta}_{4}{W}_{t}^{{\mathrm{HDD}}^{2}}(+{\delta}_{5}{\mathrm{Storage}}_{\mathrm{Level},t}+{\delta}_{6}\mathrm{\Delta}{\mathrm{Storage}}_{\mathrm{Level},t})+{\delta}_{7}{\mathrm{AUTDum}}_{t}+{\delta}_{8}{\mathrm{WINDum}}_{t}.$ As storage data is not available for the whole period considered, the number of observations included will be reduced.

As explanatory variables, we include crude oil return (${\mathrm{CRET}}_{t}$) in order to cover possibly existing spillovers between oil markets and the energy-producing sector, which are expected to affect the convenience yield. Other variables in the mean equation are the cooling and heating degree variables, denoted by ${W}_{t}^{\mathrm{CDD}}$ and ${W}_{t}^{\mathrm{HDD}}$; the risk-free rate, given as ${R}_{m,t}$; the financial market return, abbreviated as ${\mathrm{SPRET}}_{t}$; a variable visualizing the overall market situation of commodities measured via ${\mathrm{CRB}}_{t}$; and a dummy for the financial crisis (${\mathrm{CCDum}}_{t}$), which is $1$ for the period of the financial crisis ranging from 2008 to 2010, and $0$ otherwise.

In the variance equation, the cooling and heating shock variables, the squared shock (${W}_{t}^{{\mathrm{CDD}}^{2}}$ and ${W}_{t}^{{\mathrm{HDD}}^{2}}$) and the storage announcement dummy (${\mathrm{SADum}}_{t}$) are included. The storage announcement dummy is $1$ on the day that storage information about actual storage levels is released, and $0$ otherwise. Further, we include dummies covering seasonalities coming from the winter (${\mathrm{WINDum}}_{t}$) and autumn (${\mathrm{AUTDum}}_{t}$) months.

For the test of storage effects, the model is enhanced by the storage levels (${\mathrm{Storage}}_{\mathrm{Level},t}$) and the change in storage levels ($\mathrm{\Delta}{\mathrm{Storage}}_{\mathrm{Level},t}$) to explain the convenience yield’s variance (see Section 5.2). For this second model, the sample period has to be shortened, as storage data is only consistently available for the period from September 2007 to September 2010.

In the following subsections, we provide deeper insights into the effects of these variables as well as a discussion in light of the existing literature. An overview of the convenience yield’s driving factors is provided in Table 5.

(a) Mean equation | |||

Influencing factor | Abbreviation | Sign | Literature |

Cooling degree shock | ${W}_{t}^{\text{CDD}}$ | + | Mu (2007), Mansanet-Bataller and Soriano (2009) |

Heating degree shock | ${W}_{t}^{\text{HDD}}$ | + | Mu (2007), Mansanet-Bataller and Soriano (2009) |

Crude oil return | ${\text{CRET}}_{t}$ | + | Brown and Yücel (2008) |

Risk-free rate | ${R}_{m,t}$ | – | Option theory |

Financial market return | ${\text{SPRET}}_{t}$ | – | Fama and French (1987), Bailey and Chan (1993) |

Commodity index | ${\text{CRB}}_{t}$ | + | McKenzie et al (2004) |

Credit crisis dummy | ${\text{CCDUM}}_{t}$ | – | |

(b) Variance equation | |||

Influencing factor | Abbreviation | Sign | Literature |

Cooling degree shock | ${W}_{t}^{\text{CDD}}$ | + | Mu (2007), Mansanet-Bataller and Soriano (2009) |

Squared cooling degree shock | ${W}_{t}^{{\text{CDD}}^{2}}$ | + | Mu (2007) |

Heating degree shock | ${W}_{t}^{\text{HDD}}$ | + | Mu (2007), Mansanet-Bataller and Soriano (2009) |

Squared heating degree shock | ${W}_{t}^{{\text{HDD}}^{2}}$ | + | Mu (2007) |

Storage level | ${\text{Storage}}_{\text{Level},t}$ | + | Dincerler et al (2005), Mu (2007) |

Storage level change | $\mathrm{\Delta}{\text{Storage}}_{\text{Level},t}$ | – | Mu (2007) |

Autumn dummy | ${\text{AUTDum}}_{t}$ | + | Fama and French (1987), Cartea and Williams (2008) |

Winter dummy | ${\text{WINDum}}_{t}$ | + | Fama and French (1987), Cartea and Williams (2008) |

Storage announcement dummy | ${\text{SADum}}_{t}$ | + | Linn and Zhu (2004), Mu (2007) |

### 4.1 Financial factors

Stock returns are represented by the certain (national) leading stock index (denoted by ${\mathrm{SPRET}}_{t}$). For the United Kingdom, we refer to the Financial Times Stock Exchange (FTSE) 100 index, which is a share index in which the stocks of the 100 companies with the highest market capitalization on the London Stock Exchange are listed. It is one of the most widely used stock indexes. For Belgium, the BEL20 index is used. This is the Belgian benchmark stock market index of the Euronext Group, and it consists of twenty companies traded at the Brussels Stock Exchange. For the Netherlands, we refer to the AEX index, which is the Dutch stock market index (on the Euronext Amsterdam), in which the twenty-five most actively traded securities of the exchange are listed. With respect to these indexes, the model incorporates the systematic risk of financial markets. We expect this variable to have a negative impact on the convenience yields, as shown in Fama and French (1987) and Bailey and Chan (1993).

