VOLATILITY IN NATURAL GAS AND OIL MARKETS *

Transcription

1 VOLATILITY IN NATURAL GAS AND OIL MARKETS * by Rober S. Pindyck Massachuses Insiue of Technology Cambridge, MA This draf: Ocober 20, 2003 Absrac: I use daily fuures price daa o examine he behavior of naural gas and crude oil price volailiy since I es wheher here has been a significan rend in volailiy, wheher here was a shor-erm increase in volailiy during he ime of he Enron collapse, and wheher naural gas and crude oil price volailiies are inerrelaed. I also measure he persisence of shocks o volailiy and discuss is implicaions for gas- and oil-relaed coningen claims. JEL Classificaion Numbers: G13; L71, Q40 Keywords: Naural gas, oil markes, price volailiy, Enron, commodiy markes. * I am graeful o M.I.T. s Cener for Energy and Environmenal Policy Research for is financial suppor of he research leading o his paper, o Mr. Sco Byrne and he New York Mercanile Exchange for providing fuures marke daa, and o Marin Minnoni for his ousanding research assisance.

2 1 1. Inroducion This paper examines he behavior of naural gas and crude oil price volailiy since Prices of crude oil and especially naural gas rose sharply (bu emporarily) during lae 2000, and naural gas rading was buffeed by he collapse of Enron in lae 2001, suggesing o some ha volailiy in hese markes has increased. Wheher or no his is rue, volailiy has been high, and (like prices hemselves) flucuaes dramaically. Undersanding volailiy in naural gas and crude oil markes is imporan for several reasons. Persisen changes in volailiy can affec he risk exposure of producers and indusrial consumers of naural gas and oil, and aler he incenives o inves in naural gas and oil invenories and faciliies for producion and ransporaion. Likewise, volailiy is a key deerminan of he value of commodiy-based coningen claims, wheher financial or real. Thus he behavior of volailiy is imporan for derivaive valuaion, hedging decisions, and decisions o inves in physical capial ied o he producion or consumpion of naural gas or oil. In addiion, volailiy plays a role in he shor-run marke dynamics for naural gas and oil. As discussed in my earlier paper (2002), volailiy can affec he demand for sorage, and can also affec he oal marginal cos of producion by affecing he value of firms operaing opions and hus he opporuniy cos of curren producion. In paricular, greaer volailiy should lead o an increased demand for sorage, and an increase in boh spo prices and marginal convenience yield. 1 Thus, changes in volailiy can help explain changes in hese oher variables. Wih his in mind, I address he following quesions: Firs, has naural gas and/or crude oil price volailiy changed significanly since 1990, and in paricular, are here measurable rends in volailiy? Relaed o his, have he evens surrounding he collapse of Enron affeced volailiy, i.e., was here a significan shor-erm increase in volailiy around he ime of he collapse? Second, are naural gas and crude oil volailiies inerrelaed, i.e., can changes in one help predic changes in he oher? Third, alhough volailiy clearly flucuaes over ime, how persisen are he changes? If changes are very persisen, hen hey will lead o changes in he prices of opions and oher derivaives (real or financial) ha are ied o he prices of hese commodiies. 1 Using weekly daa for he peroleum complex, I show in Pindyck (2002) ha he heoreical relaionships beween volailiy and oher variables are well suppored for heaing oil, bu less so for crude and gasoline. The role of volailiy in he opporuniy cos of producion is also spelled ou and esed by Lizenberger and Rabinowiz (1995). For an inroducion o he inerrelaionships among price, invenories, and convenience yields, see Pindyck (2001).

