ONLINE APPENDIX The Effect of Uncertainty on Investment: Evidence from Texas Oil Drilling by Ryan Kellogg

Transcription

1 ONLINE APPENDIX The Effec of Uncerainy on Invesmen: Evidence from Texas Oil Drilling by Ryan Kellogg Appendix 1: Consrucion of he ime series of implied fuures price volaily This appendix describes how I consruc a ime series of he implied volaily of 18- monh NYMEX oil fuures conracs. As discussed in he main ex, I canno simply use he Black 1976) formula direcly because assumes ha he erm srucure of volaily he funcion by which he volaily of he fuure price of oil varies as ime o maury increases) is consan. My sraegy for addressing his issue proceeds in hree seps. Firs, I use he realized volaily of fuures prices o esimae he average erm srucure of volaily. Second, I use liquidly raded shor-erm fuures opions o generae a ime series of he implied volaily of one-monh fuures opion conracs. Because a one monh ime horizon is shor, his ime series is equivalen o he ime series of he implied volaily of one-monh fuures price conracs. Finally, I combine he one-monh fuures price volailies wh he esimaed erm srucure o generae he desired ime series of he implied volaily of 18-monh fuures price conracs. 1 The remainder of his appendix discusses hese hree seps in urn. Le F,τ denoe he price of a NYMEX fuures conrac raded a dae wh ime o maury τ measured in monhs. For each and τ, I calculae he realized volaily a of he τ- monh fuures conrac as he sandard deviaion of lnf s,τ / F s-1,τ ) for all daes s whin he 6 monhs prior and subsequen o. 3 Le his volaily be denoed by σ,τ. I hen esimae he erm srucure of fuures price volaily by regressing he log of σ,τ on fixed effecs for each τ and : 4 ln σ,τ = η τ + δ + ε τ, A1.1) 1 An alernaive procedure o ha used here would use he erm srucure of he implied volaily of fuures opions direcly o derive he implied volaily of 18-monh fuures prices. This approach would use he fac ha he volaily of a τ-monh fuures price is equal o he volaily of a τ-monh fuures opion plus τ imes he derivaive of he fuures opion erm srucure wh respec o τ) a τ. The use of he derivaive implies ha his approach requires a very precise esimae of he erm srucure of fuures opions implied volaily. Thin markes for fuures opions beyond 6 monhs render his procedure impracical. For example, 18-monh fuures opions are raded, on average, only 18 days each year from Time o maury in monhs is equal o he ime o maury in days divided by 365.5, muliplied by 1, and rounded o he neares whole number. 3 Observaions F s,τ for which dae s - 1 is missing for example, if s - 1 is a Sunday) are excluded. 4 I use he log of σ,τ as he dependen variable raher han he level because he levels regression does no yield an esimaed erm srucure ha is sable over ime. In levels, he erm srucure is has a seeper slope during han in he earlier par of he daa. A1

