Market Fundamentals and the Dynamics of Natural Gas Futures Volatility: An Augmented GARCH Approach Ibrahim Ergen, Federal Reserve Bank of Richmond, Baltimore, MD Islam Ridvanoglu, Zirve University, Gaziantep, Turkey July 9, 2014 Abstract We investigated the determinants of daily volatility for natural gas nearby-month futures traded in NYMEX within a GARCH framework augmented with market fundamentals. Consistent with the previous literature, we found that volatility is much higher on the storage level announcement days, on Mondays and during winters defined as De- cember, January and February. We also confirmed the previous find- ing that high volatility is associated with weather shocks in excess of seasonal norms. Samuelson’s hypothesis is also investigated and a sig- nificant time to maturity effect is detected. The impact of the (storage x season) interaction is investigated for the first time in this paper. Inclusion of this interaction term in our regressions produced very in- teresting results: The mainstream finding in the literature that lower storage levels result in higher volatility is valid only during winter. At other times, it is actually higher storage levels that results in higher volatility. Economic intuition behind this novel finding is discussed in detail. This study also fills the gap in the literature regarding the out- of-sample forecasting accuracy of GARCH models augmented with the market fundamentals. We found that augmentation with market fun- damentals improves the forecast accuracy evidenced by reduced mean absolute error and mean squared error compared to standard GARCH models with no market fundamentals. 1
Transcript
Market Fundamentals and the Dynamics ofNatural Gas Futures Volatility: AnAugmented GARCH Approach
Ibrahim Ergen, Federal Reserve Bank of Richmond, Baltimore, MDIslam Ridvanoglu, Zirve University, Gaziantep, Turkey
July 9, 2014
Abstract
We investigated the determinants of daily volatility for natural gasnearby-month futures traded in NYMEX within a GARCH frameworkaugmented with market fundamentals. Consistent with the previousliterature, we found that volatility is much higher on the storage levelannouncement days, on Mondays and during winters defined as De-cember, January and February. We also confirmed the previous find-ing that high volatility is associated with weather shocks in excess ofseasonal norms. Samuelson’s hypothesis is also investigated and a sig-nificant time to maturity effect is detected. The impact of the (storagex season) interaction is investigated for the first time in this paper.Inclusion of this interaction term in our regressions produced very in-teresting results: The mainstream finding in the literature that lowerstorage levels result in higher volatility is valid only during winter. Atother times, it is actually higher storage levels that results in highervolatility. Economic intuition behind this novel finding is discussed indetail. This study also fills the gap in the literature regarding the out-of-sample forecasting accuracy of GARCH models augmented with themarket fundamentals. We found that augmentation with market fun-damentals improves the forecast accuracy evidenced by reduced meanabsolute error and mean squared error compared to standard GARCHmodels with no market fundamentals.
1
1 Introduction
Natural gas futures contracts began trading on the New York Mer-cantile Exchange (NYMEX) in April 1990. One contract is writtenon 10,000 MMBTU of natural gas to be delivered to Henry Hub.1 2
Contracts for delivery in each month and for up to six years out aretraded at any point in time. Trading in a given contract ends threebusiness days before the first calendar day of the delivery month.
Trading in natural gas futures has skyrocketed in recent years.Open interest in Nymex natural gas futures grew at a rate of 15.2percent per year during the last decade (Chiou-Wei, Linn and Zhu,2007). Daily volume is in the order of 60,000 to 100,000 contractsfor the nearby month futures and 20,000 to 60,000 contracts for thesecond nearby futures.3 4
The volatility of natural gas prices has received increasing atten-tion in recent years. The extreme fluctuations in both spot and fu-tures prices caused researchers and market practitioners to focus onthe sources of this high volatility. Whether news about natural gasmarket fundamentals, or excessive speculation and irrational investorbehavior is responsible for the high volatility is an ongoing debate.In this paper, empirical evidence is provided that natural gas futuresprice volatility is driven by market fundamentals within a GARCHtype of dynamic volatility framework.
The response of prices to shifts in supply and demand depends onprice elasticity of the commodity. In general, natural gas markets arehighly inelastic in both the supply and demand side; hence, the priceis very responsive to short-term changes in both, which results in highvolatility. Two key fundamental pieces of information affecting thenatural gas markets are the level of working gas in storage facilities
1MMBTU= 1 Million British Thermal Units(BTU). A BTU is a unit used to describethe heat value (energy content) of fuels. A BTU is defined as the amount of heat requiredto raise the temperature of one pound of liquid water by one degree from 60 ◦F to 61 ◦Fat a constant pressure of one atmosphere. (www.wikipedia.org)
2Henry Hub is a point on the natural gas pipeline system in Erath, Louisiana. Itinterconnects with nine interstate and four intrastate pipelines. Spot and future prices setat Henry Hub are denominated in $/MMBTU and are generally regarded as the primaryprice set for the North American natural gas market. (www.wikipedia.org)
3Nearby Future Contract is the earliest maturing contract. This corresponds to thenext month delivery contract for natural gas futures.
4Second Nearby Contract is the second earliest maturing contract In natural gas mar-kets, it corresponds to the contract for delivery on the month after the next month.
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and weather changes. These are generally regarded as proxies forsupply and demand.
For supply conditions, the storage report is perceived as the mostimportant piece of information by natural gas market participants.The report is currently prepared by the Energy Information Admin-istration (EIA) of the Department of Energy (DOE). Anecdotal evi-dence on the effect of the natural gas storage report is abundant inthe financial press. The following appeared in Communications, En-ergy and Paperworkers Union of Canada (CEP) News on Thursday,October 30, 2008:Underground natural gas storage in the U.S. increased 46 billion cubicfeet (Bcf) in the week ending Oct. 24, according to the Energy Infor-mation Administration (EIA)’s weekly report on Thursday. Expecta-tions had been for a 41 Bcf increase. Following the report, natural gasprices bottomed out, and are now at new session lows of 6.528 frompre-report levels of 6.811.
