Pricing of Risk in Natural Gas Futures Contracts: A New Approach to Affine Term Structure Models Subject to Changes in Regime

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1 Pricing of Risk in Naural Gas Fuures Conracs: A New Approach o Affine Term Srucure Models Subjec o Changes in Regime Irina Zhecheva Deparmen of Economics Universiy of California, San Diego Sepember 7, 016 Absrac This paper proposes a new approach o he esimaion of Markov regime-swiching affine erm srucure models. I show how o price he ime series and cross-secion of he erm srucure of commodiies in he case of regime swiching. The maximum likelihood based mehods common in he lieraure pose significan numerical challenges, which become especially severe when here is he possibiliy of changes in regime. My approach avoids hese numerical difficulies and allows for compuaionally fas esimaion. I apply my approach o a model of naural gas fuures prices from 1994 o 013, and produce novel empirical esimaes o characerize risk premia and he erm srucure of naural gas fuures conracs. I find very srong evidence ha here are regimes in he daa. In my model, one regime corresponds o higher difference beween he longer erm and he shorer erm conracs (higher spread while he oher regime corresponds o a lower spread. I find ha he marke acs as if regimes are more persisen han hey really are. This could be a resul of hedging pressure or mispricing. I find evidence ha in he high spread regime, commercial producers are rying o hedge heir shor posiions in naural gas by selling 3-5 monh fuures conracs, while in he low spread regime, commercial producers are rying o hedge heir shor posiions by selling 6-9 monh conracs. My model implies ha i is more profiable o be long in he 3-5 monh conracs in he high spread regime, whereas i is beer o be long in he 6-9 monh conracs in he low spread regime. I also find ha a posiive one sandard deviaion shock o he facor, which represens a change in prices ha is basically consan across mauriies, decreases he expeced monhly reurn on he 3-9 monh conracs by abou 0.88%. Deparmen of Economics, UCSD, 9500 Gilman Drive, La Jolla CA The auhor can be reached a izhechev@ucsd.edu. I hank my advisors James Hamilon, Rossen Valkanov, Allan Timmermann, Ivana Komunjer, and Ruh Williams for heir invaluable guidance and suppor. This paper also benefied from helpful suggesions from Julie Cullen, Mark Machina, Asad Dossani, and seminar paricipans a UCSD. 1

2 1 Inroducion Affine erm srucure models are a popular ool for he analysis of he pricing of fixed income securiies such as bonds and fuures. Hamilon and Wu (011 show ha an affine facor srucure of commodiy fuures prices can resul from he ineracion beween arbirageurs and commercial producers seeking hedges or financial invesors seeking diversificaion. The class of Gaussian affine erm srucure models was originally developed by Vasicek (1977, Duffie and Kan (1996, Dai and Singleon (000, Duffee (00, and Piazzesi (010 o characerize he relaion beween yields on bonds of differen mauriies. Hamilon and Wu (011 adap his class of models o commodiies. Schwarz (1997, Schwarz and Smih (000, and Casassus and Collin-Dufresne (006, among ohers, also develop relaed models o describe commodiy fuures prices. The affine erm srucure framework is based on he assumpions ha he pricing kernel is exponenially affine, prices of risk are affine in he sae variables, and innovaions o he sae variables are condiionally Gaussian. Under hese assumpions, he price process is affine in he sae variables, and no-arbirage resricions consrain he coefficiens on he sae variables. Mos mehods for esimaing Gaussian affine erm srucure models in he lieraure are based on maximum likelihood esimaion and exploi boh he disribuional assumpions and he no-arbirage resricions. These mehods esimae laen pricing facors joinly wih he model parameers by explicily imposing he cross-equaion no-arbirage resricions. As a resul, hese mehods rely on numerical opimizaion, are compuaionally inensive, and pose significan numerical challenges, which become especially severe when here is he possibiliy of changes in regime. Esimaion difficulies commonly arise due o highly non-linear and badly behaved likelihood surfaces, which are fla along many direcions of he parameer space, and i is hard o achieve convergence. These problems can make esimaion by MLE very difficul or infeasible. These difficulies have been documened by muliple researchers such as Kim and Orphanides (005, Duffee (00, Ang and Piazzesi (003, Kim (008, Duffee and Sanon (008, Duffee (009, and Ang and Bekaer (00. To faciliae esimaion, I propose a new approach o he esimaion of Markov regimeswiching affine erm srucure models. I use a regression based mehod o esimae he reduced-form parameers in he firs sage, and hen esimae he prices of risk via minimumchi-square esimaion in he second sage. In his way, I bypass he numerical difficulies encounered by oher mehods and have no problem achieving convergence. I show how o price he ime series and cross-secion of he erm srucure of commodiies in he case of regime swiching. I apply my mehod o a model of naural gas fuures conracs. Anoher conribuion of his paper is providing novel empirical esimaes o characerize risk premia