3 2 Bu if changes in volailiy are highly ransiory, hey should have lile or no impac on marke variables or on real and financial opion values. Finally, exending he work in Pindyck (2002), I revisi he quesion off wheher changes in volailiy are predicable. To address hese quesions, I use daily fuures price daa for naural gas and crude oil o infer daily spo prices and daily values of he ne marginal convenience yield. From he log price changes (adjused for non-rading days) and marginal convenience yield, I calculae daily and weekly reurns from holding each commodiy. I hen esimae volailiy hree differen ways. Firs, using a five-week overlapping window, I esimae weekly series for price volailiy by calculaing sample sandard deviaions of (adjused) daily log price changes. As Campbell e. al. (2001) poin ou in heir sudy of sock price volailiy, in addiion o is simpliciy, his approach has he advanage ha i does no require a parameric model describing he evoluion of volailiy over ime. 2 Second, I esimae series for condiional volailiy by esimaing GARCH models of he weekly reurns on he commodiies, and I compare he volailiy esimaes from hese models o he sample sandard deviaions. Third, I esimae a daily series for condiional volailiy by esimaing GARCH models of he daily reurns on he commodiies. I sudy he behavior of volailiy in wo differen ways. Firs, using he esimaed weekly sample sandard deviaions, I es for he presence of ime rends, I es wheher volailiy was significanly greaer during he period of he Enron collapse, and I examine wheher gas (oil) volailiy is a significan predicor of oil (gas) volailiy. I also use hese series o esimae he persisence of changes in volailiy. Second, I address hese same quesions using weekly and daily GARCH models of commodiy reurns. For example, I es wheher a ime rend or a dummy variable for he Enron period is a significan explanaor of volailiy (and/or reurns) in he GARCH framework. Likewise, he esimaed coefficiens from he variance equaion of each GARCH model provide a direc esimae of he persisence of volailiy shocks. I focus on he volailiy of prices, bu here are oher measures of volailiy. Puing aside issues of daa availabiliy, one could insead examine he volailiy of consumpion, producion, or invenories. Tha would indeed be useful if he objecive was o explain he deerminans of invenory demand, e.g., he role of producion and/or consumpion smoohing, and producion- 2 Schwarz (1997) and Schwarz and Smih (2000) use fuures and spo prices o esimae a mean-revering price process and value commodiy-based opions. Tha approach also yields implici ime-varying esimaes of volailiy.

4 3 cos smoohing. 3 My concern, however, is wih he overall marke, and he spo price is he bes single saisic for marke condiions. Spo price volailiy reflecs he volailiy of curren as well as expeced fuure values of producion, consumpion, and invenory demand. 4 My resuls can be summarized as follows: (1) There is a saisically significan posiive rend in volailiy for naural gas (bu no for crude oil). However, his rend is of lile economic imporance; over a en year period, i amouns o abou a 3-percen increase in volailiy. (2) There is no saisically significan increase in volailiy during he period of he Enron collapse. (3) The evidence is mixed as o he inerrelaionship beween crude oil and naural gas reurns and volailiies. Using daily daa, crude oil reurns are a significan predicor of naural gas reurns (bu no he oher way around), and crude oil volailiy is a significan predicor of naural gas volailiy. Using weekly daa, however, hese resuls are less clear-cu. (4) Shocks o volailiy are generally shor-lived for boh naural gas and crude oil. Volailiy shocks decay (i.e., here is reversion o he mean) wih a half-life of abou 5 o 10 weeks. In he nex secion I discuss he daa and he calculaion of reurns and weekly sample sandard deviaions. All of he empirical work is presened in Secion 3. Secion 4 concludes. 2. The Daa I begin wih naural gas and crude oil fuures price daa covering he period May 2, 1990 hrough February 26, (The sar dae was consrained by he beginning of acive rading in naural gas fuures.) To obain a weekly series for volailiy, I use he sample sandard deviaions of adjused daily log price changes in spo and fuures prices. As discussed below, I also obain esimaes of condiional volailiy from GARCH models of weekly and daily reurns Spo Prices and Weekly Volailiy For each commodiy, I compiled daily fuures selemen price daa for he neares conrac (ofen he spo conrac), he second-neares, and he hird-neares. These prices are denoed by F1, F2, and F3. The spo price can be measured in hree alernaive ways. Firs, one can use cash prices, purporedly reflecing acual ransacions. Bu daily cash price daa are usually no 3 Pindyck (1994) addresses hese issues; also see Ecksein and Eichenbaum s (1985) sudy of crude oil invenories. 4 Furhermore, one canno acually pu aside issues of daa availabiliy. Alhough weekly daa are available for U.S. producion, consumpion, and invenories of naural gas and crude oil, daily daa are no.

