Volatility in the Natural Gas Market: GARCH, Asymmetry, Seasonality and
Announcement Effects*
Duong T. Le*
Division of Finance
Michael F. Price College of Business
University of Oklahoma
Norman, OK 73019-0450
Abstract
This paper examines the cause and behavior of natural gas volatility. Although natural
gas prices are among the most volatile, they have received very little academic scrutiny
heretofore. I theorize and find that (1) the natural gas market is characterized by volatility
persistence (2) a negative shock to the natural gas market has a smaller impact on
predicted volatility than a positive shock of the same magnitude (3) there are day-of-theweek effects in the natural gas market, (4) natural gas volatility is higher in the winter
months, (5) the weekly storage report release causes increased volatility on
announcement days and (6) the announcement has no further impact on volatility on the
subsequent days.
* This research is conducted under the guidance of Dr. Louis H. Ederington, who is also
the PhD coordinator for the Division of Finance.
1
1. Introduction
In this paper, I examine the cause and behavior of volatility in the US natural gas
market from January 1997 to December 2005. Natural gas is an essential energy source in
the US accounting for about one quarter of the nation’s energy consumption and a higher
percentage of energy production.1 Trading activity in the natural gas market has also
increased significantly in recent years. For example, the notional value of natural gas
futures contracts traded daily in 2005 ranged from $0.63 billion to $2.75 billion with an
average of approximately $1.0 billion.
Particularly since its evolution from a highly regulated market to a largely
deregulated market in which prices are driven by supply and demand, natural gas is one
of the most volatile markets. The average natural gas price in 2005 was four times the
average price in 1997 and double the average price in 2000. In 2005, the annualized
standard deviation of the daily percentage change in natural gas prices was 56.81%. By
comparison that number was 8.84% for the US dollar-Euro exchange rate, 12.36% for the
S&P 500 and 16.61% for the 10-year T-bond interest rates2.
This high volatility in natural gas prices is likely due to the short-term inelasticity
of supply and demand. Natural gas supplies are often constrained by storage levels so
natural gas suppliers are unable to increase production levels in a short time period.
Similarly, when a sharp increase in natural gas prices occurs, it is difficult for consumers
to quickly reduce their consumption, especially during winter. Since natural gas supplies
cannot rapidly adjust to match demand changes, supply and demand imbalances may
result in sharp price changes. This high variability in natural gas prices makes it
extremely difficult for natural gas consumers to forecast their costs and for natural gas
producers to forecast their profits. The desire to protect market participants against such
price fluctuations has led to the creation of and active trading in natural gas futures,
options and swaps where the market value of the latter two is determined mostly by the
natural gas volatility.
1 Natural gas has various applications, including heating more than 59 million homes and 5 million businesses,
powering industrial and agricultural production and generating a substantial amount of peak electricity needs. Source:
Natural Gas Analysis of Changes in Market Price, Report to Congressional Committees and Members of Congress.
2
The data for the US dollar-Euro exchange rate, S&P 500 and the 10-year T-bond interest rates were collected from
http://www.oanda.com, CRSP and the Department of the Treasury website (http://www.treasury.gov), respectively.
2
Volatility in other markets has been a subject of intense research activity in recent
decades. While it is difficult to forecast the direction of future price changes from past
price behavior, the absolute magnitude of price changes, i.e. volatility, has proven much
more predictable. It is generally found that highly volatile markets tend to be followed by
volatile markets and stable markets by stable markets. Whether this is also the case for
natural gas is unknown at present. The vast majority of the research on market volatility
has focused on the volatility of financial markets such as stock, bond, interest rates and
foreign exchange futures markets, etc. Despite the fact that natural gas prices tend to be
more volatile than most other prices, research into the cause and behavior of natural gas
volatility is very limited. For instance in a well-known and comprehensive study of the
volatility literature, Poon and Granger (2003) surveyed 93 articles examining volatility in
all sorts of markets but, to the best of my knowledge, not one of these examined volatility
in natural gas prices. I intend to fill this gap in our understanding.
An understanding of the cause and behavior of volatility in the natural gas market
is essential to market participants since the market value of risk management products
such as options and swaps depends mostly on volatility. In this paper, I examine volatility
in the natural gas market and attempt to answer the following questions: (1) Does
volatility in the natural gas market follow a GARCH type process in which future
volatility is partially predictable from past volatility? (2) Does an unexpected drop in the
natural gas price one day increase predicted volatility for the following day more or less
than an unexpected increase in price of similar magnitude? (3) Is there a day-of-the-week
pattern in the natural gas market? (4) Does natural gas volatility increase in winter
months? (5) Does the release of the Weekly American Gas Storage Report (hereafter
called the weekly storage report) increase natural gas volatility on the announcement
days? (6) Does the report release have any further impact on volatility on the days
following the announcement?
A few of these questions have been discussed in earlier studies but there are
important difference between their studies and mine. Using monthly data and an ARCH
type model, Susmel and Thompson (1997) found no evidence of seasonality in the natural
gas market. Moreover, they found asymmetric volatility in the natural gas market but in a
direction opposite to my findings. They estimated that a negative shock has a larger
3
impact on volatility than a positive shock of equal size whereas I find that a negative
shock has a smaller impact on volatility than a positive shock of equal size. Utilizing
intraday data and a GMM regression model, Linn and Zhu (2004) studied the impact of
the storage report announcement on the intraday natural gas volatility within the day of
the announcement and found that volatility remains higher than normal for up to 30
minutes after the announcement. In contrast, I analyze the response of volatility to the
report release not only on the days when the announcements occur but also on the days
after and in a GARCH framework. Also neither of these previous studies considered the
possibility that natural gas volatility may be higher in the winter months.
My paper has a major methodological improvement over a number of studies on
volatility in other markets (See, for example, Berument and Kiymaz (2001)). I use a
variant of a regime-switching GARCH type model outlined in Jones et al. (1998) which
allows conditional volatility to differ on each day of the week, in the winter months, on
announcement days and non-announcement days. The separation of volatility into a
persistent part and a non-persistent part allows me to implement a nice clean study of the
determinants of natural gas volatility.
To the best of my knowledge, my paper will be the first full study of persistence,
asymmetry, seasonality and announcement effects within a GARCH framework for this
market.
The paper is organized as follows. In Section 2, I propose hypotheses of the study.
The data is presented in Section 3. In Section 4, I employ several GARCH type models
for natural gas volatility. I find evidence of volatility persistence, asymmetric volatility,
days-of-the-week and winter effects in the natural gas market. Furthermore, storage
report announcements cause increased volatility but this increase does not persist to the
following day. Section 5 concludes the paper.
2. Hypotheses
In this study, I attempt to answer the following questions:
1. Are natural gas prices characterized by volatility persistence as has been
documented in other markets? It has been observed in other markets that volatile markets
tend to follow volatile markets and stable markets tend to follow stable markets. Among
4
the many studies documenting volatility persistence are: Adrian, Pagan and Schwert
(1990), Andersen, Bollerslev, Diebold and Ebens (2001), Wu (2001) and Flannery and
Protopapadakis (2002) for the stock market, Ederington and Lee (1993, 1995 and 2001)
for interest rates, Harvey and Huang (1991), Ederington and Lee (1993, 1995 and 2001),
Andersen and Bollerslev (1998) and Low and Zhang (2005) for foreign exchange markets
and Jones, Lamont and Lumsdaine (1998) for bond markets. I hypothesize that similar
volatility persistence exists in the natural gas market.
2. Is there volatility asymmetry in the natural gas market? That is do equal
positive and negative shocks have different impacts on future volatility? This hypothesis
is inspired by the generally documented evidence of an asymmetric volatility in the stock
market and, to a lesser extent, in some other markets such as Treasury bonds and
Treasury bond futures markets. French and Roll (1986), French, Schwert and Stambaugh
(1987), Campbell and Hentschel (1992), Veronesi (1999), Bekaert and Wu (2000) and
Wu (2001) found that in the stock market, an unexpected decrease in price has a bigger
impact on predicted volatility than an unexpected increase in price of equal magnitude.