3 and he erm srucure of naural gas fuures. Naural gas is one of he mos heavily raded commodiy fuures conracs in he Unied Saes. The naural gas fuures marke is highly liquid wih daily open ineres of up o abou 300,000 conracs for he one monh conrac and oal open ineres of up o 1,400,000 conracs. Naural gas accouns for 30% of U.S. elecriciy generaion and is prediced o accoun for an even larger proporion in he fuure. This commodiy has gained a lo of aenion in recen years wih he discovery of shale gas and he advancemen of drilling echnology. Naural gas spo and fuures prices end o exhibi seasonal variaions due o seasonal changes in supply and demand condiions. Naural gas producion is relaively consan hroughou he year, bu consumpion ends o peak during he winer heaing season (November hrough March as home heaing use rises and ends o be moderae in oher seasons. However, in recen years summer consumpion has increased as uiliy companies have increased heir demand for naural gas for air condiioning. Thus, naural gas consumpion follows a seasonal paern. Oher facors affecing supply and demand also affec prices. On he supply side, amoun of gas in sorage, pipeline capaciy, and imperfec informaion abou sorage affec prices. Naural gas supplies ha were pu in sorage during periods of lower demand may be used o cushion he impac on price during periods of high demand. Facors affecing demand include weaher condiions and aggregae economic condiions. In general, in he winer demand for naural gas is a is peak and shocks can cause large price flucuaions, since winer demand is inelasic and since supply is ofen unable o reac quickly o shor-erm increases in demand. Thus, conracs for delivery in he winer may exhibi high volailiy. A he same ime, he invenory sored for he winer can parially cushion he impac of shocks and hus reduce he volailiy of winer conracs. Prices of conracs for delivery in he winer can sar exhibiing high volailiy as soon as he early fall when informaion on he fuure availabiliy of naural gas is released and expecaions are formed on wheher sorage would be able o mee fuure demand (Suenaga, Smih, and Williams 008. In conras, prices of conracs for delivery in he spring end no o exhibi high volailiy, since demand is low in he spring. Mos of U.S. naural gas consumpion is from domesic producion. U.S. dry producion has been seadily increasing since 006. I reached is highes recorded annual oal in 015 and is sill rising during 016. The increases in producion were due o more efficien, coseffecive drilling echniques which have allowed for horizonal drilling in shale formaions. This has led o an unprecedened surge in supply, hereby puing downward pressure on prices and decreasing he volailiy in he spo and fuures markes. Alhough hese flucuaions have a seasonal componen, hey are far from deerminis- 3