Another important factor is given by the risk-free rate (${R}_{m,t}$), which is also part of the model to approximate the convenience yield. We expect it to have a negative relationship with the convenience yield, as the risk-free rate reduces the option values.

Moreover, a commodity index provided by Reuters Commodity Research Bureau (CRB) index, as shown in Gorton and Rouwenhorst (2004) or McKenzie et al (2004), is included. ${\mathrm{CRB}}_{t}$ is an index of commodity futures that encompasses nineteen different futures that are traded at international commodity forward exchanges.^{17}^{17}As mentioned in McKenzie et al (2004), the Reuters–Jefferies (RJ)/CRB index measures the return from investing in nearby commodity futures and rolling them forward each month, always keeping an investment close to maturity futures. It serves as an important indicator for future developments of inflation and cost trends in the manufacturing industry. Thus, we expect that it exerts a positive impact on the convenience yield.

Additionally, crude oil returns (denoted by ${\mathrm{CRET}}_{t}$) are included, as oil provides the closest substitute to natural gas. Moreover, for Europe, we expect a strong connection between the two price series because of the coupling of oil and gas prices in several retail contracts. As shown by Brown and Yücel (2008), natural gas market analysts emphasize weather and inventories as the most important drivers of natural gas prices. However, they also show that natural gas prices react sensitively to changes in close substitutes, such as oil or petroleum. Consequently, an increase in these prices can be expected to cause an increase in natural gas spot prices. So, the benefits of holding storage facilities increase, as the storage holder has the possibility to sell the natural gas, bought before at cheaper prices, at the currently existing high price; this generates additional benefits for the storage owner. All financial data is provided by Bloomberg.

### 4.2 Weather

Mansanet-Bataller and Soriano (2009) show that weather represented by the temperature is an important factor that influences gas and oil prices, as high and low temperatures directly transform into higher demand. Similarly to Mu (2007) and Mansanet-Bataller and Soriano (2009), we incorporate a weather variable that allows us to analyze whether deviations from the average temperature (measured over thirty years) affect convenience yields implied in the natural gas markets in Europe.

For this, we refer to historic weather data covering the last thirty years. This is provided by the European Climate Assessment Dataset (ECAD), which collects temperature data from various European weather stations. In a first step, the temperature is converted into cooling and heating degree days, denoted by ${\mathrm{CDD}}_{t}$ and ${\mathrm{HDD}}_{t}$, respectively, as follows:

${\mathrm{HDD}}_{t}$ | $=\mathrm{Max}(0,18-{\displaystyle \frac{({\mathrm{MaxTemp}}_{t}+{\mathrm{MinTemp}}_{t})}{2}})$ | (4.3) | ||

if $$; and | ||||

${\mathrm{CDD}}_{t}$ | $=\mathrm{Max}(0,{\displaystyle \frac{({\mathrm{MaxTemp}}_{t}+{\mathrm{MinTemp}}_{t})}{2}}-18)$ | (4.4) |

if $({\mathrm{MaxTemp}}_{t}+{\mathrm{MinTemp}}_{t})>{21}^{\circ}\mathrm{C}$, where ${\mathrm{MaxTemp}}_{t}$ and ${\mathrm{MinTemp}}_{t}$ give the maximum and minimum temperature measured on a certain day $t$. The measure for a heating degree day starts when the overall temperature falls below 15 ${}^{\circ}\mathrm{C}$, but the actual measured weather variable starts at 18 ${}^{\circ}\mathrm{C}$. This is due to the assumption that a certain level of insulation is implied in European houses. A similar procedure is used to determine the cooling degree days. Herein, ${\mathrm{CDD}}_{t}$ is measured at a maximum of 0 or 18 ${}^{\circ}\mathrm{C}$, reduced by the average temperature of a certain day. Cooling is assumed to start when the outer temperature rises to above 21 ${}^{\circ}\mathrm{C}$.

Then, the cooling or heating shock variable for each trading place can be derived as

${W}_{t}^{\mathrm{CDD}}$ | $=({\mathrm{CDD}}_{t}-{\mathrm{CDD}}_{t}^{\mathrm{AVG}})$ | (4.5) | ||

and | ||||

${W}_{t}^{\mathrm{HDD}}$ | $=({\mathrm{HDD}}_{t}-{\mathrm{HDD}}_{t}^{\mathrm{AVG}}),$ | (4.6) |

with ${\mathrm{CDD}}_{t}$ as the cooling degree days variable of a certain trading hub on day $t$ and ${\mathrm{CDD}}_{t}^{\mathrm{AVG}}$ as the thirty-year average normal cooling degree days specified for a certain date $t$.^{18}^{18}If $$, then ${\mathrm{HDD}}_{t}^{\mathrm{AVG}}=\mathrm{Max}(0,18-{\displaystyle \sum _{i=1}^{n}}{\displaystyle \frac{({\mathrm{MaxTemp}}_{i,t}+{\mathrm{MinTemp}}_{i,t})}{2}});$ and if $({\mathrm{MaxTemp}}_{t}+{\mathrm{MinTemp}}_{t})>{21}^{\circ}\mathrm{C}$, then ${\mathrm{CDD}}_{t}^{\mathrm{AVG}}=\mathrm{Max}(0,{\displaystyle \sum _{i=1}^{n}}{\displaystyle \frac{({\mathrm{MaxTemp}}_{i,t}+{\mathrm{MinTemp}}_{i,t})}{2}}-18),$ where $n=30$. The shock variable is the deviation of cooling degree days (${\mathrm{CDD}}_{t}$) over the normal level (${\mathrm{CDD}}_{t}^{\mathrm{AVG}}$). A similar line of reasoning leads to the heating shock variable (${W}_{t}^{\mathrm{HDD}}$).^{19}^{19}In contrast to Mu (2007), we create a weather variable that directly measures the deviations from the thirty-year average temperature (AVG).