The asymmetric volatility in the stock market is generally attributed to either a leverage
effect and/or a volatility feedback effect (a survey on the determinants of asymmetric
volatility in the stock market can be found in Wu (2001)). The same asymmetric
volatility was documented for the Treasury bond futures options by Simon (1997) and for
the long end of the treasury market by Brunner and Simon (1996). Hsieh (1989), Kim
(1999), McKenzie (2002) and Kwek and Koay (2006) studied volatility in the foreign
exchange market and found that although volatility in this market is not symmetric, it is
difficult to conclude on the direction of that asymmetry since a depreciation in one
currency is an appreciation of the matching currency.
Given the previous studies on asymmetric volatility, I attempt to examine whether
asymmetric volatility exists in the natural gas market and, if yes, the direction of that
asymmetry. Using monthly data, Susmel and Thompson (1997) found that in the natural
gas market, as in other markets, a negative shock has more impact on predicted volatility
than a positive shock of the same size. Contrary to the evidence from the stock and
several other markets and that presented by Susmel and Thompson (1997), I think there
are good reasons to expect a negative shock in the natural gas market to have a smaller
5
impact on the predicted volatility tomorrow than a positive shock of equal magnitude. My
reasoning for this hypothesis is based on the likely shape of the supply and the demand
curves. Consider the depictions of the supply and demand curves for natural gas in Figure
1 by Energy and Environmental Analysis Inc.
Production and Storage Gas Price
Gas Price
Inelastic Demand
P2
Distillate Switching
P1
Residual Oil Switching
P0
Deliverability
Production
Gas Supply
Quantity Consumed
Gas Demand
Figure 1
Supply and Demand in the Natural Gas market
Courtesy: Energy and Environmental Analysis, Inc.
According to these industry observers, at low volume and prices, supply is highly
elastic but once storage limits are reached, supply becomes quite inelastic as natural gas
producers, due to infrastructure constraints, can not increase their production levels
within a short time period. Also according to these industry observers, the demand curve
for natural gas also contains an elastic portion and an inelastic portion. When the natural
gas price is low (below P0), natural gas users utilize natural gas entirely to meet their
demand. When the natural gas price fluctuates in the range from P0 to P2, some industrial
and power generation facilities have the capacity to switch from natural gas to residual
fuel oil and distillate fuel oil to maintain operations. However, natural gas users can only
switch to fuel oil for short periods of time due to the limited storage capacity. Given the
6
hypothesized shape of the natural gas supply and demand curves, the same fluctuation in
demand when prices are low (at the lower parts of the supply and demand curves) should
cause a smaller change in natural gas prices than when prices are high (at the higher parts
of the supply and demand curves). Thus a positive price shock which moves the market
up the supply and demand curves is likely to presage higher future volatility than a
negative shock moving down the curves.
Evidence that a negative shock has less impact on predicted volatility than a
positive shock would be interesting since to our knowledge all studies for other markets
have found that when asymmetry exists, negative shocks have a greater impact on future
volatility than positive shocks.
3. Does volatility differ by day of the week? Some academic studies found that
volatility of financial assets returns varies across days of the week. The literature on dayof-the-week effect on volatility includes French and Roll (1986), Berument and Kiymaz
(2001) for stock market, Harvey and Huang (1991), Ederington and Lee (1993) for
interest rates and foreign exchange futures market and Jones et al. (1998) for bond
markets. I expect that consistent with the evidence in other markets, natural gas volatility
also differs across days of the week.
Also, in some financial markets, the three day volatility from Friday close to
Monday close is higher than that of a normal one day period but not as high as that for a
three weekday period because there is not much information coming out during the
weekend. I hypothesize that natural gas volatility is also higher over the weekend.
4. Is volatility higher in the winter months? A winter effect is likely unique in the
natural gas market due to the dependence of natural gas price on weather conditions. In
winter, especially when the weather is severe, the demand for natural gas may rise
dramatically. In such a situation, natural gas supplies can not increase accordingly since
suppliers can not increase production within a short time period and furthermore, the
natural gas supply is constrained by storage capacity. This observation motivates my
hypothesis that natural gas volatility is higher during winter months. To my knowledge,
this hypothesis has not been tested heretofore.
5. Does the weekly storage report release cause increased natural gas volatility on
announcement days? The Weekly American Gas Storage Survey report was compiled
7
and issued by the American Gas Association through March, 2002 and by the U.S Energy
Information Administration from April, 2002. This report is ”designed to provide a
weekly estimate of the change in inventory level for working gas in storage facilities
across the United States, using a representative sample of domestic underground storage
operators.”3 Given that natural gas supply is constrained by the storage level, the weekly
storage announcement is reportedly one of the most important news influencing the
natural gas market. Susmel and Thompson (1997) found that there is a positive
relationship between changes in predicted volatility and changes in storage capacity.
It has been documented that in other markets, prices are generally more volatile
when lots of new information is coming to the market. Ederington and Lee (1993, 1995)
found that following the releases of scheduled macroeconomic news such as the
employment report, the consumer price index (CPI) and the producer price index (CPI),
interest rate and exchange rate volatility is considerably higher than normal. Jones et al.
(1998) found that the releases of employment and producer price index data are
responsible for an increase in volatility in T-bond markets on announcement days.
Flannery et al. (2002) found that 3 macroeconomic announcements significantly increase
volatility in the stock market when they are released. Linn and Zhu (2004) documented
that natural gas volatility increases within 30 minutes after the storage report release.
I expect that consistent with the evidence in other markets, natural gas volatility is
higher on storage report announcement days.
6. If there is evidence that natural gas volatility increases on announcement days,
does the high volatility persist on the days following the announcement? In other words,
is public information about natural gas storage immediately incorporated in natural gas
prices or does the announcement impact persist over several days? If market participants
in the natural gas market finish adjusting prices according to the new information within
the day of the report release, volatility should fall all the way back to normal level on the
following day.
Studies in other markets have generally found that public information about the
macroeconomy is fully incorporated in prices within the announcement day. Ederington
and Lee (1993, 1995) documented that market prices in interest rates and foreign
3
Issue Brief 2001-03, Policy Analysis Group, American Gas Association.
8
exchange futures markets quickly incorporate the information in macroeconomic
announcements and that volatility quickly returns to preannouncement levels within one
day. Jones et al. (1998) confirmed this result by finding that the increase in predicted
volatility in T-bond market on announcement days does not persist to the following day.
Linn and Zhu (2004) found that the release of the storage report causes increased
volatility but only for 30 minutes after the report announcement. I attempt to examine
Linn and Zhu (2004)’s findings from a different perspective. By using daily natural gas
prices in a GARCH (1,1) type model, I test whether the announcement continues to cause
increased volatility or whether conditional volatility falls all the way back to normal level
on the following day.
3. Data and preliminary analysis
This study examines the natural gas volatility using daily closing prices for the
nearby and next maturity natural gas futures contracts traded on the New York
Mercantile Exchange (NYMEX)4. Natural gas futures contracts, which began trading on
the NYMEX on April 3, 1990, trade in units of 10,000 million British thermal units
(BTUs) (see Appendix for details). My sample period is January 1, 1997 to December 16,
2005 totaling 2,241 daily observations. Daily data for the nearby and next futures prices
are from Norman's Historical Data.
Futures prices are used in place of spot prices for the following reasons. First,
futures prices are the major prices in the natural gas market. When newspapers report
natural gas prices, they are actually reporting futures prices. Second, the futures market
for natural gas is big and centralized since most natural gas futures contracts are traded
on the NYMEX. On the contrary, the daily spot markets for gas are small and largely
decentralized. Third, futures prices are the prices normally used in most risk management
contracts such as swaps and options. Fourth, as documented in Ates and Wang (2005),
futures markets play the dominant role in the price discovery process in the natural gas
markets. Spot prices are also not readily available.