4 ic. In some years here is virually no winer spike in gas prices, and in ohers he high and volaile prices exend well beyond he winer monhs. This uncerainy inroduces he possibiliy ha producers or commercial users may a imes make significan use of naural gas fuures conracs for purposes of hedging. Hamilon and Wu (014 show how variaion in hedging pressure could influence he erm srucure of commodiy fuures prices. In his paper I model he level and volailiy of naural gas prices using a Markov regime-swiching model and sudy he consequences of hese changes in regime on he risk premium by generalizing he fuures-pricing model in Hamilon and Wu (014 o allow for changes in regime. Regime-swiching models for he erm srucure of ineres raes have been proposed and esimaed by Dai, Singleon, and Yang (007, Bansal and Zhou (00, and Ang and Bekaer (00 among ohers. Dai, Singleon, and Yang (007 and Ang and Bekaer (00 use maximum-likelihood based mehods based on an ieraive procedure developed by Hamilon (1989. Bansal and Zhou (00 use a wo-sep efficien mehod of momens esimaor. These mehods are subjec o he numerical issues menioned earlier, which his paper resolves. Adrian, Crump and Moench (013 and Diez de los Rios (forhcoming have recenly proposed regression based algorihms for esimaion of single-regime affine erm srucure models ha avoid he numerical difficulies associaed wih maximum likelihood esimaion. In his paper, I show how such an approach can be generalized o allow for regime swiching, and produce empirical esimaes o characerize risk premia and he erm srucure of naural gas fuures conracs. The res of he paper is organized as follows. Secions and 3 presen he model framework, Secion 4 describes he esimaion approach, Secion 5 gives empirical resuls from applying my mehod o a one facor model of naural gas fuures prices from 1994 o 013, and Secion 6 concludes. Model Le F (n be he price of a fuures conrac wih mauriy n a ime. I assume ha he log of he price is a funcion of a facor X ha follows a Gaussian auoregression: X +1 µ s + ΦX + v +1, v +1 s N(0, σ s (1 I find ha a one facor model using he firs principal componen of he log of he fuures prices wih mauriies 3 monhs o 9 monhs as a facor describes he daa well. 1 Thus, in my empirical applicaion X will be a scalar. I find ha shorer erm conracs have a 1 In secion 5. I provide some evidence on why a one facor model is appropriae. 4

5 sysemaically differen behavior and do no fi well in my framework ogeher wih he long erm conracs. The inercep parameer is regime swiching, and s denoes he regime a ime, s {1, }. I assume ha he slope parameer Φ is regime independen. The no-arbirage assumpion implies he exisence of a pricing kernel M such ha F (n E M +1 F (n 1 +1 s j ( if he regime a ime is j. Following Dai, Singleon, and Yang (007, I assume ha he pricing kernel is exponenially affine and akes he following form: M +1 exp 1 λ,s λ,s σs 1 v +1 (3 I also assume ha he marke price of risk λ is an affine funcion of he sae variable: λ,s σ 1 s (λ 0,s + λ 1 X (4 Here he price of risk λ,s is ime-varying and regime-dependen. I assume ha λ 1 does no depend on he regime. 3 No arbirage implies he exisence of an equivalen maringale measure - he risk neural measure Q. The hisoric measure P and he risk-neural measure Q are relaed hrough he pricing kernel M +1. I can compue he price P (X of an asse wih payoff g(x +1 in regime j as P (X E M +1 g(x +1 s j E Q g(x +1 s j (5 Under he Q-measure, he facor X follows a Gaussian auoregression: X +1 µ Q s + Φ Q X + v Q +1 (6 where µ Q j µ j λ 0,j (7 I also esimaed a version of he model wih Φ allowed o vary wih regime, bu found ha his led o only a rivial increase in he log likelihood for he sysem in equaions (14-(17 ha I esimae. Since he equaions are simpler and more inuiive wih Φ consan, I only discuss he simpler case in his paper. 3 When I esimaed a version of he model in which Φ varies wih he regime, I also allowed λ 1 o vary wih he regime, denoing i λ 1,s. I failed o rejec he hypohesis ha λ 1,1 λ 1, 0. Hence, I consider he simpler model in which λ 1 is regime independen. 5