2 The fixed effecs η τ represen he esimaed erm srucure while he δ conrol for he level of volaily on each dae. Given esimaes of hese fixed effecs, he prediced volaily of a τ-monh fuures price on dae is given by A expη τ ), where A = expδ + v / ) and v is he variance of he esimaed residuals. Thus, for a fixed rade dae, varying τ will race ou he erm srucure of volaily. Figure A verifies ha he erm srucure of volaily is sable over he sample by ploing wo esimaes of he erm srucure: one using daa from and anoher using daa prior o The consan erm A for each ploed esimae is se so ha he one-monh fuure price volaily is 31%, approximaely equal o he average one-monh volaily over The plos overlay each oher closely, indicaing ha he erm srucure of volaily is sable over he sample despe he subsanial increase in he overall level of volaily afer Given he esimaed erm srucure he η τ ), all ha is needed o compue expeced 18- monh fuures price volailies is a ime-series of shor-run one monh) expeced fuures price volailies. I derive his ime series from he implied volaily of shor-erm fuures opions wh a ime o maury beween 60 and 180 days. The implied volaily of opions wh a shorer ime o maury are noisy, poenially reflecing low opion values and ineger problems opions prices mus be in whole cens), while opions wh a longer ime o maury are hinly raded. For each rade dae and ime o maury whin he 60 o 180 day window, I use he Black 1976) model o find he implied volailies of he call and pu opions ha are neares o a-hemoney. 5 I hen esimae he implied volaily erm srucure by regressing he log of each opion s implied volaily on s ime o maury τ in days), a call/pu dummy, and rade dae fixed effecs δ. 6 I hen use his esimaed erm srucure he esimaed coefficien on τ) o exrapolae implied volaily back o a 30 day maury. As a validaion check on he his procedure, I compare he average, over , of he esimaed implied volailies of 30-day fuures opions o he average realized volaily of one-monh fuures prices over he same imeframe. These wo averages should be approximaely equal given he shor one monh ime o maury. The former series has an average volaily of 5 The Black 1976) model assumes ha he opions are European raher han American and ha volaily is no sochasic. Neher of hese assumpions holds here; however, heir effecs are likely o be minor and hey save considerable compuaional complexy. Hilliard and Reis 1998) demonsrae ha he American premium is no more han % of he European opion price for volailies similar o hose considered here. Sochasic volaily acs in he oppose direcion, causing he Black 1976) model o slighly over-price a-he-money opions his effec is paricularly small for he relaively shor mauries considered here); see Hull and Whe 1987), Wiggins 1987), and Poon and Granger 003). The argumen ha hese assumpions are of minor effec is suppored by he close agreemen beween he average realized and average implied volaily over he sample. 6 Inspecion of he residuals indicaes ha a linear erm srucure specificaion is appropriae. Moreover, when a squared ime o maury erm is added, is no saisically significan p-value = 0.114). A

3 30.83% while he average of he laer is 31.07%. The closeness of hese wo numbers derived from wo compleely differen daa ses) suppors he argumen ha implied volailies from onemonh fuures opions can be used as implied volailies of one-monh fuures prices. Finally, I conver he ime series of implied volailies of one-monh fuures prices o implied volailies of 18-monh fuures prices using he esimaed erm srucure of fuures price volaily he η τ ). This conversion amouns o muliplying he one-monh volaily a each rade dae by expη 18 - η 1 ). References Hull, John, and Alan Whe The Pricing of Opions on Asses wh Sochasic Volailies. Journal of Finance 4): Wiggins, James B Opion Values Under Sochasic Volaily. Journal of Financial Economics 19): A3

4 Appendix : Numerical soluion and esimaion mehods A.1 Value funcion eraion I solve he value funcion 1) on a grid of poins in P,D,σ,x) space in logs) using sandard value funcion eraion. An imporan facor in defining he grid is ha, while he price, dayrae, and volaily saes ha are realized in he daa are bounded, he sochasic processes for hese variables equaions 4, 5 and 10) imply ha agens place nonzero probabilies on realizaions ouside of hese bounds. Thus, he value funcion mus be solved for saes exending beyond he boundaries of he daa. The sae space I use exends from one-fifh of he lowes realized price and dayrae o five imes he highes price and dayrae, and from one-half he lowes realized volaily o wice he highes volaily. Wh his sae space, marginal reducions or exensions in size do no subsanially affec he esimaed parameers or he value funcion whin he range of realized observaions. I found ha a relaively dense grid was required o accuraely capure he effecs of sochasic volaily. The grid I use has 1,875,000 poins: 50 price saes by 50 dayrae saes by 15 volaily saes by 50 producivy saes. Saring from his densy, he esimaed resuls are insensive o increases or decreases in he number of grid poins. In he full esimaion rouine, he inial value funcion used for each guess of parameers is he value funcion from he previous guess. For he firs parameer guess, he inial value funcion is zero in all saes. The convergence crerion is a olerance of 10-6 on he sup norm of he value funcion he value funcion used in he compuaions is in uns of $386,501, he average drilling cos a he average dayrae). Increasing he olerance o 10-7 has essenially no affec on he parameer esimaes or value funcion. Wh he value funcion solved, I can hen find, for any given P, D, and σ, he crical producivy x such ha drilling is opimal iff xi x. Because he P, D, and σ realizaions do no coincide wh he grid saes used in he model, I use linear inerpolaion o find x. A each x i grid poin, I calculae he value funcion a he realized P, D, and σ by linearly inerpolaing he value funcion beween he saes immediaely above and below he P, D, and σ. I hen find he smalles x i grid poin such ha he value of waing exceeds he realized profs from drilling immediaely and he larges x i such ha is opimal o drill immediaely hese wo values of x i will be adjacen grid poins). Inerpolaion gives x as he producivy level for which he firm is indifferen: he value of waing equals he value of drilling immediaely. As described in he ex, he realized ime series of P, D, and σ can hen be combined wh a parameerized disribuion on he x o yield he probabily ha a given prospec will be drilled each period. A4