4 The nearby futures contract is the contract with the closest expiration date and the next futures contract is the one that
matures just after the nearby futures contract.
9
To examine natural gas volatility, I examine daily log returns5 where return is
defined as Rt=ln(Pt/Pt-1) with Pt is the price of the nearby (next) futures contract on day t
and Pt-1 is the price of the same nearby (next) futures contract the previous day. Data on
the release dates of Weekly Report of American Gas Storage were collected from the
Energy Information Administration (EIA) website6.
Of the 2,241 observations, 448 are announcement days. From January 01, 1997 to
April 01, 2002, 235 announcements were made on Monday and 31 on Tuesday. From
April 02, 2002 to December 16, 2005, 179 announcements were made on Thursday and 1
announcement was made on Monday, 1 on Wednesday and 1 on Friday. I choose to start
my sample in January, 1997 since information on storage announcement dates is
available from January, 1997.
Table 1 provides summary statistics for daily natural gas returns. As can be seen
from Table 1, the annualized standard deviation of the daily percentage change in natural
gas prices is 58.51% for the nearby futures contracts and 50.63% for the next shortest
contracts. As reported earlier, the volatilities over the same period are 16.61% for the 10year T-bond rates, 12.36% for the S&P 500 and 8.84% for the US dollar-Euro exchange
rate. So the natural gas market is characterized by very high volatility. There is also
evidence of autocorrelation in natural gas returns; the first-order correlation coefficient
for the nearby and next futures returns are -0.046 and -0.045, respectively which are both
significant at the 0.05 level7.
Table 1 also provides preliminary evidence of volatility persistence in this market
in that the first order autocorrelation coefficient for absolute daily returns for the nearby
and the next futures contracts are 0.097 and 0.072, respectively which are both significant
at the 0.01 level. For squared daily returns (not reported in Table 1), the autocorrelation
coefficients are 0.064 and 0.148 which are significant at the 0.01 level. Clearly, this
market like many others is characterized by volatility persistence.
4. Model Specification and Analysis
The daily natural gas “returns” are used to measure price changes only. These “returns” are not investment returns
since no money is actually invested.
5
6
http://www.eia.doe.gov/oil_gas/natural_gas/data_publications/natural_gas_weekly_market_update/ngwmu.html
7 These could be due to bid-ask “bounce”.
10
4.1 Volatility Persistence
Perhaps the most popular model of financial asset return volatility is the GARCH
(1,1) model proposed by Bollerslev (1986) and numerous studies have found that
GARCH (1,1) fits many financial assets returns volatility very well (See, for instance,
Engle (2001)). Comprehensive surveys of this literature can be found in Bollerslev et al.
(1992) and Poon and Granger (2003). Surveying studies that compare the volatility
forecasting ability of various time-series models, Poon and Granger (2003) reported that
a majority found that GARCH (1,1) forecasts best.
Therefore I first use a basic GARCH (1,1) specification to model volatility
persistence in this market. Since there is also evidence of first-order autocorrelation in
ordinary daily returns, I add the lagged natural gas return as an independent variable to
the standard GARCH (1,1) mean equation. So the estimated mean and variance equations
are:
Rt = μ + Φ1Rt-1+ εt,
ht = ω + αεt-12 + βht-1,
(1)8
(2)
Rt=ln(Pt/Pt-1) where Pt is the price of the nearby (next) futures contract on day t
and Pt-1 is the price of the nearby (next) futures contract on the previous day. εt is
presumed to be an independent random variable with conditional mean zero and
conditional variance ht. If volatility persistence is an attribute of the natural gas market, α
and β should be significantly positive implying that predicted volatility depends on (1)
unexpected price changes and (2) forecast volatility on the previous days.
The results for equations (1)-(2) are reported in Table 2a. As expected, the
estimates of α for both nearby and next futures daily returns are positive and significant at
the 0.001 level, implying that conditional volatility depends on the shocks in the previous
day. The estimates of β for both nearby and next futures daily returns are also positive
and significant at 0.001 level, implying that conditional volatility also depends on the
forecast volatility the previous day. Thus, highly volatile periods in the natural gas
market tend to be followed by volatile periods in the future.
8 Although Equation (1) is always estimated simultaneously with my models of the conditional variance h , I do not report its
t
parameter estimates in the tables to conserve space and focus on the main issue.
11
For comparison, I summarize the estimates of α and β in some other markets in
Table 2b. The estimates of α and β for the natural gas market are reasonable and
consistent with what have been found in other markets.
4.2 Asymmetric Volatility
While the GARCH (1,1) model in equation 2 assumes that positive and negative
shocks affect future volatility equally, previous academic studies have found evidence of
asymmetric volatility in a number of financial markets. The literature on asymmetric
volatility includes French and Roll (1986), French et al. (1987), Campbell and Hentschel
(1992), Glosten et al. (1993), Veronesi (1999), Bekaert and Wu(2000) and Wu(2001) for
the stock market, Hsieh (1989), Kim (1999), Mc Kenzie (2002) and Kwek and Koay
(2006) for the foreign exchange market, Simon (1997) for the Treasury bond futures
options and Brunner and Simon (1996) for the long end of the treasury market. Except for
the studies on asymmetric volatility in the foreign exchange market, all other studies
found evidence that a negative unexpected price change has a larger impact on volatility
than an equal positive unexpected price change.9
Consequently, I examine the question of asymmetric volatility in the natural gas
market. There are two questions to be answered in this regard. (1) In the natural gas
market, does a negative shock have a different impact on predicted future volatility than a
positive shock of the same magnitude? (2) If volatility asymmetry exists in the natural
gas market, what is the direction of that asymmetry? More specifically, I attempt to test
my hypothesis that a negative shock on the previous day has a smaller impact on
predicted volatility than a positive shock of the same magnitude. As pointed out earlier,
all other studies on asymmetric volatility found that a negative shock has a larger impact
on predicted volatility than a positive shock. Susmel and Thompson (1997) also found
that in the natural gas market, a negative shock has larger impact on volatility than a
positive shock but they used monthly data and did not provide any economic explanation
for this phenomenon. I use the GJR (Glosten, Jagannathan, and Runkle(1993)) model
9
Although asymmetric volatility exists in the foreign exchange market, the direction of asymmetry does not matter since a
depreciation in one currency is an appreciation in the matching currency.
12
estimation to answer these questions.10 Specifically, the estimated mean and variance
equations are:
Rt = μ + Φ1Rt-1+ εt,
(1)
ht = ω + αεt-12 + βht-1+ γ εt-12It-1,
(3)
where εt is a random variable with conditional mean zero and conditional variance ht; It1=1
if εt-1<0 and 0 otherwise.
Asymmetric volatility implies γ ≠ 0 in equation (3). γ >0 implies that a negative
shock increases conditional volatility more than a positive shock of the same magnitude.
The results for equations (1) and (3) are reported in Table 3. As expected, γ is
significantly different from zero implying asymmetric volatility in the natural gas market.
This concurs with earlier findings for the stock market, Treasury bond futures options and
the long end of the Treasury bond markets. However while previous studies (including
the study by Susmel and Thompson (1997) for the natural gas volatility) obtained
positive estimates of γ, in my estimation the estimated γ is significantly negative implying
that a negative shock on the previous day has less impact on the predicted volatility than
a positive shock of the same magnitude.11
This behavior of natural gas volatility can be explained by the likely shape of the
supply and demand curves as depicted in Figure 1. The same fluctuation in demand when
prices are low (at the lower part of the supply and demand curves) should cause a smaller
change in natural gas prices than when prices are high (at the higher part of the supply
and demand curves). Thus a positive price shock which moves the market up the supply
and demand curves is likely to presage higher future volatility than a negative shock
moving down the curves.