The report provides the level of total underground working gasand the historical average of this quantity for the equivalent time pe-riods of last five years.5 Although the effects of storage surprises onshort-term volatility have been emphasized in many studies (Gregoireand Boucher 2008; Mu 2007; Linn and Zhu (2004)), we did not iden-tify any research considering the effects of the interaction betweenthe storage surprises and seasonality. This paper adresses this issue.In the winter, natural gas demand spikes, and the supply is unable toreact quickly since the production of natural gas is uniform across sea-sons. When this happens, low storage levels, as compared to historicalaverages may be regarded as tight supply situations and put pressureon gas prices, which results in high volatility. Conversely, during theother seasons, higher storage levels than historical averages may in-crease concerns regarding the capacity of storage facilities and result inhighly volatile natural gas prices. In this paper, supporting evidenceis presented for this hypothesis, which implies asymmetric effect ofstorage surprises in different seasons.
For demand conditions, the key information for short-term volatil-ity dynamics is a change in the weather. Upon the observation ofunexpected cold weather during the winter months, the extra demand
5Working gas in storage is the volume of gas in the reservoir that is in addition to thecushion or base gas. Base gas is the volume of gas needed as a permanent inventory tomaintain adequate reservoir pressures. (www.doe.eia.gov)
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for heating pushes the prices up causing higher volatility since the sup-ply cannot adjust to such changes in the short run. In recent years,power generation plants have used more natural-gas-fueled technol-ogy. Natural gas usage in electricity generation rose from 12 percentto 17 percent between 1990 and 2006 (Hartley et al., 2007.a). As aresult, hotter than expected temperatures during the summer monthsincrease the demand for cooling and might have a similar effect onvolatility. Empirical evidence is presented that weather anomalies,measured as the degree days in excess of seasonal norms, result inincreased short-term volatility.
In addition to the supply-and-demand driven volatility, the pre-vious literature found significantly higher volatility on Mondays (Mu2007; Murry and Zhu 2004). The study of Fleming et al. (2004) ex-plains this effect based on the continuous weather information flowduring the long nontrading weekend period.
Additionally, time to maturity may be another determinant of fu-tures market volatility. Samuelson (1965) was the first to claim thatthe volatility of futures prices increases as the contract maturity getscloser. Using a very large futures dataset on 6,805 contracts, Daalet al. (2006) found that the maturity effect was much stronger foragricultural and energy commodity futures than it was for financialfutures. Using extreme value method, to measure the daily volatilityof natural gas futures from daily low and daily high prices, Gregorioand Boucher (2008) found that the maturity effect was significant evenafter controlling for storage surprises. On the other hand, Mu (2007)tested for the maturity effect by fitting separate GARCH models tothe nearby futures contracts and the second nearby contracts and com-paring the fitted daily volatilities. In this paper, we present empiricalevidence of the maturity effect by directly including a time to maturityvariable in the conditional variance equation of the GARCH model, asubstantially different approach from past research.
In this study, GARCH models are used as an econometric tool inorder to account for the dynamic nature of short-term market volatil-ity. Some of the more recent studies on natural gas volatility augmentthe GARCH models with market fundamentals in order to focus onthe determinants of volatility (Ates and Wang 2008; Mu 2007; Pyndick2004; Murry and Zhu 2004). However, the literature does not addressthe out-of-sample forecasting accuracy of GARCH models augmentedwith market fundamentals. In this paper, we also study the out-of-sample forecasting accuracy of several simple GARCH models together
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with augmented GARCH models using a constant-size sliding-samplemethodology.
The remaining sections of this paper are organized as follows: Insection 2, a statistical analysis of the nearby month futures returns ispresented, and natural gas volatility with respect to the market fun-damentals is analyzed without imposing any econometric structure.In section 3 the econometric model is introduced and estimation re-sults are presented. Out-of-sample volatility forecasting accuracy ofaugmented GARCH models is tested in secion 4. The Usefulness ofaugmented GARCH models for natural gas price risk measurement isevaluated in section ??. Lastly, in section 5, we discuss the conclu-sions.
2 Data and Statistical Analysis
Natural gas futures price data from February 2001 to May 2008 wereobtained from NYMEX contracts. Contract-by-contract price dataare available from DataStream. The return series for nearby monthfutures are constructed in two steps. First, returns for individualcontracts i are calculated by
rt,i = ln(Ft,i/Ft−1,i) (1)
where Ft,i is the price of the futures contract i at time t. Then, thenearby month contract return for time t is obtained as,
rt,nb = rt,j (2)
where j is the earliest maturing contract. In other words, day t is inmonth j − 1. By first obtaining the returns and then rolling over thecontracts, constructing price series from different contracts is avoided,which may distort the data. Therefore, all nearby futures returns rnb,tare tradable and realizable. At the end of this procedure, the nearbymonth futures return data are obtained that run from January 4, 2001to April 23, 2008, a total of T = 1, 823 daily observations.
Summary statistics for nearby month futures returns are presentedin Table 1. The numbers in parenthesis are the probability values forthe associated tests. The returns are right skewed for this sample pe-riod and exhibit excess kurtosis. Consequently, the Jarque-Bera testrejects the null hypothesis of normal distribution. These statisticsimply that natural gas futures returns are not normally distributed.
5
However, a comparison with the summary statistics of emerging mar-ket stock returns given in Table ?? reveals that natural gas futuresreturns are much closer to being normal despite their higher standarddeviation.