5 In mos of he esimaed models, here is no inial condions problem because he producivy shocks x are modeled as iid. An inial condions problem is presen, however, in he specificaion allowing for ime-invarian prospec heerogeney hough he specificaion ulimaely finds no evidence of such heerogeney). I address his issue by exending he simulaion back o January 199, so ha by 1993, when drilling likelihoods sar o be aken, an equilibrium is approximaely reached. This exension requires he inerpolaion of missing rig dayrae daa for he fourh quarer of 199. A. Esimaion I search for he parameers β, μ, and log ζ ha maximize he log-likelihood funcion 13) via a gradien-based search ha uses he BFGS mehod for compuing he Hessian a each sep I ake he logarhm of ζ o allow for negaive values in he parameer search). I accelerae he search by conducing in wo sages. Firs, holding β fixed, I search for he μ and log ζ ha maximize he likelihood. This sage is fas because changing μ and ζ does no require re-solving he model. The ouer-mos loop hen searches for β. The sopping crerion is a olerance on he likelihood funcion scaled down by a facor of 10,000) of for he μ and ζ loop and 10-8 for he β loop. To compue he sandard errors of he parameer esimaes, I obain he likelihood score of each observaion drilling prospec - monh) numerically. Wh respec o each parameer θ k, I lj θk εk ) lj θk εk ) calculae he derivaive of he log likelihood for observaion j as. For he ε parameers β and μ, I use a value for ε k of 0.001, and for log ζ I use a value of because he likelihood funcion is paricularly concave in his parameer. The sandard errors are robus o values of ε k ha are an order of magnude larger or smaller. I adjus he sandard errors o accoun for he fac ha he parameers of he expeced price drif funcion 11) are esimaed in a firs sage. 7 Denoing he firs-sage parameers κ p0, κ p1, and κ p ) and log-likelihood funcion by θ 1 and L 1, and denoing he second-sage parameers β, μ, and ζ) and log-likelihood funcion by θ and L, I apply he procedure of Murphy and Topel 1985) using equaion A.1), k 7 The volaily of volaily γ), he raio of dayrae volaily o oil price volaily α), and he correlaion beween dayrae and price shocks ρ) are also esimaed in a firs sage. However, I found ha hese parameers conribued only negligibly o he sandard errors of he main parameer esimaes in he reference case model. To reduce compuaional burden, he resuls presened in he paper herefore ignore hese parameers when compuing Murphy and Topel wo-sep sandard errors. In he mean-revering volaily beliefs specificaions, I also accoun for sampling error in he esimaion of he parameers governing he volaily mean reversion funcion. A5

6 R R R R R R A.1) where Σ denoes he correced variance-covariance marix for θ, and L 1 L 1 L R1 ) E E L L L ) R E E 1 1 L L L R3 ) E E A.) R 1 is simply he inverse of he variance-covariance marix from he leas-squares esimae of he price drif funcion 11), which I compue using sandard errors clusered on monh-ofsample. 8 R is he inverse of he unadjused and non-clusered) second-sage variancecovariance marix. Calculaion of R 3 requires numerical derivaives of he second-sage likelihood funcion wh respec o he firs-sage parameers. I calculae hese derivaives in he same way ha I calculae hose wh respec o he second sage parameers, as discussed above. The perurbaions I use for κ p0, κ p1, and κ p are 10-5, 10-6, and 10-3, respecively. For he specificaions ha yield esimaes of β near one, he above procedure roughly increases he esimaed sandard errors by a facor of 3, a magnude similar o ha found in several examples in Murphy and Topel 1985). The adjusmen is no subsanial for oher specificaions, however, as heir unadjused sandard errors are already large. References Murphy, Kevin M., and Rober H. Topel Esimaion and Inference in Two-Sep Economeric Models. Journal of Business & Economic Saisics 34): Clusering on year raher han monh-of-sample does no subsanially affec he esimaed sandard errors. A6