10 GJR model and EGARCH model (by Nelson (1992)) are the two most popular models of asymmetric volatility. Engle and Ng
(1993) compared different ARCH specifications using the Japanese daily stock return data from 1980-1988 and showed that the GJR
and EGARCH models are the best models to describe the asymmetric effect of negative shocks.
11 The results do not change if I use the EGARCH (Nelson(1991)) specification instead. In the EGARCH specification, the
conditional variance is: Log(h ) = ω + βlog(h )+α│ε /h │+ γ(ε /h ) where the asymmetric effect is captured by γ. If γ<0, a
t
t-1
t-1 t-1
t-1 t-1
negative shock has a bigger impact on the conditional volatility than a positive shock of the same magnitude. When I estimated the
natural gas nearby and next futures returns using the EGARCH specification, the estimates of γ are 0.0405 and 0.0505, respectively
and both are significant at the 0.001 level, implying that in the natural gas market, a positive shock has a bigger impact than a negative
shock of the same magnitude which is consistent with the results from GJR model.
13
The different impact of positive and negative shocks on predicted volatility
according to the estimates in Table 3 is depicted in Figure 2. As shown there a positive
shock to natural gas prices has a more persistent impact than an equivalent negative
shock of the same magnitude.
ht
(12%)2
(10%)2
(8%)2
(6%)2
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
εt-1
Figure 2
The asymmetric impact of the return shocks on conditional volatility in the natural gas futures
market
This news impact curve depicts how positive and negative return shocks at time t-1 impact conditional
volatility in the natural gas market according to the estimates of the GJR model. This curve measures how
the return shock this period is incorporated into volatility estimates next period.
The equation for the GJR news impact curve is: ht = ω + αεt-12 + βht-1+ γεt-12It-1
(3)
where It-1=1 if εt-1<0 and 0 otherwise, ω, α, β and γ are estimates from the GJR variance equation (Equation
3).
14
4.3 Day-of-the-week effects
Several studies have examined whether volatility differs by day of the week in
various financial markets. (See French and Roll (1986), Berument and Kiymaz (2001) for
stock market, Harvey and Huang (1991), Ederington and Lee (1993) for interest rates and
foreign exchange futures market and Jones et al. (1998) for bond markets, Hsieh (1989)
for foreign exchange market and Linn and Zhu (2004) for natural gas market).
In this study, I employ a new procedure to examine whether volatility in the
natural gas market varies by day-of-the-week. I estimate a model in which the conditional
variance follows a regime-switching GJR type process using a variant of the model
outlined in Jones et al. (1998). Specifically, the new mean and variance equations are:
Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+ δM DMONt)(1+δTDTUEt)(1+δRDTHURt)(1+δFDFRIt),
ht = ω+ αεt-12+βht-1+ γεt-12It-1,
(4)
(5a)
(6)
where st is the volatility seasonal for time t, δM, δT, δR, and δF measures the day-of-theweek effects on the conditional volatility. εt is a random variable with conditional mean
zero and conditional variance ht independent of st. DMONt, DTUEt, DTHURt and DFRIt
are zero-one dummy variables for each of the four weekdays. I choose to leave out the
dummy variable for Wednesday because of the 2,241 observations, only one
announcement was made on Wednesday.
The specification (4)-(6) allow the estimated volatility to differ by day of the
week. On Wednesday, the estimated variance of natural gas returns is ht. On Monday, the
estimated variance is ht (1+δM). On Tuesday, the estimated variance is ht (1+δT) and so on
for other days. Thus δX estimates the percentage difference between volatility on day X
(as measured by the variance) and volatility on Wednesday. I expect that some weekday
dummy estimates will differ from zero. More specifically, I expect higher volatility on
Friday-close-to-Monday-close return since this is a three day return including the
weekend. I also anticipate that volatility could be higher on Monday and Thursday since
most storage reports were released on those two dates.
Linn and Zhu (2004) also documented seasonality in the natural gas market but
this was done in an Ordinary Least Square (OLS) and Generalized Method of Moments
(GMM) framework. Hsieh (1989), Berument and Kiymaz (2001) and Ederington and Lee
15
(2001) used GARCH type models to examine day-of-the-week effects. In the three
studies mentioned above, the authors placed weekday dummies in the ht equation
(equation 6) to estimate how conditional volatility changes across weekdays. Thus using
their approach, equation (6) becomes:
ht = ω+ αεt-12+βht-1+ γεt-12It-1+ δM DMONt + δTDTUEt + δRDTHURt + δFDFRIt,
(6’)
and there is no s equation (equation 5a). In contrast, I employ a separate volatility
seasonal specification (equation 5a) to measure day-of-the-week effects. This provides a
much clearer estimation of day-of-the-week effects than if the weekday dummies are
placed directly in the ht equation as in equation (6’).
When day-of-the-week dummies are in the ht equation (as in equation 6’), the
dummy for any day of the week impacts volatilities on all days of the week through the
ht-1 term on the right-hand side of the equation. Suppose for instance that day t is
Monday. ∂ht/∂DMONt = δM. Now consider the impact of the Monday dummy on the
Tuesday (day t+1) volatility. Since ht+1 = ω + αεt2 + βht + γεt2It+ δMDMONt+1 +
δTDTUEt+1 + δRDTHURt+1 + δFDFRIt+1 and ∂ht/∂DMONt = δM, ∂ht+1/∂DMONt = βδM.
Likewise the Monday dummy impact on Wednesday’s volatility is ∂ht+2/∂DMONt = β2
δM. Thus when the day-of-the-week dummies are in the ht equation, as in equation 6’, δM
does not measure how much higher volatility is on Monday than on the left out day.
Indeed depending on the coefficient pattern, the day of the week with the highest δ
coefficient may not be the day with the highest volatility.
Separating the variance of returns into a persistent part, equation 6, and nonpersistent part, equation 5, allows me to estimate a model in which DMONt impacts only
Monday volatility. As explained above, δM measures how much higher or lower in
percentage terms the volatility is on Monday than on the left out day (Wednesday) and δ T
measures how much higher or lower volatility is on Tuesday. Thus, the introduction of a
volatility seasonal s into the specification allows me to implement a nice clean study on
how much the estimated variance differs by day of the week.
16
The results for equations (4), 5(a) and (6) are represented in Table 4. As indicated
by the likelihood ratios statistics in the last row of Table 4, volatility differs significantly
by day of the week. 12
As expected, conditional volatility over the three day period from the close on
Friday to the close on Monday is higher than that on Wednesday or any other weekday.
Specifically, the standard deviation of natural gas nearby futures returns and next futures
returns from Friday-close-to-Monday-close are 30.86% and 28.46% higher than that on
Wednesday, respectively. The difference between Friday-close-to-Monday-close
volatility and Wednesday volatility is significantly different from zero at the 0.001 level.
I see two possible explanations for the higher Friday-close-to-Monday-close volatility.
First, the Friday-close-to-Monday-close volatility is not just one-day volatility but it also
incorporates volatility over the weekend. Second, the higher Monday volatility may be
due to the fact that many storage announcements were released on Monday (From
January 1997 through March 2002, 235 out of 266 announcements were released on
Monday). However, even when I estimate the specification (4), 5(a) and (6) with the subperiod from 2002 through 2005 (only one announcement was released on Monday during
this period), the Friday-close-to-Monday-close volatility is still highest and the difference
between the Friday-close-to-Monday-close volatility and Wednesday volatility is still
significantly
different
from
zero
at
0.001
level.
(The
interaction
between
Monday/weekend effect and announcement effect will be discussed further in Section
4.6.) These results suggest that there is a Monday/weekend effect on volatility in the
natural gas market and that at least part of the high Friday-close-to-Monday-close
volatility is due to the accumulation of the new information during the weekend.