As mentioned earlier, in addition to the weather shock variable, we include their squared values (${W}_{t}^{{\mathrm{CDD}}^{2}}$ and ${W}_{t}^{{\mathrm{HDD}}^{2}}$) and two seasonal dummies ${\mathrm{AUTDum}}_{t}$ and ${\mathrm{WINDum}}_{t}$ in order to capture the seasonalities coming from the autumn and winter months.^{20}^{20}In line with Mu (2007), the inclusion of the quadratic forms of the weather shock variables is used to capture the possible nonlinear effect. This nonlinear effect of the demand shock on the volatility could potentially generate an increase in the volatility at an increasing rate.

### 4.3 Storage

For the second part of the analysis, we include in the original model the storage variable. This implies that we have to shorten the period significantly. The storage data is provided by Gas Infrastructure Europe (GIE). From September 2007 to September 2010, GIE published storage levels, ie, the amount of working gas actually stored in millions of cubic meters, on a weekly basis. In October 2009, GIE began to change their reporting frequency from weekly to daily by having a transition period of around one year, in which both volumes can be reported. Hence, at the beginning of this period, daily information covered only parts of the storage capacities installed. Consequently, for our analysis we refer to weekly data in order to avoid any bias in the data. The lowest levels of storage are observable directly after the winter months. The highest levels are measurable at the end of the injection phase.

In general, the convenience yield of a storable consumable good is expected to play a crucial role in connecting spot and futures prices via inventory levels. In this context, the Theory of Storage explains the inverse relationship between storage and convenience yields, ie, the lower the storage level, the higher the marginal storage and, consequently, the higher the convenience yield. Wei and Zhu (2006) showed that the convenience yield variability is associated with its own lagged variability and the spot price variability. In our determination, in which the convenience yield is the difference between two options written on the spot and futures products, respectively, we prescind from the possibility of including the underlyings’ volatility, explicitly in order to avoid a redundant dimensioning of the model. Instead, we incorporate the variability of the storage level, which serves as a direct measure of the price divergence in spot and futures markets but also give insights into the true availability of storage facilities. Moreover, the storage levels and changes in those levels can be seen as contributing information to market agents, which, in line with Ederington and Lee (1993) and Anderson et al (2003), should directly translate into an increase in volatility, similar to that upon the release of other news.^{21}^{21}The storage change variable ($\mathrm{\Delta}{\mathrm{Storage}}_{\mathrm{Level},t}$) covers the deviations in storage levels from one announcement date to the next, which should exert a negative impact on the convenience yield. Following Mu (2007), the convenience yield’s volatility will increase (decrease) in the event that storage levels fall below (exceed) the expectations of market participants. Further, we include a storage announcement dummy variable (${\mathrm{SADum}}_{t}$) to capture the effect of a higher degree of transparency regarding the actual state of the gas storage market (over the whole sample period). In line with Linn and Zhu (2004) and Mu (2007), we expect that ${\mathrm{SADum}}_{t}$ has a positive effect on the convenience yield.

## 5 Estimation results

In the following, we present the results of the model outlined in Section 4 using the method of quasi-maximum likelihood estimation (QMLE).^{22}^{22}Recall that the underlying distribution of convenience yields is not normally distributed (see Section 3.3); however, the QMLE is still asymptotically consistent. Further, we use Bollerslev and Wooldridge (1992) heteroscedasticity consistent robust standard errors to correct for the nonnormality in the data. We choose the number of lags included in the variance equation of the EGARCH model by minimizing the Akaike information criterion (AIC) and the Schwarz information criterion (SIC).^{23}^{23}EGARCH(1,1) models fit the convenience yield data best. We base our model selection on the AIC and SIC as well as on diagnostic checks on the serial correlation in residuals and squared residuals of the convenience yield. For each trading hub and considered maturity, the according estimation results are given in the following tables.^{24}^{24}For diagnostic checking, we report in Table 22 (see the online appendix) the ARCH–Lagrange multiplier (LM) test for heteroscedasticity in squared residuals. Further, we report the results of the EGARCH estimation including storage levels (see Section 5.2) and highlight the corresponding differences.

In addition to that, we compute the half-life time, which is determined by $\mathrm{HLT}=1-[\mathrm{log}2/\mathrm{log}\beta ]$, where $\beta $ represents the persistence in conditional volatility of the EGARCH model. According to Lamoureux and Lastrapes (1990) and Pindyck (2004), it measures the number of days (speed-of-adjustment) that it takes for the volatility to reach half of its original value. As mentioned in Oglend and Sikveland (2008), the speed at which the market absorbs the shock indicates the degree of efficiency in the market.

### 5.1 First model (with storage announcement dummy)

The estimation results of the EGARCH model based on the complete data set are reported in Tables 6–8. The EGARCH representations enable us to determine the characteristics of the convenience yield based on an analysis of the conditional variance and persistence. All results are presented separately for each hub considered and discussed in a final subsection (see Section 5.3).