7 Appendix 3: Esimaion including producivy realizaions This appendix provides deails of he process by which I use producion daa from he subse of wells for which producion is observable o esimae an expanded version of he srucural model. I firs discuss how I ransform he raw producion daa ino esimaes of each well s oal discouned lifeime producivy. I hen discuss he consrucion of an augmened likelihood funcion ha incorporaes hese producivy daa. A3.1 Calculaing discouned lifeime producivy For 160 of he 1,150 wells in he sample, I observe he well s monhly producion for he firs hree years of he well s life. The dynamic model presened in he main ex, however, is based on he producivy of each well, defined as s discouned oal lifeime producion divided by s drilling cos a he average rig dayrae. To ransform he hree years of producion daa for each well ino an esimae of discouned oal lifeime producion, I employ a decline curve analysis. The simples possible approach would be o f a hyperbolic curve o he average producion decline daa shown in figure in he paper and hen use his curve o exrapolae producion for fuure years of each well s life. However, one srong feaure of he producion daa is ha decline raes are less seep for wells ha are relaively producive. Therefore, I allow he parameers governing he hyperbolic decline o vary wh he observed hree-year producion volumes. Specifically, denoing he producion from well i in monh as q, and denoing he log of well is oal producion in s firs hree years as Q i3, I esimae he hyperbolic decline equaion A3.1) on he pooled monhly daa from all 160 wells: qi 0 1Qi 3 ) 0 1Qi 3) 1 ) A3.1) i3 Q The parameers α 1 and γ 1 allow he esimaed decline curve o seepen or flaen for more producive wells. 9 I esimae ha α 0 = 0.337, α 1 = , β = is measured in monhs since drilling), γ 0 = 3.97, and γ 1 = The negaive esimaes for α 1 and γ 1 are consisen wh a shallower decline rae for more producive wells While he β erm could in principle also be ineraced wh oal hree-year producion, becomes very difficul for he esimaor o converge when his ineracion is included. Inuively, allowing for his addional flexibily is unnecessary, as providing flexibily in he decline curve inercep hrough α 1 ) and slope hrough γ 1 ) is sufficien. 10 As an alernaive approach, I have also aemped o esimae decline curves well-by-well. However, he esimaes are generally oo noisy o be useful, especially as some wells are acually esimaed o have increasing producion over heir firs hree years, which makes impossible o projec an evenual decline. A7

8 For each of he 160 wells, I use he esimae of equaion A3.1) o exrapolae fuure producion, and I hen apply he discoun facor used in he model see secion IV.A of he paper) o obain he well s discouned oal lifeime producion. Finally, I divide his number by he well s esimaed drilling cos a he average dayrae his cos depends on he number of days needed o drill he well, as described secion I.E in he paper) o obain s realized producivy in barrels per $ of drilling cos). These realized producivy daa are ploed in figure 10 in he main ex. A3. Augmening he likelihood funcion Wh he inclusion of he realized producivy daa, he likelihood funcion mus now incorporae he probabily of each producivy realizaion which will depend on x each period and on he variance of producivy realizaions abou heir expecaion) and he probabily ha producion is observable for each drilled well which will depend on he well s producivy). The resuling likelihood funcion involves hree pieces, which I now describe in urn. The firs piece of he likelihood for each drilled well is ha given by equaion 13) in he ex: he probabily ha drilling would occur in he monh he well was acually drilled or no occur a all during he sample in he case of an undrilled prospec). This par of he likelihood does no change, and for undrilled prospecs his is he only par of he likelihood. The second piece of he likelihood applies only o wells for which producivy is observed 160 of he 1,150 wells). Some noaion is required. Le Y denoe he realized producivy of well i drilled in monh, and le y denoe s log. Le Z denoe he expeced producivy of well i drilled in monh, and le z denoe s log. This expecaion is he firm s expecaion of he well s producivy afer has made he decision o drill bu before drilling is compleed. Thus, z has a normal disribuion, wh mean μ and sandard deviaion ζ he wo parameers ha govern he disribuion of x as discussed in secion IV.B in he ex), ha is lefruncaed a he log of he producivy rigger a ime. In a sligh abuse of noaion, le x now denoe he logged producivy rigger. Finally, le a prob funcion of x. Specifically, τ 1 are parameers o be esimaed. P ) dry P x denoe he probabily of a dry hole as ) dry P x is given by equaion A3.) below, in which τ 0 and dry x ) 1 x 0 1 Given an expeced producivy z, y will be - wh probabily oherwise be normally disribued abou z wh variance, following A3.3): p ) dry A3.) P x and will A8