5 4 available, and a cash price can include discouns and premiums ha resul from relaionships beween buyers and sellers, and need no reflec precisely he same produc (including delivery locaion) specified in he fuures conrac. A second approach is o use he price of he spo fuures conrac, i.e., he conrac ha expires in monh. Bu he spo conrac ofen expires before he end of he monh, and acive spo conracs do no always exis for each monh. The hird approach, which I use, is o infer a spo price from he neares and he nex-oneares acive fuures conracs. This is done for each day by exrapolaing he spread beween hese conracs backwards o he spo monh as follows: P = F1( F1 / F2) n0 / n1 where P is he spo price on day, F, and F are he prices on he neares and nex-o-neares fuures conracs, and n0 1 2 and n1 are he number of days from o he expiraion of he firs conrac, and he number of days beween he expiraion daes for he firs and second conracs. Given hese daily esimaes of spo prices, I compue weekly esimaes of volailiy. To do his, one mus ake ino accoun weekends and oher non-rading days. If he spo price followed a geomeric Brownian moion, his could be done simply by dividing he log price changes by he square roo of he number of inervening days (e.g., hree days in he case of a weekend), and hen calculaing he sample variance. However, as is well known, on average he sandard deviaion of n-day log price changes is significanly less han one-day log price changes, when n includes non-rading days. 5 (1) n imes he sandard deviaion of To deal wih his, I sor he daily price daa by inervals, according o he number of days since he las rading day. For example, if here were no holidays in a paricular period, prices for Tuesday, Wednesday, Thursday, and Friday would all be classified as having an inerval of one day, and Monday would be assigned an inerval of hree days because of he weekend. Because of holidays, some prices are assigned o inervals of wo, four, or even five days (if a weekend was followed by a wo-day holiday). For each inerval se, I calculae he sample sandard deviaion of log price changes for he enire sample for each commodiy. Leing denoe his sample sandard deviaion for log price changes over an inerval of n days, I compue he effecive daily log price change: sˆn 5 If P follows a geomeric Brownian noion, var ( p p ) = nvar( p p ). + n + 1 p = log P follows an arihmeic Brownian noion, so ha

6 5 δτ (log Pτ log Pτ n) =. (2) sˆ / sˆ n For each week, I hen compue a sample variance and corresponding sample sandard deviaion using hese effecive daily log price changes for ha week and he preceding four weeks: 1 ˆσ = ( N 1 N 2 δ τ δ ), (3) 1 τ = 1 where N is he number of effecive days in he five-week inerval. Eqn. (3) gives he sample sandard deviaion of daily percenage price changes; o pu i in weekly erms, I muliply by 30 / 4 = 7.5. The resuling weekly series is a measure of volailiy, σ Daily and Weekly Reurns An imporan advanage of using he weekly esimaes of volailiy discussed above (besides is simpliciy) is ha i does no require a parameric model of he evoluion of volailiy over ime. However, here are also disadvanages. The firs is ha he use of overlapping inervals inroduces serial correlaion as an arifac, which makes i more difficul o discern he imeseries properies of volailiy. A second disadvanage is ha even he use of a five-week inerval yields imprecise esimaes of he sandard deviaion. Hence, I also esimae volailiy from GARCH models of commodiy reurns. These models can include parameers ha es for ime variaion (such as rends or an Enron effec ), and have he addiional advanage ha he imeseries properies of volailiy (he ARCH and GARCH componens, which deermine he persisence of volailiy shocks) are esimaed along wih he volailiy iself. Marginal Convenience Yield. One par of he oal reurn on he commodiy is he ne marginal convenience yield, ψ, i.e., he value of he flow of producion- and deliveryfaciliaing services from he marginal uni of invenory, ne of sorage coss. Ne marginal convenience yield can be measured from spo and fuures prices as follows: ψ = (1 + r) P, (4) where F 1 is he fuures price a ime for a conrac mauring a ime + 1, and r is he oneperiod riskless ineres rae. I calculae values of F1 ψ for every rading day using he fuures price corresponding as closely as possible o a 1-monh inerval from he spo price. (When here are few or no rades of he neares fuures conrac, as someimes occurs wih naural gas, he nex-

7 6 o-neares conrac is used insead.) Also, I use he yield on 3-monh Treasury bills, adjused for he number of days beween P and F 1, for he ineres rae r. In wha follows, I use boh daily and weekly series for he marginal convenience yield, so I conver ψ ino daily erms, i.e., dollars per uni of commodiy per day. For days followed by anoher rading day (e.g., a Monday), I simply divide he values of ψ calculaed above by he number of days beween P and F 1. For days followed by n non-rading days, I muliply hese values by n+1. (Thus for a Friday, which is ypically followed by n = 2 non-rading days, he convenience yield is he flow of value from holding a marginal uni of invenory over he nex 3 days.) This daily series is used o compue daily reurns from holding he commodiy. To obain a weekly series, I use he calculaed values of ψ for he Wednesday of each week, and muliply hose values by 7 so ha he convenience yield is measured in dollars per uni of commodiy per week. (If Wednesday is a holiday, I use Thursday s price.) I hen use his weekly series o calculae weekly reurns from holding he commodiy. Calculaing Reurns. The oal reurn from holding a uni of a commodiy over one period is he capial gain or loss over ha period, plus he dividend, which is he ne marginal convenience yield, i.e., he flow of benefis o producers or consumers from holding he marginal uni of invenory, ne of sorage coss. I calculae a series of daily (weekly) reurns by summing he effecive daily log price changes over each day (week) and adding o his he esimae of daily (weekly) convenience yield. The weekly reurn, for example, is calculaed as: R T δτ ψ (5) τ = 1 = + where δτ is given by eqn. (2), and T is he number of days in he week. A series for he daily reurn is calculaed by using he effecive daily log price change for each effecive rading day and adding he daily flow of marginal convenience yield. (Because I use effecive rading days, he daily series will have abou 20 daa poins per monh.) 3. The Behavior of Volailiy and Prices To examine he behavior of naural gas and crude oil price volailiy, I firs use he weekly ime series of sample sandard deviaions of adjused log price changes. These ime series show lile evidence of eiher a rend in volailiy or a significan increase in volailiy during he period