As indicated in Table 4, volatility is higher on Thursday and the difference
between volatility on Thursday and on Wednesday is significantly different from zero at
the 0.05 level. However, as noted above many of the storage announcements occur on
Thursday and when I estimate the specification (4)-(6) with the sub-period from January
1997 through March 2002 (no storage announcement was made on Thursday during this
12
The likelihood ratio is based on the log likelihood for estimation 4, 5(a) and 6 and estimation 1 and 3
where all weekday dummies coefficients are forced to be zero, i.e., there are no weekday effects in the
natural gas market. For instance, for the natural gas nearby futures returns, the likelihood ratio is:
2(5974.469-5914.264) = 121.041. This statistic is distributed as a chi-square with 4 degrees of freedom.
17
period), the volatility on Thursday is not significantly different from the volatility on
Wednesday. This suggests that volatility is higher on Thursday because for the later
period (from April 2002 to December 2005), most announcements were made on
Thursday, i.e., that the high Thursday volatility is due to an announcement effect.
Volatility is lower on Friday than on Wednesday and all other days of the week
except Tuesday and this difference is significantly different from zero at the 0.001 level.
The standard deviation of natural gas prices on Friday is 21.49% lower than that on
Wednesday. This is opposite to the findings for other markets. For example, Harvey and
Huang (1991) reported higher volatility in interest rate and foreign exchange futures
market on Friday. Ederington and Lee (1993) further support these results. Jones et al.
(1998) and Berument and Kiymaz (2001) found similar evidence in the bond and stock
markets. This contradiction may be due to the fact that many economic news releases
which impact volatility in other markets are often announced on Friday whereas this is
not the case in the natural gas market.
4.4 Winter effect
It has been documented that returns in some markets differ by month-of-the-year.
(see, for instance, Keim (1983), Lakonishok and Smidt(1984) for stock markets and
Jordan and Jordan (1991) for corporate bond market). However, little attention has been
given to whether volatility differs by month-of-the-year. I expect to find that natural gas
volatility is higher in the winter months due to the fact that natural gas demand often
increases during the winter months and if the weather becomes severe, this increase may
be dramatic. If the supply is constrained by storage and natural gas suppliers can not
increase production within a short time period, when this happens sharp price swings
could result (this can be seen as the intersection of the inelastic portions of both the
supply and demand curves in Figure 1). In order to test the hypothesis of winter effects, I
employ a regime-switching model similar to that in section 4.3. First, I include a zero-one
dummy variable, WIN, for the winter months (which include the time period from
November through February) in the volatility seasonal equation. The specification is:
Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+ δW WINt),
(4)
(5b)
18
ht = ω+ αεt-12+βht-1+ γεt-12It-1,
(6)
Specification (4), (5b) and (6) allows conditional volatility to differ by month of
the year. Specifically, natural gas conditional volatility is ht(1+ δW ) in the winter months
and ht in other months of the year. A significantly positive parameter estimate of δW
implies that volatility is higher in the winter months than in other months.
The results from specification (4), (5b) and (6) are represented in Table 5. As
expected, I find that volatility is higher in the winter months than in other months of the
year. Specifically, the estimate in Table 5 indicates that the standard deviations of natural
gas futures returns in the winter months is 12.09% higher than in other months for the
nearby contracts and 11.83% higher for the next contracts and this difference is
significant at the 0.001 level.
To provide a more detailed estimation of how natural gas volatility differs by time
of year, I expand equation (5b) to include the dummies for each month of the year. The
revised specification is:
Rt = μ + Φ1Rt-1 + st1/2εt,
(4)
st = (1+ δJJANt)(1+ δFFEBt)(1+ δMarMARt)(1+δAAPRt)(1+ δMayMAYt)(1+ δJuly
JULYt)(1+ δAugAUGt)(1+ δSSEPt)(1+ δOOCTt)(1+ δNNOVt)(1+ δDDECt),
ht = ω+ αεt-12+βht-1+ γεt-12It-1,
(5c)
(6)
In this specification (4), (5c) and (6), conditional volatility differs in each month
of the year. In June, the conditional volatility is simply ht and the volatility switches to
ht(1+ δX) in month X. Consistent with the results from specification (4), (5b) and (6)
above, the results from (4), (5c) and (6) confirm that natural gas volatility is higher in
winter months. Natural gas volatility is higher in the winter months from November
through February than in June and the differences are significant at the 0.05 level. A
somewhat surprising time-of-the-year effect is that September has significantly higher
conditional volatility than other months of the year.13
13 When I estimate the specification 4, 5(c) and 6 without the 2005 data, the estimate of September dummy
is still high and significantly positive, implying that the high September natural gas volatility can not be
explained only by the hurricane Katrina impact.
19
4.5 Impact of the Storage Report
It has been documented that volatility in some financial markets is higher on days
when important reports are released, i.e., announcement days (See, for instance,
Ederington and Lee (1993, 1995), Jones et al. (1998), Linn and Zhu (2004)). On
announcement days, new information comes to the market and thus, market participants
adjust prices according to the new information. I expect that consistent with the findings
in earlier studies, natural gas volatility is higher on days the storage report is released.
To examine whether the storage report release increases natural gas volatility, I
introduce a zero-one announcement dummy, DAt, into the volatility seasonal equation.
The specification is:
Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+ δ0DAt),
ht = ω+ αεt-12+βht-1+ γεt-12It-1,
(4)
(5d)
(6)
Again, the above specification allows conditional volatility to differ on
announcement days and on non-announcement days. On non-announcement days, the
conditional volatility is simply ht and on announcement days, conditional volatility
switches to ht(1+δ0). I hypothesize that the storage report release increases the conditional
volatility and thus, the estimate of δ0 should be significantly positive. As explained later
in Section 4.7, the introduction of an announcement dummy in a volatility seasonal
equation allows me to implement a clear examination of announcement impact on
volatility.
The results of specification (4), (5d) and (6) are represented in Table 7. As
predicted, conditional volatility is higher on announcement days and this is significant at
the 0.001 level. This result is consistent with what has been documented by Ederington
and Lee (1993, 1995), Jones et al. (1998) and Flannery et al. (2001) for other markets and
by Linn and Zhu (2004) for natural gas. As indicated in Table 7, the standard deviation of
natural gas returns increases 18.02% for nearby contracts and 19.57% for next contracts
on announcement days.
20
4.6 The interaction of day-of-the-week effects and announcement effects
As discussed in section 4.3, volatility in the natural gas market is significantly
higher on Monday and Thursday and significantly lower on Friday. As seen in section
4.5, natural gas volatility is significantly higher on announcement days. Since most of the
storage announcements occur on Monday and Thursday, a question arises whether
volatility is higher on Monday and Thursday due to the announcement effect or due to a
day of the week effect. To answer this question, I estimate the specification 4, 5(a) and 6
for two sub-samples. The first sub-sample ranges from January, 1997 through March,
2002 (during this period, 235 out of 266 storage announcements were released on
Monday) and the second sub-sample ranges from April, 2002 through December, 2005
(during this period, 179 out of 182 storage announcements were released on Thursday).
The results are presented in Table 8.
As indicated in the second and third columns of Table 8, in the period from
January 1997 through March 2002 (no storage announcement was released on Thursday
during this period), natural gas volatility on Thursday was not significantly different from
the volatility on Wednesday. I conclude that higher Thursday volatility is due to
announcement effect because for the later period (from April 2002 to December 2005),
most announcements were made on Thursday.
Also in the fourth and fifth columns of Table 8, in the period from April, 2002
through December, 2005 (only one announcement was made on Monday during this
period), the Friday-close-to-Monday-close natural gas volatility is still highest and the
difference between the Friday-close-to-Monday-close natural gas volatility and
Wednesday volatility is still significantly different from zero at the 0.001 level. These
results lead me to the conclusion that there is a weekend effect on natural gas volatility as
the Friday-close-to-Monday-close natural gas volatility also incorporates volatility over
the weekend.