#### 5.1.1 National Balancing Point

In the mean equation, the number of significant explanatory variables decreases with increasing maturity. Overall, we see that only a few driving factors are able to describe the dynamics of the convenience yield. For the one-month convenience yield (${\text{CY}}_{\text{1m}}$), we observe that ${W}_{t}^{\mathrm{HDD}}$, ${R}_{m,t}$ and ${\mathrm{CCDum}}_{t}$ exert a significant impact. Nevertheless, only the risk-free rate and the credit crisis dummy show the expected negative impact on the convenience yield. The negative sign of ${W}_{t}^{\mathrm{HDD}}$ is astonishing. In a similar context, Cartea and Williams (2008) illustrate that the short-term market price of risk depends on how the injections and withdrawals occur compared with what the market expects them to be. Thus, market participants forecast injections and withdrawals from the inventories relative to seasonal need quite well, ie, harsh weather conditions do not increase the benefits of holding storage. For the two-month maturity, we observe that only ${R}_{m,t}$ and ${\mathrm{CCDum}}_{t}$ are still significant with the expected sign pattern, while for the three-month convenience yield only the credit crisis dummy turns out to be significant.

In the variance equation, only a few explanatory variables are able to explain the conditional variance of the EGARCH(1,1) model. This is contrary to findings in recent literature (see, for example, Lamoureux and Lastrapes 1990; Kalev et al 2004; Mu 2007) on the persistence of the conditional variance, which suggests that the persistence might be generated by exogenous influencing factors that are correlated with the conditional variance.^{25}^{25}This strand of literature suggests that the inclusion of exogenous variables should reduce the volatility persistence considerably. Thus, for the one-month convenience yield, only the estimated weather variables for cooling (${W}_{t}^{\mathrm{CDD}}$ and ${W}_{t}^{{\mathrm{CDD}}^{2}}$) and the autumn dummy variable (${\mathrm{AUTDum}}_{t}$) are statistically significant. ${W}_{t}^{\mathrm{CDD}}$ shows a positive impact on the volatility (ie, it generates higher volatility levels). One reason for this could be the deviation from the preplanned yearly gas cycle, which implies that gas is injected over the summer months and withdrawn in winter. Thus, depleting gas levels during the warmer seasons of the year interferes with the designed injection process, which leads to higher variance levels (see, for example, Cartea and Williams 2008). In contrast to Mu (2007), we do not find a positive impact of the included nonlinear effect on the conditional volatility. Further, there is no evidence that during the winter months the conditional variance is higher than in other seasons.

For the two- and three-month maturities, the estimated weather variables for heating (${W}_{t}^{\mathrm{HDD}}$/${W}_{t}^{{\mathrm{HDD}}^{2}}$) turn out to be significant, while for the two-month convenience yield (${\text{CY}}_{\text{2m}}$) the storage announcement dummy (${\mathrm{SADum}}_{t}$) also exerts a positive effect. The latter finding is in line with Linn and Zhu (2004) and Murry and Zhu (2004), who suggest that on the day storage information about the actual levels is released, significantly more volatility in the natural gas markets is created.

Overall, the effects of the weather variables on the conditional variance of the convenience yield are mixed and vary over time.

(a) Mean equation | ||||||
---|---|---|---|---|---|---|

${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | ||||

Coef | Prob | Coef | Prob | Coef | Prob | |

Constant | 0.0156*** | 0.0000 | 0.0094*** | 0.0016 | 0.0099** | 0.0165 |

${W}_{t}^{\text{CDD}}$ | 0.0002 | 0.3800 | $-$0.0010 | 0.1045 | $-$0.0003 | 0.6083 |

${W}_{t}^{\text{HDD}}$ | $-$0.0008*** | 0.0002 | 0.0000 | 0.7766 | 0.0000 | 0.9082 |

${\text{CRET}}_{t}$ | $-$0.0217 | 0.3068 | $-$0.0046 | 0.4813 | $-$0.0006 | 0.9466 |

${R}_{m,t}$ | $-$0.1789*** | 0.0009 | $-$0.0833** | 0.0257 | $-$0.0760 | 0.1078 |

${\text{SPRET}}_{t}$ | 0.0326 | 0.3543 | $-$0.0021 | 0.8826 | $-$0.0029 | 0.7375 |

${\text{CRB}}_{t}$ | 0.0113 | 0.6542 | $-$0.0005 | 0.9753 | 0.0011 | 0.9190 |

${\text{CCDum}}_{t}$ | $-$0.0064*** | 0.0082 | $-$0.0047** | 0.0226 | $-$0.0093*** | 0.0091 |

(b) Variance equation | ||||||

${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | ||||

Coef | Prob | Coef | Prob | Coef | Prob | |

$\omega $ | $-$3.8643*** | 0.0000 | $-$1.8347*** | 0.0000 | $-$1.7029*** | 0.0000 |

$\alpha $ | 1.2867*** | 0.0000 | 1.0339*** | 0.0000 | 0.9543*** | 0.0000 |

$\gamma $ | 0.0039 | 0.9431 | $-$0.0121 | 0.8087 | $-$0.0122 | 0.7908 |

$\beta $ | 0.6406*** | 0.0000 | 0.8670*** | 0.0000 | 0.8640*** | 0.0000 |

${W}_{t}^{\text{CDD}}$ | 1.3982** | 0.0243 | 0.6828 | 0.3820 | 0.7046 | 0.2501 |

${W}_{t}^{{\text{CDD}}^{2}}$ | $-$0.3464** | 0.0125 | $-$0.1531 | 0.3778 | $-$0.1785 | 0.1743 |

${W}_{t}^{\text{HDD}}$ | $-$0.0174 | 0.3720 | $-$0.0084 | 0.5710 | $-$0.0353** | 0.0101 |

${W}_{t}^{{\text{HDD}}^{2}}$ | 0.0032 | 0.3622 | 0.0041* | 0.0634 | $-$0.0098*** | 0.0010 |