8 7 of he Enron collapse. In addiion, changes in volailiy appear o be highly ransiory, wih a half-life of several weeks. As an alernaive way of measuring volailiy, I esimae GARCH models of he weekly reurns o holding he commodiy, which yields esimaes of he weekly condiional sandard deviaions. I es for changes in volailiy over ime by inroducing a ime rend and an Enron dummy variable in he variance equaions of he GARCH models, and obain resuls ha are similar o hose obained from he weekly sample sandard deviaions. Finally, I use he daily adjused reurn series o esimae daily GARCH models. These provide esimaes of condiional sandard deviaions on a daily basis, and are also used o es for ime rends and an Enron effec, and o esimae he persisence of changes in volailiy Weekly Sample Sandard Deviaions Figures 1 and 2 show he weekly series for he spo price and volailiy, where volailiy is measured as he sample sandard deviaions of adjused log price changes. Noe ha for boh commodiies, volailiy is high, and is iself volaile. The mean values of volailiy are 12.8 percen per week for naural gas and 5.9 percen per week for crude oil; he corresponding sandard deviaions are 7.0 percen for naural gas and 3.2 percen for crude oil. Naural gas and crude oil volailiies are correlaed, bu only weakly; he coefficien of correlaion for he wo series is As expeced, boh volailiy series have high degrees of skewness and kurosis; he skewness coefficien and degree of kurosis are 1.60 and 6.99 respecively for naural gas, and 1.76 and 7.77 for crude oil. For he log of volailiy, hese coefficiens are 0.46 and 3.99 for naural gas, and 0.23 and 2.84 for crude oil, which are roughly consisen wih a normal disribuion. However, a Jarque-Bera es rejecs normaliy a he 1-percen level in boh cases. As Figures 1 and 2 illusrae, periods of unusually high volailiy end o accompany sharp increases in he spo price. In he case of crude oil, for example, volailiy was high in lae 1990 and early 1991 following he Iraqi invasion of Kuwai, as spo prices reached $40 per barrel. However, here were also periods of high volailiy ha accompanied unusually low spo prices, e.g., during 1998 for boh commodiies. Overall, volailiy and price are moderaely correlaed; he correlaion (in levels) is.27 for naural gas and.37 for crude oil. Was here a significan increase in volailiy during he period of he Enron collapse? The Enron bankrupcy sharply reduced spo and forward rading in naural gas and elecriciy, and also led o speculaion over ne long and shor posiions in naural gas. This probably caused

9 8 increased uncerainy over naural gas prices, which could have spilled over ino crude oil. Pinpoining he beginning of he Enron collapse is difficul, bu clearly by Sepember 2001 analyss began quesioning Enron s valuaion. (On Sepember 26, 2001, Kenneh Lay made his famous announcemen o employees ha he sock is an incredible bargain. ) On Ocober 16, 2001, Enron repored a $638 million hird-quarer loss and disclosed a $1.2 billion reducion in shareholder equiy. Furher financial saemen revisions were announced during Ocober and November, and Enron filed for Chaper 11 bankrupcy proecion on December 2. I defined he period of he Enron collapse as Augus 29 o December 5, 2001, and creaed a dummy variable equal o 1 during his period and 0 oherwise. Figure 3 shows naural gas and crude oil price volailiy from he middle of 2000 hrough he middle of 2002, wih he Enron period shaded. Naural gas volailiy reached a peak during his period of 38 percen per week, and crude oil volailiy was also above average. I examine he significance of hese increases in volailiy in he conex of forecasing regressions. Using daa for crude oil, heaing oil and gasoline, I have shown elsewhere (Pindyck (2002)), ha price volailiy canno be forecased using marke variables for ha commodiy (such as producion, invenories, or convenience yields), or using macroeconomic variables (such as ineres raes). As menioned above, here is a conemporaneous posiive correlaion beween volailiy and he price level iself (and hus beween volailiy and he conemporaneous convenience yield), bu lile or no correlaion wih lagged prices or oher marke variables. As discussed below, he only variables ha do have forecasing power for volailiy are is own lagged values (i.e., volailiy can be modeled as an ARMA process), and possibly lagged values of volailiy for anoher commodiy (e.g., crude oil in he case of naural gas). Table 1 shows simple forecasing regressions for volailiy. In columns (1) and (4), he explanaory variables are 6 lags of volailiy and he Enron dummy variable. For naural gas, he Enron dummy is marginally significan, and for crude oil i is insignifican. Bu even for naural gas i has lile economic significance, emporarily adding abou 1.5 percen o an average volailiy of abou 20 percen. In columns (2) and (5), a ime rend is added; in boh cases i is insignifican, and has almos no effec on he oher esimaed coefficiens. Finally, columns (3) and (6) es wheher lagged values of crude oil volailiy help explain naural gas volailiy, and vice versa. For naural gas, he answer is ambiguous: an F-es on he join significance of he lagged crude oil volailiy erms in column (3) has a value of 1.84, which is significan a he 10-