6 and Φ Q Φ λ 1 (8 and v+1 s Q j Q N(0, σj. The above relaions are derived in Appendix A. Le f (n ln F (n. Equaions (1, (, and (3 ogeher imply ha he log of he fuures price is affine in he sae variable: f (n A (n s + B (n X + u (n (9 From equaion (8 and equaion (A-8, i follows ha he facor loadings B (n are regime independen. This ensures exac closed-form soluions for he fuures prices, and is consisen wih Dai, Singleon, and Yang (007. The inercep erm is allowed o change wih he regime. Le rx (n 1 +1 denoe he one-monh log holding reurn of a fuures conrac mauring in n periods: rx (n 1 +1 f (n 1 +1 f (n (10 The holding reurn is he reurn on buying a fuures conrac wih mauriy n monhs in monh and hen selling i as an (n 1 period fuures conrac in monh + 1. In my applicaion, I assume ha here are regimes ha govern he dynamic properies of he facor X. The unobserved regime variable s is presumed o follow a -sae Markov chain, wih he risk-neural probabiliy of swiching from regime s j o regime s +1 k given by π Q jk, 1 j, k, wih πq jk 1, for j 1,. I assume ha he risk-neural ransiion probabiliies π Q jk and he real-world ransiion probabiliies π P jk are regime independen. I allow π P jk π Q jk. Agens are presumed o know he hisory of he facor X and of he regime. The Markov process governing regime changes is assumed o be condiionally independen of he X process for racabiliy. 3 No arbirage condiions for fuures conrac prices Under my assumpions, he log of he fuures price is affine in he facor X : f (n A (n s + B (n X + u (n (11 6

7 My model implies he following cross-equaion non-arbirage resricions on he parameers A (n j, j 1, and B (n characerizing he fuures conrac price: A (n j log ( π Q jk e A(n 1 k + B (n 1 (µ j λ 0,j + 1 B(n 1 σ j (1 for j 1, and B (n B (n 1 (Φ λ 1 (13 Equaions (1 and (13 are derived in Appendix B in equaions (A-7 and (A-9. They are very similar o he sandard recursions for affine erm srucure models in he bond pricing lieraure (see for example Ang and Piazzesi (003. In bond pricing, he recursion for he inercep adds a erm δ 0 corresponding o he ineres earned each period. For commodiies, such a erm does no appear since here is no iniial capial invesmen. The recursions above represen non-linear cross-equaion no arbirage resricions. These resricions are no used or imposed in he iniial reduced-form esimaion, bu are exploied in he second sage of inference described below. 4 Esimaion procedure I assume ha he facor X is observed, and i is he firs principal componen of he logs of he fuures conracs wih mauriies from 3 monhs o 9 monhs. Based on equaions (9 and (1, I propose he following wo-sep mehod for esimaing he parameers of he model. 4.1 Esimaion of reduced-form parameers via regime-swiching VAR s Firs, I esimae he following regime-swiching regressions: f (n A (n s + B (n X + u (n, n 3,..., 9 (14 7

8 where and u (3 u (4 u (5 u (6 u (7 u (8 u (9 s N(0, Ω (15 Ω ( Ω ( Ω ( Ω Ω ( Ω ( Ω (8 0 ( Ω (9 joinly wih he regime-swiching regression for X : X +1 µ s + ΦX + v +1, v +1 s N(0, σ s (17 as a vecor sysem of regime-swiching equaions. The ime-series regressions in equaion (14 esimae exposures of he fuures prices wih respec o he conemporaneous pricing facor. The regime-swiching regression in equaion (17 serves o decompose he pricing facor ino a predicable componen and a facor innovaion by regressing he facor on is lagged level. The esimaion is done via he EM algorihm and is explained in deail in Appendix C. The general vecor version of he EM algorihm is found in Hamilon (forhcoming. 4. Minimum-chi-square esimaion of srucural parameers I use a minimum-chi-square approach o esimae he price of risk parameers λ 0,1, λ 0,, and λ 1. I choose he values of λ 0,1, λ 0,, and λ 1 ha mos closely fi he no arbirage resricions in equaions (1 and (13. Minimum-chi-square esimaion is described in Hamilon and Wu (01. Le π denoe he vecor of reduced-form parameers (VAR coefficiens, variance of he facor, measuremen error variances, and P-measure regime-swiching probabiliies. Le L(π; Y denoe he log likelihood for he enire sample, and le ˆπ arg max L(π; Y de- 8