${\text{SADum}}_{t}$ | 0.3423 | 0.1224 | 0.6107** | 0.0258 | 0.3411 | 0.1534 |

${\text{AUTDum}}_{t}$ | 0.2956* | 0.0715 | 0.1097 | 0.1426 | 0.1043 | 0.1337 |

${\text{WINDum}}_{t}$ | 0.0675 | 0.5215 | 0.0332 | 0.7140 | 0.0898 | 0.1728 |

LOGL | 8307.1460 | — | 8485.8060 | — | 8346.5680 | — |

AIC | $-$5.2500 | — | $-$5.3626 | — | $-$5.2750 | — |

SIC | $-$5.2116 | — | $-$5.3223 | — | $-$5.2366 | — |

HLT | 2.5565 | — | 5.8569 | — | 5.7415 | — |

#### 5.1.2 Zeebrugge

For ZEE, a slightly different picture occurs. In the mean equation, the number of significant explanatory variables varies with the futures maturity. Similar to NBP, ${W}_{t}^{\mathrm{HDD}}$ and ${\mathrm{CCDum}}_{t}$ reveal a significant impact on the one-month convenience yield. Consistent with our previous finding for ${W}_{t}^{\mathrm{HDD}}$, the expected impact of the estimated weather variable for ZEE is not in line with recent research in the area of weather effects on energy commodities (see Mu 2007; Mansanet-Bataller and Soriano 2009). In addition, our findings confirm that for both trading hubs (NBP and ZEE) ${\mathrm{CCDum}}_{t}$ plays a significant role. The ${\mathrm{CRB}}_{t}$ index exerts a significant positive impact on ${\text{CY}}_{\text{2m}}$ in addition to the variables already mentioned for ${\text{CY}}_{\text{1m}}$. For the three-month convenience yield (${\text{CY}}_{\text{3m}}$), solely the ${\mathrm{CRB}}_{t}$ index is important. For the variance equation of the EGARCH(1,1) model, differences to NBP are observable. The one-month convenience yield’s volatility is not driven by any of the included exogenous variables. This clearly contradicts the recent literature on the influence of exogenous variables on the persistence of the conditional variance. While the one-month convenience yield’s volatility is not driven by any of the included variables, the volatility of the convenience yield for longer maturities is influenced by weather and storage. Explicitly, for the two-month convenience yield (${\text{CY}}_{\text{2m}}$), ${W}_{t}^{{\mathrm{CDD}}^{2}}$, ${\mathrm{SADum}}_{t}$ and ${\mathrm{AUTDum}}_{t}$ play a significant role, while for the three-month convenience yield, ${W}_{t}^{\mathrm{HDD}}$ and ${W}_{t}^{{\mathrm{HDD}}^{2}}$ also turn out to be significant, which is a sign of higher withdrawals of gas during the colder season of the year and, thus, a deviation from the preplanned yearly gas cycle. In line with NBP, these deviations do not increase the volatility level. Similar to NBP, there is no evidence that the variance is higher in the winter months.

(a) Mean equation | ||||||
---|---|---|---|---|---|---|

${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | ||||

Coef | Prob | Coef | Prob | Coef | Prob | |

Constant | 0.0052*** | 0.0043 | 0.0061** | 0.0150 | $-$0.0038 | 0.5329 |

${W}_{t}^{\text{CDD}}$ | 0.0003 | 0.2629 | 0.0000 | 0.7283 | 0.0000 | 0.9458 |

${W}_{t}^{\text{HDD}}$ | $-$0.0004*** | 0.0030 | $-$0.0001* | 0.0986 | $-$0.0001 | 0.2527 |

${\text{CRET}}_{t}$ | 0.0196 | 0.2711 | 0.0093 | 0.3009 | $-$0.0157 | 0.1503 |

${R}_{m,t}$ | 0.0232 | 0.6526 | $-$0.0251 | 0.7014 | 0.1043 | 0.4576 |

${\text{SPRET}}_{t}$ | 0.0170 | 0.4815 | $-$0.0178 | 0.2603 | $-$0.0057 | 0.6353 |

${\text{CRB}}_{t}$ | 0.0153 | 0.5775 | 0.0622*** | 0.0046 | 0.0547*** | 0.0048 |

${\text{CCDum}}_{t}$ | $-$0.0055*** | 0.0002 | $-$0.0046** | 0.0157 | 0.0019 | 0.6867 |

(b) Variance equation | ||||||

${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | ||||

Coef | Prob | Coef | Prob | Coef | Prob | |

$\omega $ | $-$2.9973*** | 0.0000 | $-$2.3068*** | 0.0000 | $-$2.4522*** | 0.0000 |

$\alpha $ | 1.0698*** | 0.0000 | 1.1589*** | 0.0000 | 1.1474*** | 0.0000 |

$\gamma $ | 0.1207*** | 0.0093 | 0.0754 | 0.1351 | 0.0006 | 0.9876 |

$\beta $ | 0.7283*** | 0.0000 | 0.8064*** | 0.0000 | 0.7813*** | 0.0000 |

${W}_{t}^{\text{CDD}}$ | $-$0.0316 | 0.7206 | 0.1584 | 0.1798 | 0.2564** | 0.0360 |

${W}_{t}^{{\text{CDD}}^{2}}$ | 0.0015 | 0.9226 | $-$0.0409** | 0.0385 | $-$0.0575*** | 0.0030 |