10 9 percen level. Lagged values of naural gas volailiy, however, are no significan explanaors of crude oil volailiy: he corresponding F-saisic for column (6) is The boom of Table 1 shows he sum of he auoregressive coefficiens along wih he implied half-life for volailiy shocks. The half-life is abou five o six weeks for naural gas, and eleven o welve weeks for crude oil. Thus, alhough volailiy iself flucuaes considerably, shocks o volailiy appear o be quie ransiory, paricularly for naural gas. The volailiy series shown in Figures 1 o 3 and used in he regressions in Table 1 suffer from wo main problems. Firs, he sample sandard deviaions are esimaed from daily log price changes for overlapping five-week inervals, so ha he series are serially correlaed by consrucion. Second, even wih five-week inervals, each sample sandard deviaion is based on a mos weny-five observaions. One way o ge around hese problems is o esimae GARCH models of he commodiy reurns hemselves GARCH Models of Weekly Reurns I esimae models of he following form. The weekly reurn o holding he commodiy is: RET = a + atbill + a σ + a ENRON + a TIME + b DUM j + ε, (6) j j= 1 11 where DUM j are monhly dummy variables. In his equaion, he Treasury bill rae should affec he reurn because i is a large componen of he carrying cos of holding he commodiy. Likewise, we would expec he reurn o increase wih is own riskiness, so σ, he sandard deviaion of he error erm ε, is included in he equaion. Finally, I also include he Enron dummy variable and a ime rend o es for any sysemaic ime variaion in reurns. The second equaion explains he variance of ε as a GARCH (p,q) process: p q = + j j + j j + 1ENRON + 2TIME j= 1 j= 1 σ α α ε β σ γ γ (7) The Enron dummy and a ime rend are included o es for ime variaion in volailiy. 6 Noe ha when lagged values of volailiy for he second commodiy are added o he regression, he Enron dummy becomes insignifican. This may simply reflec he fac ha volailiy for boh commodiies was unusually high during he Enron period.

11 10 Table 2 shows maximum likelihood esimaes of his model. Because he reurn includes he curren and previous week s price, he model is esimaed wih and wihou a firs-order moving average error erm in eqn. (6). In all cases he number of lags in eqn. (7) is chosen o minimize he Akaike informaion crieria. The resuls for crude oil (columns 3 and 4 of Table 2) are consisen wih he basic heory of commodiy reurns and sorage. Reurns have a srong posiive dependence on he ineres rae and on volailiy (i.e., he sandard deviaion of ε ). For naural gas, however, boh he ineres rae and volailiy are saisically insignifican in he reurns equaion. For boh commodiies, he ime rend is insignifican in he reurns equaion, bu is posiive and significan in he variance equaion, and he Enron dummy is posiive bu saisically insignifican in he variance equaion. Thus, I find a saisically significan posiive rend in volailiy for boh gas and oil, bu no separae impac of he Enron evens. However, his rend is no economically significan. For naural gas, he ime rend coefficien is abou , which implies a 10-year increase in he average variance of The mean value of volailiy (sandard deviaion of reurns) is abou.13 for naural gas, so he mean variance is abou.017, so he rend represens a roughly 2- percen increase in he variance over a decade. Table 2 also shows esimaes of he half-life of volailiy shocks. This is deermined by he sum of he ARCH and GARCH coefficiens in he variance equaion, i.e., Half-life = log(.5) / log( α + β ). j j (8) The half-life of volailiy shocks is abou seven o en weeks for naural gas, and seven o eigh weeks for crude oil. These numbers differ slighly from he esimaes in Table 1, bu overall, shocks o volailiy again appear ransiory for boh commodiies. We can compare he volailiy esimaes from hese GARCH models (i.e., he condiional sandard deviaion of ε ) wih he sample sandard deviaions. Using he GARCH models ha include he moving average erm, i.e., columns 2 and 4 of Table 2, he simple correlaion of he wo volailiy series is.593 for naural gas and.665 for crude oil. Figure 4 shows he wo volailiy series for naural gas. The wo series generally rack each oher, bu he GARCH volailiy is lower on average (a mean of 8.7 percen vs percen for he sample sandard deviaion) and has a higher degree of kurosis.