4.7 Impact of the Storage Report on subsequent days
The results from section 4.5 imply that the storage announcement report does
increase conditional volatility on announcement days. The remaining question is whether
this increased volatility is persistent or just transitory. Ederington and Lee (1993, 1995)
21
suggested that announcement shocks do not have impact on the volatility on the
subsequent days for the interest rate and foreign exchange futures markets. Jones et al.
(1998) further supported these results for the bond markets.
Linn and Zhu (2004)
documented that natural gas volatility increases within only 30 minutes after the report
announcement.
To examine the impact of the report release on natural gas volatility on the days
following the announcement, I expand specification (4), (5c) and (6) to accommodate the
possible impact of the report release on natural gas volatility on subsequent days after the
announcements. The specification is:
Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+ δ0DAt),
ht = ω+ αεt-12+βht-1+ γεt-12It-1+ δ1DAt,
(4)
(5d)
(7)
When announcement dummy is in the ht equation (equation 7), the announcement
dummy impacts volatilities on the day after the announcement and all subsequent days
through the ht-1 term on the right-hand side of the equation. Suppose for instance that day
t is the announcement day. From equation 5(d) and equation 7, ∂st/∂DAt = δ0 and
∂ht/∂DAt = δ1. Now consider the impact of the announcement dummy on the following
day (day t+1) volatility. ∂st+1/∂DAt = 0 and ∂ht+1/∂DAt = βδ1. Likewise the announcement
dummy impact on day t+2 volatility are: ∂st+2/∂DAt = 0 and ∂ht+2/∂DAt = β2δ1. Thus,
specification 4, 5(d) and 7 allows storage announcement to affect volatility in two parts: a
persistent part (through δ1 in equation 7) and a non-persistent part (through δ0 in equation
5d).
A number of previous studies which just placed announcement dummy into the ht
equation (without the s equation) are inadequate to model the impact of announcement
shocks on volatility since the announcement impact is forced to persist on the subsequent
days. (∂ht/∂DAt = δ1, ∂ht+1/∂DAt = βδ1, ∂ht+2/∂DAt = β2δ1 and so on). On the contrary,
separating the announcement impact into a persistent part, equation 5(d), and a nonpersistent part, equation 7 allows me to estimate a model in which the announcement
dummy has a separate impact on announcement days and on subsequent days. Given that
the estimate of δ0 in equation (5d) is significantly positive (volatility increases on
announcement days), a significantly positive estimate of δ1 in equation (7) implies that
22
announcement impact is accumulated in the estimated volatility on subsequent days after
the announcement. On the contrary, if the estimate of δ1 is not significantly positive, we
fail to reject the hypothesis that announcement shocks do not have impact on subsequent
days following the announcement.
The results of specification (4), (5d) and (7) are reported in the second and third
columns of Table 9. Although both δ0 and δ1 are positive, I fail to reject the hypothesis
that the report release has no impact on the conditional volatility on the days after the
announcement. Even though volatility increases on announcement days, the market
participants apparently complete most of the adjustment of natural gas prices within the
announcement days and volatility falls most of the way back to normal level on the
following day. It is likely that since the weekly storage report release is a scheduled event
(most of the announcements occur on Mondays from 1997-March 2002 and on Thursdays
from April 2002-2005), natural gas market participants are well prepared to receive and
analyze the new information in the report and thus, the market will incorporate the
information into prices quickly.
When I estimate volatility using the specification which does not include the s
equation and in which the announcement dummy is in the ht equation only, the estimate
of δ1 is significantly positive, implying that conditional volatility increases on
announcement days and this increase will persist on the subsequent days. The results are
presented in the fourth and fifth columns of Table 9. These results demonstrate that the
latter specification is inadequate to model the impact of announcement shocks on
volatility since the announcement impact is forced to persist on the subsequent days
whereas the former specification provides a much clearer approach to estimate the
announcement impact on volatility since volatility is separated into a persistent part and a
non-persistent part.
The results from specification 4, 5(d) and 7 concur with the findings in
Ederington and Lee (1993, 1995), Jones et al. (1998) and Linn and Zhu (2004) that
market prices quickly incorporate the information contained in the relevant
announcements and that volatility quickly returns to preannouncement levels.
23
5. Summary and Conclusions
The contribution this paper makes is to provide an empirical examination of the
causes and behavior of volatility in natural gas market. Daily returns data from January
1997 to December 2005 are used to estimate different variants of GARCH type model. In
summary, I find that volatility persistence and volatility asymmetry are the attributes of
volatility in the natural gas market. Also, there is evidence of day-of-the-week and winter
effects on natural gas volatility. Consistent with the findings in other markets, natural gas
volatility is higher on days storage report announcements are released but announcement
shocks do not continue to impact natural gas volatility on subsequent days.
The natural gas futures market is characterized by volatility persistence where
highly volatile periods are followed by highly volatile periods and stable periods are
followed by stable ones. Also, positive and negative returns shocks have different impact
on future volatility. Contrary to previous studies of volatility in other financial markets, I
find that in the natural gas market, a positive shock has a bigger impact on predicted
volatility than a negative shock of the same magnitude. This is one of the main
contributions of this paper since this is the first time volatility asymmetry is found in this
direction. This can be explained by the hypothesized shape of the natural gas supply and
demand curves. Since the natural gas supply and demand curves are both inelastic when
prices are high, a fluctuation in natural gas demand when prices are high causes a larger
change in volatility than the same fluctuation when prices are low.
Consistent with the findings in other financial markets including that of Linn and
Zhu (2004), there are day-of-the-week effects in the natural gas market where natural gas
volatility is highest on Monday and lowest on Friday. The high Monday volatility is
mainly due to the accumulation of information over the weekend. In contrast to the
evidence in the stock, bond, interest rate and foreign exchange futures markets, natural
gas volatility is lower on Friday. Natural gas volatility is also higher on Thursday but this
is mainly because Thursday is the announcement days in almost half of all observations.
A finding which has not appeared in previous literature on the natural gas market
is that the natural gas volatility is significantly higher in the winter months than in other
months of the year. Given that natural gas supply is constrained by storage limit, a large
24
increase in demand due to severe weather may cause a dramatic swing in natural gas
volatility.
Consistent with the findings in Linn and Zhu (2004), natural gas volatility is
higher on report announcement days. However, the increase in natural gas volatility is
only transitory since volatility falls most of the way back to normal level on the following
days.
A major methodological contribution of this paper is the separation of natural gas
volatility into a persistent part and a non-persistent part using the regime-switching GJR
model with a seasonal volatility equation. This model allows me to implement a clean
study of the determinants of natural gas volatility since volatility is allowed to differ by
day of the week, month of the year and on announcement days.
25
Appendix
Contract Specification
Trading Unit
Trading Hours (All times are
New York Time)
Trading Months
Natural Gas
10,000 million British thermal units (mmBtu).
Open outcry trading is conducted from 10:00 AM until
2:30
PM.
Electronic trading is conducted from 6:00 PM until
5:15 PM via the CME Globex® trading platform,
Sunday through Friday. There is a 45-minute break
each day between 5:15PM (current trade date) and
6:00 PM (next trade date).
72 consecutive months commencing with the next
calendar month (for example, on January 6, 2004,
trading occurs in all months from February 2004
through January 2010).
Minimum Price Fluctuation
$0.001 (0.1¢) per mmBtu ($10.00 per contract).
Maximum Daily Price
Fluctuation
$3.00 per mmBtu ($30,000 per contract) for all
months. If any contract is traded, bid, or offered at the
limit for five minutes, trading is halted for five
minutes. When trading resumes, the limit is expanded
by $3.00 per mmBtu in either direction. If another halt
were triggered, the market would continue to be
expanded by $3.00 per mmBtu in either direction after
each successive five-minute trading halt. There will be
no maximum price fluctuation limits during any one
trading session.