${W}_{t}^{\text{HDD}}$ | $-$0.0142 | 0.2090 | $-$0.0077 | 0.5140 | $-$0.0276* | 0.0879 |

${W}_{t}^{{\text{HDD}}^{2}}$ | 0.0031 | 0.2207 | $-$0.0031 | 0.2171 | $-$0.0067** | 0.0176 |

${\text{SADum}}_{t}$ | $-$0.1008 | 0.4985 | 0.5887*** | 0.0040 | 0.7431*** | 0.0001 |

${\text{AUTDum}}_{t}$ | 0.1065 | 0.2341 | $-$0.3110*** | 0.0002 | $-$0.2685*** | 0.0036 |

${\text{WINDum}}_{t}$ | 0.0588 | 0.4591 | 0.1455 | 0.1493 | 0.0240 | 0.8190 |

LOGL | 7687.0080 | — | 7418.5330 | — | 7601.6750 | — |

AIC | $-$5.5450 | — | $-$5.3997 | — | $-$5.4840 | — |

SIC | $-$5.5000 | — | $-$5.3543 | — | $-$5.4412 | — |

HLT | 3.1860 | — | 4.2214 | — | 3.8084 | — |

#### 5.1.3 Title Transfer Facility

For TTF, the one-month convenience yield’s mean is explained by ${W}_{t}^{\mathrm{HDD}}$ and the financial market return (${\mathrm{SPRET}}_{t}$), while for ${\text{CY}}_{\text{2m}}$ the risk-free rate (${R}_{m,t}$), the ${\mathrm{CRB}}_{t}$ index and the credit crisis dummy (${\mathrm{CCDum}}_{t}$) also turn out to be significant. The expected sign for ${\mathrm{SPRET}}_{t}$ is only in line with the literature for ${\text{CY}}_{\text{1m}}$. Further, ${W}_{t}^{\mathrm{HDD}}$ exerts a negative impact, which is also highlighted by NBP and ZEE. The sign of ${\mathrm{CRB}}_{t}$ is negative and, therefore, contradicts our findings from ZEE and the expectations deduced from the literature.

Similar to ZEE, no incorporated variable has a significant impact for the one-month conditional variance equation of TTF. For the two-month convenience volatility, ${W}_{t}^{{\mathrm{HDD}}^{2}}$ becomes significant. Again, this strongly indicates that the persistence of the conditional variance is not influenced by exogenous variables.

(a) Mean equation | ||||
---|---|---|---|---|

${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | |||

Coef | Prob | Coef | Prob | |

Constant | 0.0032 | 0.1255 | 0.0067*** | 0.0012 |

${W}_{t}^{\text{CDD}}$ | 0.0000 | 0.8341 | 0.0001 | 0.6519 |

${W}_{t}^{\text{HDD}}$ | $-$0.0004*** | 0.0038 | $-$0.0003*** | 0.0002 |

${\text{CRET}}_{t}$ | $-$0.0046 | 0.7539 | $-$0.0065 | 0.5327 |

${R}_{m,t}$ | $-$0.0578 | 0.2063 | $-$0.0949** | 0.0138 |

${\text{SPRET}}_{t}$ | $-$0.0276** | 0.0325 | 0.0149* | 0.0965 |

${\text{CRB}}_{t}$ | $-$0.0309 | 0.1000 | $-$0.0273*** | 0.0067 |

${\text{CCDum}}_{t}$ | 0.0005 | 0.7374 | $-$0.0030* | 0.0647 |

(b) Variance equation | ||||

${\text{??}}_{\text{??}}$ | ${\text{??}}_{\text{??}}$ | |||

Coef | Prob | Coef | Prob | |

$\omega $ | $-$2.4301*** | 0.0000 | $-$2.0172*** | 0.0000 |

$\alpha $ | 1.0452*** | 0.0000 | 0.9965*** | 0.0000 |

$\gamma $ | $-$0.0359 | 0.5016 | $-$0.0891** | 0.0505 |

$\beta $ | 0.8047*** | 0.0000 | 0.8631*** | 0.0000 |

${W}_{t}^{\text{CDD}}$ | 0.2123 | 0.4564 | 0.5091 | 0.1499 |

${W}_{t}^{{\text{CDD}}^{2}}$ | $-$0.0579 | 0.2091 | $-$0.0825 | 0.1311 |

${W}_{t}^{\text{HDD}}$ | $-$0.0098 | 0.4801 | 0.0112 | 0.5466 |

${W}_{t}^{{\text{HDD}}^{2}}$ | $-$0.0003 | 0.9332 | 0.0092*** | 0.0005 |

${\text{SADum}}_{t}$ | 0.2602 | 0.1913 | 0.2760 | 0.1327 |

${\text{AUTDum}}_{t}$ | 0.0059 | 0.9553 | 0.1462 | 0.1276 |

${\text{WINDum}}_{t}$ | $-$0.0920 | 0.3521 | $-$0.1377 | 0.2249 |

LOGL | 5333.8500 | — | 5471.0550 | — |

AIC | $-$6.2663 | — | $-$6.4281 | — |

SIC | $-$6.2022 | — | $-$6.3640 | — |

HLT | 4.1909 | — | 5.7076 | — |

#### 5.1.4 Summary

Overall, the resulting significant explanatory variables that drive the mean and conditional variance of the convenience yield estimates from the European trading places differ. Surprisingly, the variable for the crude oil return (${\mathrm{CRET}}_{t}$) shows no significant influence for all trading hubs and considered maturities, which is the opposite of what we expected.^{26}^{26}This effect is illustrated by Neumann et al (2008), who reveal that oil prices have no influence on the basis of spot and futures contracts traded at NBP, TTF and ZEE. Moreover, we are not able to confirm earlier results from the literature that suggest the inclusion of exogenous variables in the conditional variance equation of the EGARCH setting reduces the volatility (persistence) considerably. Overall, the findings for the impact of the demand shock/weather variables are mixed. Thus, no clear conclusions can be drawn as to whether these variables increase or decrease the volatility of the convenience yield.^{27}^{27}Additionally, we perform a Wald restriction test on the null hypothesis that all coefficients of the exogenous variables are equal to zero. We reject the null hypothesis for all trading hubs and considered maturities, beside the one-month model for ZEE and the two-month model for TTF, indicating that the included exogenous variables have a small impact on the conditional volatility.