Mean -0.09538 LBQ(5) 5.31 (0.379)
Median -0.09935 LBQ(10) 7.526 (0.675)
Variance 12.7719 LBQ2(5) 61.81 (5.13e-12)
Standard deviation 3.5738 LBQ2(10) 74.195 (6.82e-12)
Table 1: Summary Statistics for Nearby Month Futures Returns
The Ljung-Box test at lags 5 and 10 does not reject the null hy-pothesis of no autocorrelation in raw returns. However, for squaredreturns, the null hypothesis of no autocorrelation is rejected strongly.This suggests that there is strong volatility persistence in the data.Therefore, the LM-ARCH test (Engle, 1982) is administered and thenull hypothesis of no ARCH effects is strongly rejected. An augmentedDickey Fuller test (Dickey and Fuller, 1979) is administered with aconstant and 12 lags in the unit root regression without a time trend.The null hypothesis of non-stationarity is rejected, so non-stationarityis not a problem in the analysis.
The gas storage report was being announced by the American GasAssociation (AGA) until May 2002 on Wednesdays at 2:00 pm. Sincethen, the report is released every Thursday at 10:30 am by the EnergyInformation Administration (EIA). The report provides informationon storage levels the Friday before, net weekly changes in storagelevels, and the storage levels one-year before. In addition, the five-year historical average for the equivalent time period and the differencebetween the current level and the five-year average are reported. Thedeviation from the historical average is the key variable in this study.
The storage data are publicly available from the EIA website.6
From all the above-mentioned variables, the downloadable data onlyincludes the storage levels. we followed a two-step procedure to con-struct the deviations from historical levels. First, the weekly data are
6The data are downloadable from http://tonto.eia.doe.gov/dnav/ng/ng stor wkly s1 w.htm
6
interpolated to obtain a daily storage level data.7 This is needed sothat storage data for the same day in each of the previous five yearsare available. In the second step, the storage deviation variable SDt
is constructed as
SDt = St,s −1
5
i=5∑i=1
St,s−i (3)
where St,s is the level of storage on day t in year s. This two-stepprocedure is the same methodology followed by the EIA to constructthe five-year averages and the deviations from historical averages whilepreparing the storage report.8
A time series plot of the working gas in underground storage ispresented in Figure 1 panel A.
The demand is highly seasonal, and the supply is uniform acrossseasons in the natural gas market. As such, storage levels also exhibitvery strong seasonality because it balances the difference between thesupply and demand providing a buffer to the market. In Figure 1panel B, the time series plot of the storage deviation from the five-year historical average is provided. Note that during the winters of2001 and 2003 the storage levels were significantly lower than theirfive-year historical averages, and these periods also coincide with highvolatility in natural gas markets.
we obtained the daily realized temperature data running from Jan-uary 1960 to April 2008 from the trading floor of a very active natu-ral gas trading firm. The dataset includes daily minimum and dailymaximum temperatures for seven locations: Atlanta Hartsfield Air-port, Chicago Midway Airport, Chicago O’Hare Airport, Dallas ForthWorth Airport, New York Central Park, New York JFK Airport, andNew York LaGuardia Airport. The discussion of the weather modelingis left to Section 3.2.
2.1 A First Look at the Natural Gas Volatility
In this section, we analyze the nearby month futures volatility with re-spect to natural gas market fundamentals without imposing any econo-metric structure. The previous literature emphasizes higher volatility
7Simple linear interpolation is used here.8A complete documentation of the EIA methodology can be found at the link:
Figure 1: Working Gas In Underground Storage and Its Deviation From FiveYear Historical Mean
on Mondays and storage report announcement days. In some stud-ies, winter was found not to be associated with higher volatility after
8
controlling for other factors. we believe this result is driven by usinga broad definition for winter: November to March. This coincideswith the period when withdrawal from storage is higher than the in-jection to storage; hence, known as the withdrawal season. Here, werestrict the definition of winter to include only December, January,and February.
In Table 2, the standard deviations of nearby month futures re-turns are presented for several subgroups based on certain character-istics. There are several important patterns in this table. In panelA, standard deviations are calculated for Mondays, storage report an-nouncement days (SDDAYs), and all other days along with winter andnon-winter days.
Panel-A: By Mondays SDDAYs and WinterWinter Non-Winter All Seasons
Monday 6.30 4.24 4.79SDDAY 4.37 3.66 3.85Other Days 3.43 2.79 2.96All Days 4.27 3.31 3.57
Panel B: By Winter and BidweekBidweek Non-Bidweek All Days
Table 2: Standard Deviation of Daily Returns Broken into Groups
The standard deviation is highest for Monday returns, 4.79. Al-though lower than Mondays, the standard deviation on storage reportannouncement days, 3.85, is higher than the standard deviation forall other days, 2.96. The winter effect is also evident. The returnvolatility for winter days, 4.27, is higher than the return volatility for
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(a) Days and Winter (b) Winter and Bidweek (c) Winter and Storage Deviation
Figure 2: Standard Deviations of Futures Returns Across Days, Seasons,Bidweeks and Storage Levels
non-winter days, 3.31. The reason for higher volatility on storage re-port announcement days is obvious. The storage report is regarded asthe most important piece of information by market practitioners, andit is priced as soon as it becomes available. Using intraday data, Linnand Zhu (2004) showed that the new information is absorbed into theprices within minutes. The volatility of the 10:30-10:35 am intervalis much higher than any other five-minute interval during the tradingday. However, when Thursdays were excluded from their sample, thiseffect is completely gone. Fleming et al. (2004) explain the Mondayeffect with the continuous flow of weather information. They find thatthe variance ratios of trading to nontrading periods are significantlylower for weather sensitive markets compared to equity markets. Theyattribute the difference to the continuing flow of weather informationover the nontrading period, whereas the information flow for equitymarkets is reduced in the nontrading period. Additionally, the ra-tios get even lower over the weekend compared to weekdays, whichsupports their hypothesis further.