12 GARCH Models of Daily Reurns An advanage of esimaing GARCH models of weekly reurns is ha he resuling esimaes of he condiional sandard deviaions can be compared o he weekly sample sandard deviaions. However, hese weekly models do no make use of all of he available daily daa. Thus I also esimae GARCH models of daily reurns. These models also ake he form of eqns. (6) and (7), excep ha I do no include monhly dummy variables in he reurns equaion. The number of lags is again chosen o minimize he Akaike informaion crierion. As wih he weekly GARCH models, he resuls for crude oil, bu no naural gas, are consisen wih he heory of commodiy reurns and sorage. (See Table 3.) Crude oil reurns have a srong posiive dependence on he ineres rae and on volailiy, bu boh variables are insignifican in he equaion for naural gas reurns. And as wih he weekly models, here is no saisically significan impac of he Enron evens on volailiy for eiher commodiy. The ime rend for volailiy is now only marginally significan for naural gas and insignifican for crude oil, bu even for naural gas i is only of marginal economic imporance. (Using an average esimae of for he rend coefficien, he 10-year rend increase in he variance of daily reurns would be.00020, which is abou 9 percen of he mean daily variance of ) The esimaes of he half-life of volailiy shocks vary across he differen specificaions, bu overall are close o hose in Tables 1 and 2. The half-life is abou 6 o 9 weeks for naural gas, and 3 o 11 weeks for crude oil. Once again, shocks o volailiy appear o be largely ransiory Reurns and Volailiies Across Markes I urn nex o he inerrelaionship beween crude oil and naural gas reurns and volailiies. The resuls in Table 1, based on he 5-week sample sandard deviaions, provide some evidence ha crude oil volailiy has predicive power wih respec o naural gas volailiy (bu no he oher way around). To explore his furher, I run Granger causaliy ess beween gas and oil using he sample sandard deviaions, and he weekly and daily volailiies from he GARCH models. I also run hese ess on weekly and daily gas and oil reurns. These ess are simply F- ess of he exclusion resricions b1 = b2 =... = b L = 0 in he regression equaion L i i y = a + a y + bx 0 i i i= 1 i= 1 L. A failure o rejec hese exclusion resricions is a failure o rejec

13 12 he hypohesis ha x Granger-causes y. When running hese ess, I use 2, 4, and 6 lags for he weekly regressions, and 4, 6, 10, 14, 18, and 22 lags for he daily regressions. The resuls are shown in Table 4. The firs wo panels show ess for he weekly and daily reurns. The weekly reurns show no evidence of causaion in eiher direcion, bu for he daily reurns, I can rejec he hypohesis ha here is no causaliy from oil o gas. Given ha oil prices are deermined on a world marke, if here is causaliy in eiher direcion we would expec i o run from oil o gas, and no he oher way around. The nex hree panels show es resuls for volailiy. The ess based on he weekly sample sandard deviaions and he daily GARCH models show evidence of causaliy from oil o gas, and no from gas o oil, as expeced. However, he resuls using he volailiy esimaes from he weekly GARCH models show jus he opposie. Bu noe ha he simple correlaions of he oil and gas volailiies are much higher for he weekly sample sandard deviaions and he daily GARCH esimaes (.170 and.146, respecively) han for he weekly GARCH esimaes (.092), so I discoun hese laer resuls. Overall, hese ess (along wih he regressions in Table 1) provide some evidence ha crude oil volailiy is a predicor of naural gas volailiy. 4. Summary and Conclusions My resuls can be summarized as follows. Firs, here is a saisically significan posiive ime rend in volailiy for naural gas, and o a lesser exen for oil. The rends, however, are small, and no of grea economic imporance. Given he limied lengh of my sample, here are cerainly no conclusions ha can be drawn abou long-erm rends. As for he demise of Enron, i does no appear o have conribued o any significan increase in volailiy. Second, here is some evidence ha crude oil volailiy and reurns has predicive power for naural gas volailiy and reurns, bu no he oher way around. Bu his predicive power is quie limied; for pracical purposes, volailiy can be modeled as a pure ARMA process. Third, alhough volailiy flucuaes considerably, shocks o volailiy are shor-lived, wih a half-life on he order of 5 o 10 weeks. This means ha flucuaions in volailiy could cerainly affec he values of financial gas- or oil-based derivaives (such as opions on fuures conracs), because such derivaives ypically have a duraion of only several monhs. Bu flucuaions in volailiy should no have any significan impac on he values of mos real opions (e.g., opions o inves in gas- or oil-relaed capial), or he relaed invesmen decisions. Of course hese