### 5.2 Second model (with storage levels)

In this section, we focus on the processing of information dealing with storage levels and changes in storage levels on the conditional variance. As described in the previous section, we fit EGARCH(1,1) models to determine the impact of different explanatory variables on the mean and the conditional variance. Overall, the resulting significant driving factors are different to the outcomes from the model incorporating the whole sample period (see Table 19 in Appendix 1, available online).

However, for all three trading hubs we observe that ${\mathrm{Storage}}_{\mathrm{Level},t}$ plays a significantly positive role. In line with Murry and Zhu (2004) and Linn and Zhu (2004), this pattern indicates that the release of the weekly storage data through GIE generates significant volatility; this confirms our findings for the storage announcement dummy from the model, which were outlined in Section (5.1). Further, changes in the storage levels ($\mathrm{\Delta}{\mathrm{Storage}}_{\mathrm{Level},t}$) also influence the conditional variance of the considered trading hubs.^{28}^{28}Except for ${\text{CY}}_{\text{3m}}$ at NBP and ${\text{CY}}_{\text{2m}}$ at TTF.

Comparing Tables 6 and 8 with Table 19 in Appendix 1 (available online), it becomes obvious that the release of storage news, as given by the storage announcement dummy or the level of storage (as well as the change in levels), has a significant effect on the conditional variance of the convenience yield. All included storage variables are significant; however, the explanatory power and model fit in terms of AIC and persistence of the shock improve slightly for the second model.^{29}^{29}We also implement a Wald test to examine whether all coefficients of the exogenous variables in the conditional variance equation are zero (ie, we test the nullity of parameters). The null hypothesis is rejected for all trading hubs and maturities. This indicates that the exogenous variables in the variance equation have an influence on the persistence of the variance, even though it might only be small.

### 5.3 Discussion of the results

The results of our analyses of the driving factors on the mean and conditional variance of the EGARCH(1,1) model for the European trading places differ significantly, especially for the models with and without storage level information. For the mean equation, we observe that for each trading place at least one of the financial market variables (${R}_{m,t}$, ${\mathrm{SPRET}}_{t}$ or ${\mathrm{CCDum}}_{t}$) is significant, whereas the significance of the other variables varies considerably. From the conditional variance equation, we conclude that although the storage announcement dummy (${\mathrm{SADum}}_{t}$) has a significant effect, the inclusion of more information (in terms of storage levels) only slightly improves the overall model performance. Consequently, the new information in the second model (see Section 5.2) seems to be limited; this may be due to the limited storage access in Europe, or the level of market efficiency, which may hinder appropriate information processing. The reasons for this can be manifold, but they have one commonality: the not completely efficient natural gas market. Reports from, for example, Neumann et al (2008), and reports issued by the European Commission (see European Commission 2007b, 2007a), state that market participants face a substantial lack of storage facilities and restricted access to the existing facilities (as well as insufficient information and a missing secondary market for unused capacities), which creates possible obstacles in the appropriate use of storage.^{30}^{30}A sector-wide inquiry carried out in the EU illustrates that the access to storage is scarce, and occasionally nonexistent; therefore, market participants do not have the necessary flexibility to create a competitive market. This is the exact opposite of the state of the storage market in the United States, where storage is currently a highly traded commodity and, thus, provides more benefits than just backup and seasonal supply (see von Hirschhausen 2008). Although it is commonly accepted that NBP and ZEE now develop closer together because of their connecting pipelines, no perfect integration is given. This is mentioned in, for example, Neumann et al (2006), who apply a Kalman filter to daily price data from the NBP, Zeebrugge and Henry Hub. They show that the time-varying market integration coefficient moves toward $1$ for the NBP–Henry Hub and Zeebrugge–Henry Hub pairs, revealing a rising degree of integration in these markets. Nevertheless, in recent years TTF has been increasingly becoming a main competitor, especially in the over-the-counter sector (see Miriello and Polo 2015). They ascribe this development to the abandonment of entry barriers, which were caused by the different types of gas quality traded at this hub until 2009. For a good overview on the development of the EU gas market until 2011, see Heather (2012); for a more recent description that covers developments beyond the considered time horizon of this paper, see Miriello and Polo (2015). Also, in the most recent report by the European Commission, the lack of competition is mentioned. They emphasize that, although regulation on gas markets has made significant progress during the last few years, many European gas markets are still not sufficiently liquid, and regulation requires further improvements.^{31}^{31}In order to account for these changes, a time-varying model could be helpful, but this also has several shortcomings. We refer the interested reader to Kremser and Rammerstorfer (2014), who investigate the dynamic behavior of the leading European natural gas trading hub by implementing a time-varying estimation technique given by the state-space model (SSM), using the Kalman filter and the implied risk premium in these markets. They find that, for the European natural gas trading market, the corresponding futures and spot prices share a common stochastic trend, and that futures prices are biased predictors. Further, they investigate that the efficiency of the leading European natural gas trading hub is time-dependent and has been improving significantly over time. Hence, the gap between the realized and expected spot price (also known as forecast error) has nearly diminished over time and can only be partly explained by the included influencing variables.