The volatilities for subgroups generated by the Cartesian productof days and seasons also make complete sense. Standard deviationsfor different days have the same order both in and outside of winter,
10
and winter volatility is consistently higher than non-winter volatilityin any subgroup of days. A bar plot of the statistics presented in Table2 panel A is presented in Figure 2(a).
Panel B of Table 2 analyzes the standard deviations of the returnsin the same way, but now the sample is split as the bidweek andnon-bidweek days across winter and non-winter days. In the naturalgas market, the largest volume of spot trading occurs in the last fivebusiness days of every month known as “bidweek.” This is whenproducers are trying to sell their core production and consumers aretrying to buy for their core natural gas needs for the upcoming month(see www.naturalgas.org). The average prices set during bidweek arecommonly the prices used in physical contracts over the next month.Since the trading in futures contracts terminates on the third businessday before the first business day of the next month and bidweek is thelast five business days of the month, the last three business days oftrading for the nearby month contract coincides with the bidweek.In analyzing the maturity effect first proposed by Samuelson (1965),we use bidweek as a natural cutoff point. In panel B of Table 2,the standard deviation of the returns on the bidweek days is 4.03,whereas the standard deviation of returns not on the bidweek days is3.49. The maturity effect is particularly strong during winter. Thestandard deviation of bidweek days in winter is 5.57, but the standarddeviation of non-bidweek days in winter is 4. On the other hand, thereis only a marginal difference between the standard deviations outsideof winter, 3.34 for bidweek days and 3.31 for non-bidweek days. Abar plot of the statistics presented in Table 2 panel B is provided inFigure 2(b).
In panel C, SD is the storage deviation variable constructed by(3). The short-term volatility studies in the literature found that lowerthan expected storage levels results in increased volatility because thissignals a tight supply situation to the natural gas market (Mu 2007).However, to my knowledge, there are no studies analyzing the effectsof the interaction of storage levels with seasonality. In panel C, thefinding of the previous literature is first confirmed and then challenged.The standard deviation of the periods in which SD < 0; that is,storage level is lower than five-year historical average, is 3.8, whereasthe standard deviation of the periods in which SD > 0 is 3.51. Thisconfirms the previous literature. However, a more careful examinationof the table reveals that this relationship is valid only during the wintermonths when supply tightness is really a big problem. During the
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winter, those periods with SD < 0 have a standard deviation of 5.66,whereas those periods with SD > 0 have a standard deviation of3.73. During the non-winter months, the effect is just the opposite:Returns of those periods with SD > 0 have a standard deviationof 3.42, whereas the returns of those periods with SD < 0 have astandard deviation of 2.84. This should be because of the concernsregarding the storage capacities. Very high storage levels during non-winter months when demand is minimal increase the concerns aboutwhether there will be enough storage space to store the production forwinter demand. This puts a pressure on the price of storage space,which naturally spills over to natural gas prices, causing excessivevolatility. Lee Van Atta (2008) cites the excess volatility as one of themost important reasons leading to excessive storage construction overthe past few years. This view is consistent with the finding here. Abar plot of the statistics presented in Table 2 panel C is presented inFigure 2(c).
2.2 Levene Tests for Variance Equality
we formally test for the equality of variances among some subsamplesof the data in this section. The robust Brown-Forsythe (1974) typeLevene (1960) test statistics and associated probability values are pre-sented in Table 3. In each row of the table the null hypothesis of equalvariances across the J groups in the second column is tested. In thefirst row, the null hypothesis of equal variances for winter and non-winter returns is rejected. In the second row, equality of variancesfor Mondays, storage report announcement days, and all other days isrejected. In the third row, six groups are constructed as the Cartesianproduct of the winter groups in the first row and days in second row.The equality of variances across these groups is rejected.
12
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13
These results are consistent with those presented in Table 2 andFigure 2. The result in row four—equal variances for bidweek daysand non-bidweek days cannot be rejected—is somewhat surprising. Inrow five, four groups are produced from the Cartesian product of win-ter groups in row one and bidweek groups in row four. The equalityof variances is rejected in this case. However, this may be becauseunequal variances of winter and non-winter dominating the analysis.To get rid of that effect, we test the equality of variances for thosebidweek and non-bidweek days only in winter. The results in row sixstill cannot reject the equality of variances, although the probabilityvalue gets much smaller compared to that in row four. Therefore,there is not strong evidence for unequal variances for bidweek andnon-bidweek days. In row seven, the equality of variances for peri-ods with positive storage deviation and negative storage deviation istested, and the equal variance hypothesis can not be rejected. Con-structing four groups based on the Cartesian product of winter andthe sign of storage deviation, the equal variance hypothesis is rejected.In the last row, to control for the winter effect, we constructed twogroups with positive and negative storage deviations only from winterreturns. Now, the null hypothesis of equal variances is rejected. This isconsistent with my hypothesis that storage deviation has asymmetriceffects during winter and outside of winter.