9 y z ~ z p N log Pdry ) wh prob. P, p wh prob. 1 dry P dry A3.3) I is he variance erm ha allows for noise in he producion realizaions so ha hey can be p raionalized by he model. Also noe ha he disribuion of y is designed so ha E[Y ] = Z. Le fy z ) denoe he disribuion of y condional on z and on he well no being dry i.e., fy z ) is he second par of A3.3)). Le gz x ) denoe he runcaed normal disribuion of z, condional on x. The conribuion of producion realizaion y o he likelihood is hen: P dry if y 0 dry x 1 P ) f y z ) g z x ) dz if y 0 The hird and final piece of he likelihood conribuion from each drilled well is he probabily ha producion from he well is observable in he daa. There are wo componens of his probabily, which I denoe by P obs. The firs is he probabily ha he well can be mached o a lease name in he producion daabase. I ake his probabily, which I denoe by P mach, o be exogenous o he model and fix o equal he observed mach rae in he daa, 57/1150. The second componen addresses selecion. The probabily of observing a drilled well s producion should increase if s realized producivy is low relaive o expecaions, and should also increase if x is high, since fewer wells are drilled when he rigger producivy is high and rigger producivies are serially correlaed). I herefore specify P obs per equaion A3.4) below, in which λ 0, λ 1, and λ are parameers o be esimaed: y x ) x 0 obs, ) mach, ) mach 1 1 P y x P s y x P A3.4) Noe ha equaion A3.4) implies ha he probabily of observing a dry hole, for which y = -, is equal o P mach. Thus, for all wells for which producion is observed, he final componen of heir likelihood is given by P y x. obs, ) For he drilled wells for which I do no observe producion, I mus compue he probabily ha producion is unobserved, condional on he rigger producivy x a he ime A9

10 of drilling. This compuaion requires a double inegral over realized producivy condional on expeced producivy and over expeced producivy condional on x. The probabily ha producion is unobserved is herefore given by equaion A3.5): P x P P x P x s y x f y z dy g z x dz unobs ) 1 mach dry ) 1 dry )), ) ) ) x is given by A3.5) For all wells for which producion is unobserved, he final componen of heir likelihood P x. This complees he likelihood. unobs ) Esimaion involves six parameers no presen in he reference case model: he parameers τ 0 and τ 1 ha dicae how he dry hole probabily varies wh x, he parameer σ p ha dicaes he variance of he realized producion daa, and he parameers λ 0, λ 1, and λ ha dicae which producivy observaions are likely o be observable. The esimaes of hese parameers, which correspond o he esimaes presened in column V of able 3 in he ex, are: τ 0 : ) τ 1 : ) σ p : ) λ 0 : ) λ 1 : ) λ : ) The esimaes of τ 0 and τ 1 are imprecise bu consisen wh a modes decrease in he probabily of a dry hole as x increases. A he sample average x of -.69, he esimaes imply ha he probabily of a dry hole is 1.4%. For comparison, I observe 7 dry holes ou of he 160 observed wells and 1,150 oal wells. The σ p parameer is large and precisely esimaed, consisen wh he noise in realized producivy ploed in figure 10 in he paper. The λ 0, λ 1, and λ esimaes are consisen wh he probabily of observing producion varying negaively wh realized producivy and posively wh x. The parameers imply ha a one sandard deviaion increase in realized producivy from he sample mean x of -.69 o σ p reduces he probabily of observing he well s producion from 1.1% o 8.0%. For reference, I observe producion for 13.9% =160/1150) of he drilled wells. Thus, given he parameer esimaes, observed producivy realizaions mus on average be lower han he sample mean x in order o aain he 13.9% observaion rae. A10

11 Volaily %, annual) Wells per monh, unized fields Wells per monh, all fields Figure A1: Infill drilling in sole operaed vs. all fields All infill wells Infill wells in sole-operaed fields Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan Figure A: Esimaed erm srucures of fuures price volaily pre Time o maury in monhs Noes: The figure displays wo erm srucures, one esimaed using daa from before 1999, he oher using daa from Volaily of a one-monh fuure is se o 31.0% for boh erm srucures. A11