9 noe he full-informaion maximum likelihood esimae. If ˆR is a consisen esimae of he informaion marix, R T 1 L(π; Y E (18 π π hen θ can be esimaed by minimizing he chi-square saisic T ˆπ g(θ ˆR ˆπ g(θ (19 As noed by Hamilon and Wu (01,he variance of ˆθ can be approximaed wih T 1 (ˆΓ ˆRˆΓ 1 for ˆΓ g(θ θ θˆθ. In my case, I wan o minimize he disance beween he unresriced maximum likelihood esimaes of he coefficiens A (n j and B (n (from he regime-swiching regressions and he values of A (n j and B (n implied by he no arbirage resricions. According o equaions (1 and (13, hese are prediced o be funcions of θ, a vecor of srucural parameers summarized in equaion ( below. likelihood esimaes from he regime-swiching VAR: ˆπ Le ˆπ be he vecor of he unresriced maximum ( µ 1, µ, Φ, vec(ã, vec( B, σ 1, σ, Ω (3, Ω (4, Ω (5, Ω (6, Ω (7, Ω (8, Ω (9, π P 11, π P (0 where µ 1, µ, Φ, σ 1, and σ are he unresriced maximum likelihood esimaes of he parameers µ 1, µ, Φ, σ 1, and σ from he regime-swiching auoregression for he facor in equaion (17, and B Ã Ã (3 1 Ã (4 1 Ã (5 1 Ã (6 1 Ã (7 1 Ã (8 1 Ã (9 1 Ã (3 Ã (4 Ã (5 Ã (6 Ã (7 Ã (8 Ã (9 (1 ( B(3, B (4, B (5, B (6, B (7, B (8, B (9 are he unresriced maximum likelihood esimaes of he coefficiens of he regime-swiching regressions for he fuures prices in equaion (14, Ω (n, n are he unresriced maximum likelihood esimaes of he measuremen error variances Ω (n, n in equaion (16, and π P 11 and π P are he unresriced maximum likelihood esimaes of he regime swiching probabiliies π P 11 π P from he regime-swiching VAR. and 9

10 Le θ ( µ 1, µ, Φ, A (3 1, A (3, B (3, σ 1, σ, Ω (3, Ω (4, Ω (5, Ω (6, Ω (7, Ω (8, Ω (9, π P 11, π P, λ 0,1, λ 0,, λ 1, π Q 11, π Q ( and g(θ ( µ 1, µ, Φ, vec(a (θ, vec(b(θ, σ 1, σ, Ω (3, Ω (4, Ω (5, Ω (6, Ω (7, Ω (8, Ω (9, π P 11, π P (3 Here A(θ A (3 1 A (3 A (4 1 A (4 A (5 1 A (5 A (6 1 A (6 A (7 1 A (7 A (8 1 A (8 A (9 1 A (9 (4 and B(θ ( B (3, B (4, B (5, B (6, B (7, B (8, B (9 (n. A j in vec (A (θ and B (n in vec (B(θ for n 4,..., 9 are defined by he no arbirage resricions from equaions (1 and (13: A (n j log ( π Q jk e A(n 1 k + B (n 1 (µ j λ 0,j + 1 B(n 1 σ j (5 for j 1, and B (n B (n 1 (Φ λ 1 (6 For n 3, g(a (3 1 A (3 1, g(a (3 A (3, and g(b (3 B (3. Then ˆθ is obained as ˆθ argmin θ {T ˆπ g(θ ˆR ˆπ g(θ} (7 In his way I obain esimaes of he prices of risk λ 0,1, λ 0,, and λ 1 and of he risk-neural ransiion probabiliies π Q 11 and π Q as par of he vecor ˆθ. I also obain second-sage esimaes of µ 1, µ, Φ, σ1, σ, π P 11, π P, A (3 1, A (3, B (3, and Ω (n, n Equaions (5 and (6 represen resricions implied by he model. By using he esimaors for λ 0,1, λ 0,, and λ 1 oulined above, I obain he price of risk parameers ha mos closely fi hese resricions. 10