3 Empirical Model and Estimation Re-
sults
The Ljung-Box test statistics for squared returns in Table 1 suggestthat there is strong volatility persistence for natural gas nearby monthfutures returns. Consequently, the LM-ARCH tests confirmed the ex-istence of ARCH effects. In order to take this persistence into account,a GARCH volatility model is adopted as the econometric tool in thissection. The focus of this study is completely on the estimation andout-of-sample prediction of daily volatility. Therefore, no structure isspecified for the mean equation of the GARCH model. Instead, zeroexpected return is assumed. Since the day-ahead return is very dif-ficult to forecast, this approach is common for volatility forecastingstudies. Consequently, the specification of the empirical model is as
14
follows:
rt = σt zt
σ2t = ω + αr2
t−1 + βσ2t−1 + γXt (4)
where rt is the nearby month futures return on day t given by (1)and (2), σt is the conditional volatility, zt is the shocks to the datagenerating process with E[zt] = 0, E[z2
t ] = 1. Lastly, Xt is a vector ofexogenous variables capturing the dynamics of the natural gas marketvolatility. The parameters of the model are obtained by maximizingthe following log–likelihood function:
logL(ω, α, β, γ) ∝t=n∑t=1
(logσ2
t (ω, α, β, γ)− r2t
σ2t (ω, α, β, γ)
). (5)
This likelihood function assumes that the shocks zt are normally dis-tributed.
3.1 Day-of-the-Week, Seasonality and Matu-rity Modeling
Inspired by the statistical analysis presented in panel A of Table 2,the following variables are included in the model.SDDAYt: A dummy variable for the storage report announcementdaysMONt: A dummy variable for MondaysWINt: A dummy variable for winter days, with the winter defined asDecember, January, and FebruaryAdditionally, panel B of Table 2 presents preliminary evidence regard-ing the maturity effect on futures volatility. However, the Brown-Forsythe type Levene test does not confirm the unequality of thevariances for the bidweek and non-bidweek days. Therefore, usinga dummy variable for bidweek is not justified. Instead, we constructmore general variables to capture the maturity effect:TTM : The number of business days to the maturity of nearby monthfutures contractTTMWIN : Time to maturity variable on winter days. It is con-structed as TTMWINt = TTMt ∗WINt to model the asymmetricmaturity effect across seasons.
The results of the estimations including these first set of exoge-nous variables are presented in Table 4. The first estimation is for a
15
simple GARCH(1,1) model. Starting with the second estimation, onemore variable is included in the model at each time. The estimationresults are consistent with the previous data analysis. Volatility issignificantly higher on storage report announcement days, Mondays,and winter days. In estimation-5, the time to maturity (TTM) vari-able is significant and has the correct negative sign. So, the volatilityof futures returns increases as the maturity gets closer. However, itlost its significance in estimation-6 after TTMWIN is also includedin the estimation. This is consistent with the previous idea that thematurity effect is present only during winter months. More formally,the coefficient of time to maturity variable during non-winter monthsis γ4. On the other hand, during the winter months, it is γ4 + γ5.Therefore, testing for the conditional hypothesis that the maturity ef-fect is present only during the winter requires a statistical test of thenull hypothesis H0 : (γ4 = 0 and γ4 +γ5 < 0). The t statistic for γ4 is-0.15. Also, the t statistic for γ4 + γ5 is calculated using the variance-covariance matrix of estimated parameters and reported in Table 4as -8.416. This confirms the null hypothesis that the maturity effectis present only in winter. In the final estimation, the non-significantTTM variable is dropped from the estimation.
16
Model
ωα
βγ
1γ
2γ
3γ
4γ
5H
-L
0.25
50.
077
0.90
8(1
)(3.5
3∗∗
∗)
(8.6
2∗∗
∗)
(75.9
4∗∗
∗)
45.8
6
0.50
00.
073
0.90
24.
211
(2)
(−2.3
0∗∗
)(7.0
2∗∗
∗)
(62.9
9∗∗
∗)
(3.9
9∗∗
∗)
27.3
8
-0.9
970.
080
0.87
34.
323.
69(3
)(-
0.9
97)
(9.3
9∗∗
∗)
(66.0
8∗∗
∗)
(4.5
6∗∗
∗)
(6.3
2∗∗
∗)
14.4
-1.0
20.
078
0.85
94.
094.
470.
341
(4)
(−5.5
3∗∗
∗)
(6.0
0∗∗
∗)
(44.6
5∗∗
∗)
(4.7
3∗∗
∗)
(7.7
3∗∗
∗)
(4.3
5∗∗
∗)
10.6
5
-0.5
030.
070
0.87
44.
314.
010.
162
-0.0
47(5
)(−
1.6
6∗)
(5.6
1∗∗
∗)
(44.0
9∗∗
∗)
(4.9
6∗∗
∗)
(6.8
1∗∗
∗)
(1.6
45∗)
(−2.6
9∗∗
∗)
12.0
2
-1.0
60.
073
0.86
34.
694.
442.
40-0
.002
9-0
.217
(6)
(−3.1
1∗∗
∗)
(5.6
8∗∗
∗)
(38.1
3∗∗
∗)
(5.3
8∗∗
∗)
(6.9
8∗∗
∗)
(7.1
8∗∗
∗)
(−0.1
5)
(−6.7
3∗∗
∗)
10.4
8
-1.1
00.
073
0.86
54.
674.
412.