14 13 flucuaions migh lead one o hink ha financial or real opions should be valued using a model ha accouns for sochasic volailiy. However, he numerical analyses of Hull and Whie (1987), among ohers, suggess ha reaing volailiy as non-sochasic will make lile quaniaive difference for such valuaions. Sharp (bu emporary) increases in he prices of crude oil and naural gas, along wih he collapse of Enron, have creaed a percepion ha volailiy has increased significanly, increasing he risk exposure of energy producers and consumers. This does no seem o be he case. The increases in volailiy ha I measure are oo small o have economic significance, and flucuaions in volailiy are generally shor-lived.

15 14 References Campbell, John Y., Buron Malkiel, Marin Leau, and Yexiao Xu, Have Individual Socks Become More Volaile? An Empirical Exploraion of Idiosyncraic Risk, Journal of Finance, Feb. 2001, 56, Ecksein, Zvi, and Marin S. Eichenbaum, Invenories and Quaniy-Consrained Equilibria in Regulaed Markes: The U.S. Peroleum Indusry, , in T. Sargen, ed., Energy, Foresigh, and Sraegy, Washingon, D.C.: Resources for he Fuure, Hull, John, and Alan Whie, The Pricing of Opions on Asses wih Sochasic Volailiies, Journal of Finance, June 1987, 42, Lizenberger, Rober H., and Nir Rabinowiz, Backwardaion in Oil Fuures Markes: Theory and Empirical Evidence, Journal of Finance, 1995, 50, Pindyck, Rober S., Invenories and he Shor-Run Dynamics of Commodiy Prices, The RAND Journal of Economics, Spring 1994, 25, Pindyck, Rober S., The Dynamics of Commodiy Spo and Fuures Markes: A Primer, The Energy Journal, 2001, 22, Pindyck, Rober S., Volailiy and Commodiy Price Dynamics, M.I.T. Cener for Energy and Environmenal Policy Research Working Paper, Augus, Schwarz, Eduardo S., The Sochasic Behavior of Commodiy Prices: Implicaions for Valuaion and Hedging, The Journal of Finance, July 1997, 52, Schwarz, Eduardo S., and James E. Smih, Shor-Term Variaions and Long-Term Dynamics in Commodiy Prices, Managemen Science, 2000.

16 15 Table 1: Forecasing Equaions for Volailiy (1) (2) (3) (4) (5) (6) Dep. Var. NG NG NG CRUDE CRUDE CRUDE Cons (5.71) (4.62) (4.38) (3.40) (2.40) (1.40) NGSIG (-1) (28.12) (28.06) (27.69) (-1.67) NGSIG (-2) (-1.74) (-1.73) (-1.53) (1.38) NGSIG (-3) (-1.17) (-1.17) (-1.11) (0.70) NGSIG (-4) (3.81) (3.81) (3.68) (-0.20) NGSIG (-5) (-9.82) (-9.81) (-9.81) (0.32) NGSIG (-6) (7.84) (7.79) (7.80) (-0.80) ENRON (2.00) (1.88) (1.73) (1.63) (1.44) (1.18) TIME 3.54E E E E-06 (0.60) (0.70) (0.99) (0.79) CSIG (-1) (2.33) (30.10) (30.03) (29.86) CSIG (-2) (-0.68) (-2.96) (-2.95) (-2.82) CSIG (-3) (-0.35) (-0.24) (-0.24) (-0.31) CSIG (-4) (0.98) (2.13) (2.13) (2.06) CSIG (-5) (0.67) (-9.09) (-9.09) (-9.15) CSIG (-6) (-0.56) (10.96) (10.91) (10.95) R Σ AR(i) Half-Life (weeks)