Last Trading Day
Trading terminates three business days prior to the
first calendar day of the delivery month.
Physical
The Sabine Pipe Line Co. Henry Hub in Louisiana.
Pipeline specifications in effect at time of delivery.
Settlement Type
Delivery
Grade and Quality
Specifications
Position Accountability Levels
and Limits
Any one month/all months: 12,000 net futures, but not
to exceed 1,000 in the last three days of trading in the
spot month.
Note: Contract Specifications are obtained from NYMEX webpage:
http://www.nymex.com
26
Table 1
Summary Statistics: Natural gas daily returns
The second (fourth) columns present the summary statistics for natural gas returns:
Rt=ln(Pt/Pt-1) where Pt is the price of the nearby (next) futures contract on day t and Pt-1 is
the price of the nearby (next) futures contract on the previous day. The third (fifth)
columns present the summary statistics for absolute values of daily returns. (***), (**),
(*) designate estimates significantly different from zero at the 0.001, 0.01 and 0.05 levels,
respectively. The sample extends from January 01, 1997 through December 31, 2005.
Mean
Maximum
Minimum
Std Dev
Annualized Std Dev
Skewness
Kurtosis
Rho (First order
autocorrelation coefficient)
Nearby futures contracts
Next futures contracts
Returns Absolute Returns Returns Absolute Returns
0.001
0.027
0.001
0.024
0.324
0.324
0.188
0.188
-0.199
0.000 -0.155
0.000
0.037
0.026
0.032
0.021
0.585
0.406
0.506
0.332
0.344
2.547
0.077
1.836
7.91
16.65
5.043
8.345
-0.046*
0.097*** -0.045*
0.072**
27
Table 2a
The GARCH (1,1) model of natural gas volatility
Rt = μ + Φ1Rt-1+ εt,
(1)14
2
ht = ω + αεt-1 + βht-1,
are presented where Rt is the natural gas daily returns, εt is an independent random
variable with conditional mean zero and conditional variance ht. Returns are expressed in
percent. Standard errors are shown in parentheses. (***), (**), (*) designate estimates
significantly different from zero at the 0.001, 0.01 and 0.05 levels, respectively. The
sample extends from January 01, 1997 through December 31, 2005.
Estimates of the model:
ω
α
β
Log likelihood
Nearby futures
0.3941***
(0.0733)
0.0943***
(0.0082)
0.8809***
(0.0109)
Next futures
0.3054***
(0.0744)
0.0656***
(0.0098)
0.9040***
(0.0142)
-5979.147
-5675.611
14 Although Equation (1) is always estimated simultaneously with my models of the conditional variance h , I do not report its
t
parameter estimates in the tables to conserve space and focus on the main issue.
28
Table 2b
GARCH (1,1) estimates in representative previous studies
Coefficient estimates
Type of asset
Omega
Alpha
Beta
0.00084** 0.07906** 0.90501** Stock
Period
Source
1963-1986
0.0009**
Akgiray (1989)
Berument and Kiymaz
(2001)
Hsieh (1989)
Hsieh (1989)
Hsieh (1989)
Hsieh (1989)
Hsieh (1989)
Jones et al. (1998)
Jones et al. (1998)
Jones et al. (1998)
Ederington and Lee
(2001)
Ederington and Lee
(2001)
Ederington and Lee
(2001)
0.0008*
0.0021**
0.0031**
0.0369**
0.1907**
0.1263*
0.2296*
0.3708**
0.1723*
0.050*
0.051**
0.037**
0.9567**
0.8056**
0.8511*
0.7329*
0.7138**
0.8229*
0.938*
0.937**
0.955**
Stock
BP
CD
DM
JY
SF
5-yr T-bond
10-yr T-bond
30-yr T-bond
1973-1997
1974-1983
1974-1983
1974-1983
1974-1983
1974-1983
1979-1995
1979-1995
1979-1995
0.1809**
0.2925**
0.5939**
Eurodollar
1989-1993
0.1325**
0.2058**
0.6938**
T-bond futures
1989-1993
0.0667**
0.1768**
0.7522**
Deutschmark
1989-1993
29
Table 3
GJR model of asymmetric volatility
Estimates of the model: Rt = μ + Φ1Rt-1+ εt,
ht = ω + αεt-12 + βht-1+ γ εt-12It-1,
are presented where Rt is the natural gas daily returns, εt is an independent random
variable with conditional mean zero and conditional variance ht. It-1=1 if εt-1<0 and 0
otherwise. Standard errors are shown in parentheses. (***), (**), (*) designate estimates
significantly different from zero at the 0.001, 0.01 and 0.05 levels, respectively. The
sample extends from January 01, 1997 through December 31, 2005.
Nearby futures
0.3280***
(0.0673)
0.1117***
(0.0091)
0.8947***
(0.0107)
Next futures
0.2833***
(0.0665)
0.0954***
(0.0127)
0.9097***
(0.0129)
γ
-0.0539***
(0.0098)
-0.0657***
(0.0122)
Log Likelihood
-5974.469
-5666.718
ω
α
β
30
Table 4
GJR model with Day-of-the-week effects
Estimates of the model: Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+ δM DMONt)(1+δTDTUEt)(1+δRDTHURt)(1+δFDFRIt),
ht = ω+ αεt-12+βht-1+ γεt-12It-1,
are presented where Rt is the natural gas daily return. It-1=1 if εt-1<0 and 0 otherwise, st is
the volatility seasonal for time t, δM, δT, δR, and δF measure the day-of-the-week effects
on the conditional volatility. εt is a random variable with conditional mean zero and
conditional variance ht independent of st. DMONt, DTUEt, DTHURt and DFRIt are zeroone dummy variables for each of the four weekdays. Standard errors are shown in
parentheses. (***), (**), (*) designate estimates significantly different from zero at the
0.001, 0.01 and 0.05 levels, respectively. Asterisks on the likelihood ratios indicate
rejection at the 0.01 level of the null hypothesis that volatility does not vary by day of the
week. The sample extends from January 01, 1997 through December 31, 2005.
Nearby
futures
0.324095***
(0.0773)
0.1108***
(0.0126)
0.8821***
(0.0133)
Next futures
0.1998***
(0.0591)
0.0745***
(0.0122)
0.9135***
(0.0129)
-0.0425**
-0.0290*
(0.0142)
(0.0130)
δM
0.7125***
(0.1114)
0.6503***
(0.1245)
δT
-0.0912
(0.0677)
-0.0647
(0.0712)
δR
0.2331*
(0.0929)
0.3010**
(0.1032)
δF
-0.3829***
(0.0470)
-0.3234***
(0.0530)
Log likelihood
-5914.264
-5621.373
Likelihood ratio
120.410*
90.690*
ω
α
β
γ
31
Table 5
GJR model with a Winter effect
Estimates of the model: Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+ δW WINt),
ht = ω+ αεt-12+βht-1+ γεt-12It-1,
are presented where Rt is the natural gas daily return. It-1=1 if εt-1<0 and 0
otherwise, st is the volatility seasonal for time t, δW measures the winter effect on the
conditional volatility. εt is a random variable with conditional mean zero and conditional
variance ht independent of st. DWINt is a dummy variable which equals to 1 if the
observation is in November, December, January and February. Standard errors are shown
in parentheses. (***), (**), (*) designate estimates significantly different from zero at the
0.001, 0.01 and 0.05 levels, respectively. The sample extends from January 01, 1997
through December 31, 2005.