38-0
.215
(7)
(−5.5
0∗∗
∗)
(5.6
8∗∗
∗)
(38.8
4∗∗
∗)
(5.3
5∗∗
∗)
(7.3
0∗∗
∗)
(9.3
8∗∗
∗)
(−8.2
9∗∗
∗)
10.8
3
t.stat(γ̂
4+γ̂
5)
=−
8.41
6(I
nE
stim
atio
n-6
)
Sig
nifi
cance
Codes
:*
10%
,**
5%,
***
1%
r t=
σtz t
σ2 t
=ω
+αr2 t−
1+βσ
2 t−1
+γ
1SDDAYt+γ
2MON
t+γ
3WIN
t+γ
4TTM
t+γ
5TTMWIN
t
Tab
le4:
GA
RC
HE
stim
atio
nR
esult
sF
orN
earb
yM
onth
Futu
res
Ret
urn
s
17
A high level of persistence in natural gas futures volatility, as mea-sured by the sum α + β, is evident in these estimations. The volatil-ity literature suggests that the persistence in volatility might be theresult of driving exogenous variables that are persistent themselves.Therefore, such variables should reduce the level of volatility persis-tence once they are included in the conditional variance equation ofGARCH models. This kind of behavior is observed in the parameterestimates with exogenous variables in the volatility equation. Whileα + β = 0.985 in the simple GARCH estimation, it reduced to 0.938in the final estimation. The half-life of a volatility shock is defined asthe time it takes for half of the shock to vanish and is given by:
Half − Life = log(0.5)/log(α+ β) (6)
The half-lives estimated for all models are presented in the finalcolumn of Table 4. After including the persistent covariates, the half-life of a volatility shock decreased to 10.83 days from the 45.86 daysin the simple GARCH model.
3.2 Storage and Weather Modeling
In order to incorporate the asymmetric effects of storage levels duringdifferent seasons, two additional variables are constructed: SDt andSDWINt. The first variable SDt is the same variable constructedin Section 2. It is the deviation of storage level from its five-yearhistorical average. Once SDt is calculated for a storage report an-nouncement day, the same number is used on the following days untilthe next storage report announcement. This is different from the pre-vious literature that included the storage surprise variable only for theannouncement days (Mu 2007; Gregoire and Boucher 2008). This isbecause we regard this variable not only as a proxy for storage sur-prise but as a proxy for supply tightness during winter and as a proxyfor tightness in storage space supply in other seasons. The secondvariable SDWINt is a proxy for supply tightness during winter. Itis constructed as SDWINt = SDt ∗ WINt. This variable enablesus to model the asymmetric effect of storage levels for different sea-sons. The expectation is a positive coefficient for SDt and a highernegative coefficient for SDWINt to confirm the hypothesis that lowstorage levels increase the volatility in winter, whereas high storagelevels increase the volatility at other times.
18
The weather modeling is accomplished by the well-known degreeday variables. These are quantitative indices used to reflect the de-mand for energy. Experience shows that there is no need for heatingor cooling if the outside temperature is 65oF . Consequently HeatingDegree Days and Cooling Degree Days variables are defined as:
where Tave,t is the average of the maximum and minimum observedtemperature on day t. There are two common ways of modelingweather shocks, either with ex-post forecast errors or with tempera-ture anomalies, defined as the deviation of degree days variables fromtheir seasonal norms (Mu 2007). Here, we follow the second approachbecause the forecast data are not available. The following weathershock variables are constructed: HDD.Shockt: This is defined as thedeviation of Heating Degree Days from the seasonal norms over theforecasting horizon. Following Mu (2007), the forecasting horizon ischosen to be seven days since the weather forecasts from the publicmedia are typically broadcast for seven days ahead.9
HDD.Shockt =i=t+7∑i=t+1
(HDDi −HDD.Normi), (8)
where HDD.Normt is the historical 30-year average of HDD on day t.The historical 30-year average is the definition of the National WeatherService (NWS) for the seasonal norm. Since we do not have the actualforecast data, the realized HDD is used in creating this variable.CDD.Shockt: This is defined as the deviation of CDD from the sea-sonal norms and calculated in the same way as:
CDD.Shockt =i=t+7∑i=t+1
(CDDi − CDD.Normi), (9)
where CDD.Normt is the 30-year historical average of CDD on dayt.
Both weather variables are constructed for Chicago, New York,Atlanta, and Dallas. Then, the natural gas consumption weightedaverage of these locations is calculated.10 These national averages for
9Results are robust to the choice of a forecasting horizon as 8, 9, or 10 days.10we used the same weights used in Mu (2007): 0.42 for Chicago, 0.28 for New York,
0.17 for Atlanta, and 0.13 for Dallas.
19
weather shock variables are plotted in Figure 3. Heating degree dayshocks are closer to zero in summer months, whereas cooling degreeday shocks are closer to zero in winter months since both the thirty-year historical averages and the actual realizations get closer to zero.
Estimation results including the storage and weather variables arepresented in Table 5. In estimation-8, only the two storage variablesare added to the last estimation in Table 4. Both variables have thecorrect sign and are significant at all conventional levels. The positivesign for the coefficient of SDt suggests that high storage levels increasethe short-term volatility during the non-winter period. Also, the neg-ative coefficient for SDWINt is greater than the positive coefficientof the SDt, which suggests that it is the low storage levels resulting inhigh volatility during winter months. This asymetric effect is testedmore formally later in estimation-11.
20
(a) HDD Shocks
(b) CDD Shocks
Figure 3: National Weather Shock Variables
21
Model
ωα
βγ
1γ
2γ
3γ
4γ
5γ
6γ
7γ
8H
-L
-1.2
90.
070.
839
5.58
5.43
3.54
-0.2
80.
0006
2-0
.001
7(8
)(-
5.46
)(5
.09)
(31.
85)
(6.5
5)
(7.9
4)
(9.2
4)
(-7.4
9)
(3.7
2)
(-5.0
0)
7.24
***
***
***
***
***
***
***
***
***
-1.0
50.
076
0.85
05.
034.
362.
74-0
.24
0.02
70.
062
(9)
(-5.
47)
(5.6
7)(3
5.08
)(5
.87)
(6.2
9)
(7.5
4)
(-6.6
7)
(1.6
9)
(1.8
5)
9.01
***
***
***
***
***
***
***
**
-1.0
50.
073
0.88
84.
643.
040.