17 16 Table 2: GARCH Models of Weekly Reurns (1) (2) (3) (4) Dep. Var. NG NG CRUDE CRUDE Cons (1.12) (1.11) (-9.55) (-5.75) σ (-0.71) (-0.90) (5.21) (3.28) TBILL (-1.43) (-1.52) (12.24) (8.01) ENRON (-1.56) (-1.63) (-0.87) (-0.45) TIME 1.42E E E E-07 (1.05) (1.28) (-1.39) (0.04) MA (1) (-2.96) (10.87) VARIANCE EQUATION CONST E E-05 (4.78) (3.93) (0.57) (0.68) ARCH (1) (2.34) (2.08) (4.79) (1.32) ARCH (2) (-1.14) (-1.13) (3.58) (3.15) ARCH (3) (0.62) (0.99) (3.96) (-1.02) ARCH (4) (1.72) (1.48) (5.60) (5.06) ARCH (5) (-1.87) (-1.63) GARCH (1) (9.08) (8.79) (1.40) (3.13) GARCH (2) (0.32) (0.07) (-2.00) (-2.52) GARCH (3) (-1.01) (-0.98) (-4.71) (0.48) GARCH (4) (3.65) (2.16) (7.44) (2.09) GARCH (5) (-3.40) (-2.39) ENRON (1.06) (0.98) (0.77) (1.09) TIME 7.56E E E E-06 (4.98) (4.20) (2.58) (4.37) Half-Life (weeks) Noe: Regression equaions for weekly reurns include monhly dummy variables, which are no repored. Number of ARCH and GARCH erms chosen o minimize Akaike informaion crierion.

18 17 Table 3: GARCH Models of Daily Reurns (1) (2) (3) (4) (5) (6) Dep. Var. NG NG NG CRUDE CRUDE CRUDE CONST (-0.15) (1.35) (-0.13) (-1.46) (-1.52) (-1.77) σ (-1.79) (-1.70) (-1.60) (3.98) (6.51) (4.15) TBILL (0.47) (0.51) (0.51) (2.52) (2.24) (2.39) ENRON (-0.80) (-1.19) (-4.95) (-5.05) TIME 2.54E E E E-07 (1.98) (1.81) (0.96) (1.44) VARIANCE EQUATION: GARCH (p,q) (p,q) (8,7) (4,8) (5,8) (5,9) (4,9) (4,9) CONST 2.08E E E E E E-06 (111.93) (54.78) ( ) (0.53) (3.63) (0.80) ENRON (1.78) (1.57) (0.97) (0.91) TIME 7.32E E E E-08 (2.43) (1.55) (1.19) (1.16) Half-Life (weeks) Noe: Number of ARCH and GARCH erms chosen o minimize Akaike informaion crierion. ARCH and GARCH coefficiens are no shown.

19 18 Table 4: Granger Causaliy Tess Variable Lags NG Crude Crude NG Weekly Reurns 2 No No (Simple Corr. =.095) 4 No No 6 No No Daily Reurns 4 Yes* Yes* (Simple Corr. =.028) 6 No Yes** 10 No Yes** 14 No Yes** 18 No Yes* 22 No No Weekly Volailiy, 2 No Yes* Sample Sand. Dev. 4 No Yes* (Simple Corr. =.170) 6 No No Weekly Volailiy, 2 Yes** No GARCH 4 Yes** No (Simple Corr. =.092) 6 Yes* No Daily Volailiy, 4 No No GARCH 6 No Yes* (Simple Corr. =.146) 10 No No 14 No Yes** 18 No Yes** 22 No Yes** Noe: Tes of x y is an F-es of he exclusion resricions b1 = b2 =... = b L = 0 in he regression L i i y = a + a y + bx 0 i i i= 1 i= 1 L. A no implies a failure o rejec he hypohesis ha he b i s equal 0, and a yes implies rejecion a he 5% (*) or 1% (**) level.

20 19 Figure 1 NATURAL GAS: WEEKLY SPOT PRICE AND VOLATILITY.5 $/mcf SPOT PRICE VOLATILITY (percen/week) Figure 2 CRUDE OIL: WEEKLY SPOT PRICE AND VOLATILITY.25 $/BARREL SPOT PRICE VOLATILITY (percen/week)

21 20 Figure 3 NATURAL GAS AND CRUDE OIL PRICE VOLATILITY NATURAL GAS VOLATILITY PEAK N.G. VOLATILITY: 9/26/ CRUDE OIL VOLATILITY : : : :07.5 Figure 4 NATURAL GAS PRICE VOLATILITY.4 (percen/week) Sample Sandard Dev. GARCH Esimaes