Nearby futures
0.3835***
(0.0848)
0.1166***
(0.0121)
0.8837***
(0.0137)
Next futures
0.2647***
(0.0745)
0.0734***
(0.0127)
0.9142***
(0.0144)
γ
-0.0669***
(0.0122)
-0.0394***
(0.0118)
δW
0.2564***
(0.0591)
0.2505***
(0.0684)
Log likelihood
-5967.707
-5666.000
ω
α
β
32
Table 6
GJR model with Month-of-the-year effects
Estimates of the model: Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+δJJANt)(1+δFFEBt)(1+ δMarMARt)(1+δAAPRt)(1+
δMayMAYt)(1+ δJuly JULYt)(1+ δAugAUGt)(1+ δSSEPt)(1+ δOOCTt)(1+ δNNOVt)(1+
δDDECt),
ht = ω+ αεt-12+βht-1+ γεt-12It-1,
are presented where Rt is the natural gas daily returns. It-1=1 if εt-1<0 and 0
otherwise, st is the volatility seasonal for time t, δJ….δD measures the monthly effect on
the conditional volatility. εt is a random variable with conditional mean zero and
conditional variance ht independent of st. JANt…. DECt are zero-one dummies for each
month of the year. Standard errors are shown in parentheses. (***), (**), (*) designate
estimates significantly different from zero at the 0.001, 0.01 and 0.05 levels, respectively.
Asterisks on the likelihood ratios indicate rejection at the 0.01 level of the null hypothesis
that volatility does not vary by month. The sample extends from January 01, 1997
through December 31, 2005.
Nearby futures
0.3872***
(0.0925)
0.0691***
(0.0129)
0.9108***
(0.0140)
Next futures
0.1479**
(0.0476)
0.0542***
(0.0098)
0.9503***
(0.0090)
γ
-0.0439***
(0.0129)
-0.0457***
(0.0103)
δJ
0.3743*
(0.1600)
0.3223*
(0.1508)
δF
0.3922**
(0.1264)
-0.0538
(0.0924)
δMar
-0.2274*
(0.0920)
-0.2545**
(0.0913)
δA
-0.3328***
(0.0775)
-0.2959***
(0.0805)
δMay
-0.1131
(0.1102)
-0.1132
(0.1086)
ω
α
β
33
δJuly
-0.2173*
(0.0914)
-0.1636
(0.0975)
δAug
0.2282
(0.1491)
0.2434
(0.1420)
δS
0.8150***
(0.2059)
0.2273
(0.1407)
δO
0.2298
(0.1447)
0.1680
(0.1405)
δN
0.3733*
(0.1700)
0.3687*
(0.1700)
δD
0.5124*
(0.2072)
0.6410**
(0.2267)
Log likelihood
-5938.173
-5645.123
Likelihood ratio
72.592*
43.190*
34
Table 7
GJR model with Announcement effect
Estimates of the model: Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+ δ0DAt),
ht = ω+ αεt-12+βht-1+ γεt-12It-1,
are presented where Rt is the natural gas daily returns. It-1=1 if εt-1<0 and 0
otherwise, st is the volatility seasonal for time t, δ0 measures the announcement effect on
the conditional volatility. εt is a random variable with conditional mean zero and
conditional variance ht independent of st. DAt is announcement dummy. Standard errors
are shown in parentheses. (***), (**), (*) designate estimates significantly different from
zero at the 0.001, 0.01 and 0.05 levels, respectively. The sample extends from January
01, 1997 through December 31, 2005.
Nearby futures
0.2860***
(0.0696)
0.1228***
(0.0097)
0.8879***
(0.0122)
Next futures
0.1981***
(0.0573)
0.0784***
(0.0112)
0.9166***
(0.0126)
γ
-0.0687***
(0.0105)
-0.0390***
(0.0109)
δ0
0.3928***
(0.0960)
0.4298***
(0.0983)
Log likelihood
-5963.098
-5657.818
ω
α
β
35
Table 8
GJR model with Day-of-the-week effects for two sub- periods (January 1997 to March
2002 and April 2002 to December 2005)
Estimates of the model: Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+ δM DMONt)(1+δTDTUEt)(1+δRDTHURt)(1+δFDFRIt),
ht = ω+ αεt-12+βht-1+ γεt-12It-1,
are presented where Rt is the natural gas daily return. It-1=1 if εt-1<0 and 0 otherwise, st is
the volatility seasonal for time t, δM, δT, δR, and δF measure the day-of-the-week effects
on the conditional volatility. εt is a random variable with conditional mean zero and
conditional variance ht independent of st. DMONt, DTUEt, DTHURt and DFRIt are zeroone dummy variables for each of the four weekdays. Standard errors are shown in
parentheses. (***), (**), (*) designate estimates significantly different from zero at the
0.001, 0.01 and 0.05 levels, respectively.
January 1997 to March 2002
Nearby futures
Nearby futures
0.2426**
0.2426**
(0.0875)
(0.0875)
0.0952***
0.0952***
(0.0209)
(0.0209)
0.8937***
0.8937***
(0.0173)
(0.0173)
April 2002 to December 2005
Next futures
Next futures
0.7445***
0.3781*
(0.2142)
(0.1704)
0.1547***
0.0723***
(0.0244)
(0.0185)
0.8188***
0.8844***
(0.0321)
(0.0334)
-0.0375**
-0.0375**
-0.1138***
-0.0375**
(0.0131)
(0.0131)
(0.0256)
(0.0131)
δM
0.4468**
(0.1593)
0.4468**
(0.1593)
1.0928***
(0.2071)
1.0386***
(0.2280)
δT
-0.0592
(0.0950)
-0.0592
(0.0950)
-0.1224
(0.1025)
0.0050
(0.1169)
δR
0.1650
(0.1168)
0.1650
(0.1168)
0.3787*
(0.1719)
0.5625**
(0.1917)
δF
-0.4032***
(0.0647)
-0.4032***
(0.0647)
-0.3395***
(0.0716)
-0.3240***
(0.0780)
-3480.403
-3480.403
-2424.402
-2325.587
ω
α
β
γ
Log
likelihood
36
Table 9
GJR model with announcement effects on announcement days and on subsequent days
Estimates of the model: Rt = μ + Φ1Rt-1 + st1/2εt,
st = (1+ δ0DAt),
and the model:
(4)
(5d)
ht = ω+ αεt-12+βht-1+ γεt-12It-1+ δ1DAt,
(7)
Rt = μ + Φ1Rt-1 + εt,
ht = ω+ αεt-12+βht-1+ γεt-12It-1+ δ1DAt,
(4’)
(7)
are presented where Rt is the natural gas daily return. It-1=1 if εt-1<0 and 0 otherwise, st is
the volatility seasonal for time t, δ0 measures the announcement effect on the conditional
volatility on announcement days, δ1 measures the announcement effect on the conditional
volatility on subsequent days. εt is a random variable with conditional mean zero and
conditional variance ht independent of st. DA is announcement dummy. Standard errors
are shown in parentheses. (***), (**), (*) designate estimates significantly different from
zero at the 0.001, 0.01 and 0.05 levels, respectively. The sample extends from January
01, 1997 through December 31, 2005.
Estimates of specification 4, 5(d) and 7
Nearby futures
Next futures
0.1743
0.4193**
(0.1971)
(0.1554)
0.1215***
0.0754***
(0.0095)
(0.0110)
0.8838***
0.9148***
(0.0123)
(0.0127)
Estimates of specification 4’ and 7
Nearby futures
Next futures
0.1932
0.0012
(0.1452)
(0.1213)
0.1136***
0.0989***
(0.0095)
(0.0132)
0.8940***
0.9065***
(0.0110)
(0.0133)
γ
-0.0612***
(0.0113)
-0.0337**
(0.0104)
-0.0584***
(0.0099)
δ0
0.3214**
(0.1107)
0.5562***
(0.1399)
δ1
0.6916
(0.9201)
1.0752
(0.7405)
2.6420***
(0.6596)
1.5161**
(0.5320)
-5657.807
-5969.276
-5663.831
ω
α
β
logL -5964.902
-0.0703***
(0.0126)
-
-
37
38