027
0.05
5(1
0)(-
5.62
)(6
.47)
(49.
87)
(5.1
4)
(4.8
9)
(3.7
5)
(2.0
4)
9.01
***
***
***
***
***
***
**
-1.2
80.
068
0.83
35.
735.
543.
77-0
.29
0.00
064
-0.0
017
0.01
70.
015
(11)
(-5.
13)
(4.9
8)(3
0.34
)(6
.64)
(7.3
5)
(8.6
6)
(-6.9
8)
(3.3
4)
(-4.1
9)
(0.8
42)
(0.4
07)
6.65
***
***
***
***
***
***
***
***
***
t.stat(γ̂
5+γ̂
6)
=−
3.09
4(I
nes
tim
atio
n-1
1)
Sig
nifi
cance
Codes
:*
10%
,**
5%,
***
1%
r t=
σtz t
σ2 t
=ω
+αr2 t−
1+βσ
2 t−1
+γ
1SDDAYt
+γ
2MON
t+
γ3WIN
t+γ
4TTMWIN
t+
γ5SD
t+
γ6SDWIN
t
+γ
7HDD.Shockt
+γ
8CDD.Shockt
Tab
le5:
GA
RC
HE
stim
atio
nR
esult
sw
ith
Sto
rage
and
Wea
ther
Var
iable
sF
orN
earb
yM
onth
Futu
res
Ret
urn
s
22
In estimation-9, only the two weather variables are added to thelast estimation in Table 4. They are both significant at the 10 percentconfidence level with a correct positive sign. Higher degree days thanthe seasonal norms increase short-term volatility. One unexpectedresult is that the significance of CDD.Shock is stronger than the sig-nificance of HDD.Shock. This might be due to the other variablesaccounting for higher volatility in winter. If the winter dummy vari-able and its interaction term with the maturity were excluded from themodel as in estimation-10, the significance of HDD.Shock becomesstronger than the significance of CDD.Shock. In this estimation,HDD is significant at the 1 percent level and CDD is significant at the5 percent level. Lastly, in estimation-11, we include the storage andweather variables together. The inference for the storage variablesremains the same. The asymetric effect of storage deviation variableSDt across seasons can be formally tested with the null hypothesis ofH0 : (γ5 > 0 and γ5 + γ6 < 0). The t statistic for γ5 is 3.34. Also,the t statistic for γ5 + γ6 is calculated using the variance-covariancematrix of estimated parameters and reported in Table 5 as -3.094.This confirms the asymetric effect of the storage variable across sea-sons. As for weather variables, they still have the correct positivesigns, but they lost their significance after the addition of the stor-age variables. Note that the storage variables were included in themodel as a proxy for supply, and the weather shocks were includedas a proxy for demand in the natural gas market. However, storagelevels could be thought of as the result of the combination of supplyand demand forces, thereby providing an explanation for the reduc-tion in the significance of weather variables. The winter dummy isanother factor that reduces the significance of weather variables dueto nonorthogonality as discussed before.
4 Out-of-Sample Forecast Accuracy
Recent papers employing GARCH models to investigate the effects ofnatural gas market fundamentals on the volatility dynamics of natu-ral gas futures report results only for in-sample estimations. Theseestimation results provide valuable information for understanding thevolatility dynamics of natural gas futures. However, the out-of-samplepredictive power of these augmented GARCH models has not beentested. In this section, day-ahead volatility predictions are made for
23
natural gas nearby month futures using simple GARCH models, aswell as their augmented counterparts, and the accuracy of these fore-casts are compared.
The empirical methodology followed here is known as a sliding win-dow scheme. To make a prediction for day t where t ∈ {501, 502, ..., T},only the returns {rt−1, rt−2, ...rt−500} are used. So, the length ofthe sliding window is chosen as 500 observations. That is, returns1 through 500 are used to predict the volatility for day 501; returns2 through 501 are used to predict the volatility for day 502, and soon. Since T = 1, 823, there are T − 500 = 1, 323 volatility predic-tions. One problem in out-of-sample forecasting is that the variablesHDD.Shockt and CDD.Shockt are using information from the fu-ture. On day t, the volatility for day t + 1 is being forecasted, butat that time these variables are not available yet in the informationset. To solve this problem, first weather shock forecasts are obtainedby fitting an ARIMA(1,2,1) model to the last 500 calendar days oftemperature data for all four cities. The ARIMA(1,2,1) is chosenbased on the Schwarz information criterion (SIC). The natural gasconsumption weighted average of weather shock variables across thefour cities is calculated as the final weather shock forecasts. Then,the weather shock forecasts are used in augmented GARCH modelsto forecast the volatility. A simpler approach is to use the appropriatelags of weather shock variables. This can be regarded as forecastingthe next seven days’ weather shocks as being equal to the last sevendays’ weather shocks. The presented results are from the ARIMAforecasting approach, but using a simpler lag approach provides thesame results.
4.1 Other Simple Models for Forecasting
Random Walk Model:With a random walk assumption, the volatility forecast for day t isthe realized volatility on day t−1. It is used as the benchmark model.
σ̂t = |rt−1| . (10)
Moving Average Model:Moving average models are widely used by natural gas market prac-titioners. In this paper, 20-day and 60-day moving averages are used
24
that correspond to one-month and three-month trading days.