In order to shed light on this hypothesis, we provide an efficiency test for NBP, ZEE and TTF. The convenience yield patterns, the explanatory models and the half-life time (model with storage effects) argue in favor of the following efficiency ranking: NBP $\to $ ZEE $\to $ TTF.

In general, the efficiency of futures markets can be tested by the following approach, which is provided in Modjtahedi and Movassagh (2005) and Geman (2007):

$${S}_{T}=\alpha +\beta {F}_{t,T}+{\u03f5}_{T},$$ | (5.1) |

where ${S}_{T}$ is the spot price at time $T$, ${F}_{t,T}$ denotes the futures price at time $t$ with maturity $T$, ${\u03f5}_{T}$ is the error term with zero mean and finite variance and $\alpha $ and $\beta $ are coefficients that have to be determined (see Kremser and Rammerstorfer 2014). In line with Fama (1970), the weak form of market efficiency holds in case $\alpha $ and $\beta $ in (5.1) are $0$ and $1$. In this case, ${F}_{t,T}$ contains all relevant information to forecast the future spot price,^{32}^{32}The relation between ${F}_{t,T}$ and ${S}_{T}$ is called unbiasedness. ie, market participants are not able to pursue any strategy that allows additional benefits from speculating on the future spot price by trading the futures product.^{33}^{33}Changes in the spot price should be unpredictable, so only new information will lead to a change in the future spot price.

Here, in order to test for market efficiency, we refer to the Johansen procedure based on Johansen (1988) and Johansen and Juselius (1990).^{34}^{34}Engle and Granger (1987) state that price series, which are individually dominated by a unit root, are cointegrated if their linear combination has a lower order of integration. If there exists such a cointegration relation, then the series considered are in a long-run equilibrium relationship. The test design implemented follows three steps. First, the unit root test is conducted (see Table 2). If the financial price series are non-stationary, a cointegration test is performed to study whether ${S}_{T}$ and ${F}_{t,T}$ are in a long-run relationship. In case the cointegration test is not able to detect a cointegration relation between the related price series, the test for unbiasedness is not viable. In case of cointegration between spot and futures prices, the restrictions imposed on the coefficients ($\alpha =0$ and $\beta =1$) can be tested using the likelihood ratio test.^{35}^{35}The analysis of the cointegration relation and a test of the estimates of the cointegration vector are conducted under the specifications of (1, $-\beta $, $-\alpha $) (see, for example, Lai and Lai 1991).

The results of the cointegration vector and the restrictions tested on $\alpha $ and $\beta $ are exemplarily reported for NBP and a maturity of one month in Figures 2 and 3. The outer line gives the optimum that could be approached in case of an efficient market. Obviously, the existence of an efficient relationship between spot and futures prices cannot fully be confirmed, although efficiency seems to improve over time. Also, for ZEE and TTF, a similar picture occurs. This finding is in line with Neumann et al (2008) and Dockner et al (2015), and it is also suggested by our former results from the half-life time measure.

## 6 Conclusion

This paper sheds light on the effects of storage, weather and other driving factors on the conditional variance of the convenience yield implied in the European gas markets. The results can be summarized as follows. First, we highlight that the time series properties of spot and futures contracts traded at NBP, TTF and ZEE argue in favor of the closed-form solutions of the geometric average floating-strike Asian put option to approximate the convenience yield. Second, in line with previous literature, we illustrate that the convenience yield estimates exhibit significant autocorrelation patterns in levels as well as squared levels for all considered trading hubs, which strongly indicates the existence of a time-varying volatility; thus, the implementation of (E)GARCH$(p,q)$ models is crucial to capture the dynamic behavior of the convenience yield series. Third, we find that, for all trading hubs considered, the Engle and Ng (1993) asymmetry test argues in favor of asymmetric responses of the volatility to negative and positive shocks and, thus, supports the implementation of an EGARCH model. Fourth, in contrast to several papers, we do not find that crude oil has a significant or consistent impact on the convenience yield, which indicates that there is no significant spillover effect between gas and oil for the considered time horizon. Fifth, we demonstrate that the inclusion of exogenous variables in the conditional variance equation does not considerably reduce the volatility persistence; this is contrary to the finding provided in Mu (2007) that a major share of the variance is explained by the included driving factors. Sixth, we highlight that the release of storage news, as given by the storage announcement dummy or the level of storage (as well as the change in levels), has a significant effect on the conditional variance of the convenience yield. Seventh, we illustrate that the mixed evidence and the absence of a clear pattern to explain the convenience yield’s characteristics underline the transition state of the European natural gas market.

In general, our findings contribute to the question of what drives the volatility of the convenience yield implied in certain commodity markets. This is of major importance, as not only the volatility of the prices but also the volatility of the related facilities may play a key role in financial decisions. For example, the valuation of commodity-based options and risk-hedging decisions rely on assumptions about volatilities and their evolution over time. Moreover, it is of extraordinary importance for traders in inefficient markets to determine the best information set available, even when storage facilities cannot be exhausted by every single trader. Our findings underline the ongoing debate on the lack of efficiency in the European gas markets that still hinders the implementation of efficient and profitable trading strategies. However, our results also reveal that the European markets are on the best path to improvement.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

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