11 5 Empirical resuls 5.1 Daa I esimae a one facor model using daa on prices of naural gas fuures conracs raded on NYMEX wih mauriies 3 monhs o 9 monhs for he period from January, 1994 o Augus, 013. The facor is consruced as he firs principal componen exraced from he demeaned log prices of hese conracs. The daa is obained from Daasream. Naural gas conracs expire hree business days prior o he firs calendar day of he delivery monh. Figure 1 shows he log of he observed 3 monh fuures price. I use a cross-secion of N 7 mauriies in my esimaion. I esimae he reduced-form parameers {A j, µ j, σj }, j 1,, B, Ω, and Φ, and he probabiliies π P 11, π P, and ρ 1 in he firs sep of he esimaion procedure, and hen esimae he prices of risk λ 0,j and λ 1 and he risk-neural probabiliies π Q 11 and π Q in he second sep. Here ρ 1 is he probabiliy ha he iniial sae is regime Esimaion resuls The firs principal componen used as facor in my model capures 98.73% of he variaion in fuures prices. I use he Eigenvalue Raio and Growh Raio ess proposed in Ahn and Horensein (013 in order o esimae he number of facors in my model. Le Y be he T N marix conaining he demeaned fuures price daa, wih T34 and N 7, and le ˆλ k denoe he k h larges eigenvalue of he covariance marix (Y Y /NT. The Eigenvalue Raio crierion funcion ER(k is he raio of wo adjacen eigenvalues of (Y Y /NT : ER(k ˆλ k ˆλ k+1, k 1,,..., k max (8 where k is he number of facors used, and k max is a specified maximum number of facors. The Growh Raio crierion funcion GR(k is given by GR(k log(1 + ˆλ k log(1 + ˆλ k+1 (9 where V (k m jk+1 ˆλ j and ˆλ k ˆλ k /V (k. The esimaors of he rue number of facors r are he maximizers of ER(k and GR(k: ˆr ER max 1 k kmax ER(k (30 ˆr GR max 1 k kmax GR(k (31 11

12 These esimaors are called he ER and GR esimaors, respecively. Boh of hese esimaors yield 1 as he number of facors ha need o be used. This jusifies my use of a one facor model. Table 1 shows he esimaes of he reduced-form parameers and he hisorical ransiion probabiliies from he firs sage of he esimaion. Table shows second sage esimaes, including he esimaes of he marke prices of risk and he risk-neural ransiion probabiliies. I find ha he facor is very persisen, wih ˆΦ The facor is a saionary sochasic process under he P-measure. The loadings of he fuures prices on he firs principal componen are basically consan across mauriies. Thus, he facor essenially represens a parallel change in prices. Because of is effec on price levels, I refer o his facor as he level facor. This is commonly done in he lieraure using principal componen facor models. Table 3 shows Wald -saisics for he hypohesis ess esing wheher here is regime swiching in he various parameers. I find very srong evidence ha here are regimes in he daa. The null hypoheses ha he level of he facor µ and he levels A (3, A (4, A (5, A (7, A (8, A (9 of he conracs are no regime swiching are srongly rejeced. The level µ of he facor is higher in regime 1 han in regime, and µ 1 µ is saisically significanly differen from 0. The levels of he conracs wih mauriy 3-5 monhs are saisically significanly lower in regime 1, while he levels of he conracs wih mauriy 7-9 monhs are saisically significanly higher in regime 1. The esimaed variances in he wo regimes are no saisically significanly differen. The coefficiens ˆB and he variance of he measuremen errors ˆΩ are regime independen by assumpion. Figure shows he spread beween he 9 monh conrac and he 3 monh conrac ploed agains he smoohed probabiliy of regime. I observe ha in regime 1 he spread is higher and ends o be posiive, whereas in regime he spread is lower and ends o be negaive. Therefore, I refer o regime 1 as he high spread regime and regime as he low spread regime. Figures 3 and 4 show he log of he observed 3 monh fuures price and he facor, respecively, wih shaded areas represening he low spread regime. I es he resricions π P 11 π Q 11 and π P π Q, and find ha hey are rejeced. Thus, my resuls sugges ha π P π Q. By allowing for π P π Q in my model, I have anoher channel hrough which risk preferences can affec expeced reurns. I find ha π Q 11 > π P 11 and π Q > π P. This suggess ha invesors ac as hough regimes are more persisen han hey really are. This could be a resul of hedging pressure or mispricing. The raio πp jk can be inerpreed as relaed o he marke price of regime shif risk. π Q jk Consider a securiy which pays $1 if he regime changes nex monh. This securiy has 1