σ̂t =
√√√√ 1
m
m∑i=1
r2t−i . (11)
Implied Volatility:Annualized implied volatilities for the closest-to-the-money call op-tions are obtained from the trading floor of a vey active natural gastrading firm. Dividing annualized implied volatility by
√250, the daily
implied volatilities are obtained. Forecasting can be done as follows:
σ̂t = σimp
t−1/√
250 . (12)
4.2 Forecast Accuracy Results
After making the forecast and observing the realized volatility thenext day, the volatility forecast error can be defined as
FE = σt − σ̂tFE = |rt| − σ̂t ,
where σ̂t is the forecasted volatility, and σt is the realized volatilityfor day t. The latter equality follows because, in the absence of in-traday data, the most common approach in the literature is to usethe absolute value of return as the realized volatility. Three statis-tical measures are used for measuring forecast accuracy. These aremean absolute error (MAE), mean squared error (MSE), and Theil’sU statistic.
MAE =1
n
n∑t=1
∣∣∣|rt| − σ̂t∣∣∣ , (13)
MSE =1
n
n∑t=1
(|rt| − σ̂t
)2, (14)
Theil′s U =
√∑nt=1
(|rt| − σ̂t
)√∑n
t=1
(|rt| − |rt−1|
) , (15)
Theil’s U statistic can be thought as a relative accuracy measure.It is the ratio of the root mean squared error of the chosen model to
25
the root mean squared error of the random walk model. Out-of-sampleforecasting accuracy measures for the simple and augmented GARCHmodels as well as other simple forecasting methods are presented inTable 6.
Table 6: Accuracy Statitics for Out-of-Sample Forecasting
Model 7 is based on the estimation-7 in Table 4. The forecastingaccuracy of this model is slightly worse than the simple GARCH modelgiving marginally higher MAE, MSE, and Theil’s U statistics. Model8 and 9 are based on estimations 8 and 9 in Table 5. Both modelsperform better than the simple GARCH model. Moreover, storagevariables in model 8 improve the performance much more than theweather variables in model 9. Lastly, model 11 includes all variablesand among all the GARCH models, it provides the lowest MAE, MSE,and Theil’s U statistics, making a great improvement on the simpleGARCH model.
So, the evidence is not conclusive that any augmentation of theGARCH models would increase the accuracy in out-of-sample fore-casting as in the case of model 7. However, with carefully chosenvariables to account for natural gas market fundamentals, it is possi-ble to increase the forecast accuracy as in models 8, 9, and 11.
The simple forecasting schemes generally perform very poorly withthe exception of the 20-day moving average. MA(20) method ranksfirst with a very small margin in terms of MAE and third in termsof MSE and Theil’s U. The MSE measure penalizes models with thesquare of their errors, and the MA(20) does not perform as good aswith the linear penalty function used in MAE calculation. Therefore,
26
this simple model is producing very good forecasts in general, butonce it is wrong it is way of the target. On the other hand, model 11that includes all fundamental variables within a GARCH frameworkranks second in terms of MAE and first in terms of MSE and Theil’sU, thereby consistently providing good forecast accuracy.
5 Conclusion
Recently, a new literature emerged on modeling short-term volatilitydynamics of natural gas futures. This research focuses on the augmen-tation of GARCH models and its variants with the natural gas marketfundamentals in order to understand the sources of high volatility innatural gas prices. In this paper, several new findings contributing tothis literature have been presented, and more importantly forecastingthe accuracy of these models is analyzed for the first time.
First of all, the effect of storage levels on short-term volatilityis asymmetric across the seasons. During the winter months, lowerstorage levels than the five-year historical average were found to beincreasing the short-term volatility. In contrast, it is the high lev-els of storage causing excess volatility in other seasons. This canbe attributed to the changing concerns of market players at differentseasons. In the winter, low storage levels are perceived as a tightsupply situation causing excess volatility. At other times, the marketis mainly concerned about the storage space supply. Therefore, highlevels of storage cause excess volatility.
Secondly, the maturity effect for natural gas nearby month fu-tures is found to be a significant determinant of volatility only in thewinter months. This result is confirmed by both data analysis andeconometric estimation of the GARCH models, including the matu-rity variable and its interaction with the seasonality in the volatilityequation. Since winter is the season when demand is highly inelastic,traders might be overreacting to new information arrival closer to thematurity and this causes excess volatility.
In addition to these new findings, this study confirms some ofthe previous results in the literature. Higher volatility is observedon Mondays possibly due to the accumulation of weather informationover the non-trading weekend. Storage report announcement daysalso exhibit higher volatility than other days since new arriving in-formation is priced very fast in this case. Volatility on winter days,
27
defined as December, January, and February, is found to be higherthan other seasons. Lastly, weather shocks in excess of the seasonalnorms increase the short-term volatility. However, when storage vari-ables and the winter dummy is included in the model, they take thesignificance of weather variables. This might be because the storagevariables are taking care of both supply and demand dynamics and/ornon-ortgonality with the winter dummy.
As for forecasting accuracy, the augmented GARCH models withcarefully chosen fundamental variables have the potential to decreaseMAE, MSE, and Theil’s U statistics. The model using storage andweather variables in addition to other variables capturing the Monday,storage report announcement, winter, and maturity effects providesthe best forecasting accuracy, thereby greatly improving on the simpleGARCH model forecasts. This is a very important finding becausebetter volatility forecasts can be used in option pricing and hedgingnatural gas exposures. Fleming et al. (2001) suggests focusing on theeconomic significance of time varying predictable volatility insteadof evaluating the statistical performance of volatility models. Futureresearch can be conducted in such applications of augmented GARCHmodels for natural gas volatility.
Lastly, we found that a simple GARCH model provides very goodbacktesting result in risk estimation, and there is no room for im-provement by augmenting the model with market fundamentals. Thisis because natural gas futures return distribution does not exhibit veryfat tails. Overall, the results suggest that the volatility forecasting per-formance can be increased by augmentation of the GARCH models,whereas a simple GARCH model can be preferable for the risk mea-surement of linear portfolios considering the simplicity advantage.
28
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