13 payoff 1 {s+1 k} and has exposure only o he risk of shifing from regime j in monh o regime k in monh + 1. Condiional on he curren regime s j, is curren price is P j E Q 1 {s+1 k} s j π Q jk Therefore, is log expeced reurn is log EP 1 {s+1 k} s j P j ( π P jk log π Q jk ( π Thus, log P jk gives he log expeced reurn per uni of regime shif risk exposure, and π Q jk can herefore be inerpreed as he marke price of regime shif risk from regime j o regime k. Denoe ( π P jk Γ jk log π Q jk Since π P jk and π Q jk are saisically significanly differen, he marke price of regime shif risk is nonzero, i.e. regime shif risk is priced. Using my esimaes for π P jk and π Q jk, I find ha he esimaed marke prices of regime shif risk are ˆΓ , ˆΓ , ˆΓ , ˆΓ For insance, a securiy which pays off $1 if he regime swiches from regime (low spread oday o regime 1 (high spread nex monh is priced a P ( E Q 1 {s+1 1} s π Q 1 1 π Q $ Thus, invesors are willing o pay only abou cens o hedge agains he regime swiching from he low spread regime o he high spread regime nex monh. The expeced payoff of he securiy is E P 1 {s+1 1} s π P 1 1 π P $ So he securiy pays off abou 19 cens on average. The premium invesors are willing o pay o hedge agains regime shif risk is very low, reflecing he fac ha hey hink he low spread regime is considerably more persisen han i acually is. Similarly, a securiy which pays off $1 if he regime changes from regime 1 (high spread oday o regime (low spread nex monh is priced a P (1 E Q 1 {s+1 } s 1 π Q 1 1 π Q 11 $0.04 Thus, invesors are willing o pay abou cens o hedge agains he risk of he regime swiching from high spread oday o low spread nex monh. On average, he securiy pays 13 (3 (33

14 off E P 1 {s+1 } s 1 π P 1 1 π P 11 $ i.e. abou 9 cens. Once again, he premium invesors are willing o pay o hedge agains he risk of he regime changing is low, bu i is closer o he acual expeced payoff of he securiy. To summarize, I find ha agens ac as if boh regimes are more persisen han hey are, wih he perceived overesimaion of he regime persisence being even higher for he low spread regime. The expeced reurn EF (n 1 +1 s j F (nj E F (n 1 +1 s j F (nj is πp jk e A (n 1 k πq jk e A (n 1 k This expression is derived in Appendix B in equaion (A-1. e B(n 1 σ j λ,j (34 Figure 5 shows he expeced reurns for each conrac in he high spread regime and in he low spread regime, averaged over ime. In he high spread regime, buying he 3-5 monh conracs oday and selling hem nex monh on average yields a profi. So does shoring he 6-9 monh conracs oday and closing ou he posiion nex monh. In he low spread regime, shoring he 3-5 conracs oday and closing ou he posiion nex monh on average yields a profi. So does going long on he 6-9 monh conracs oday and selling hem nex monh. Thus, i is more profiable o be long in he 3-5 monh conracs in he high spread regime, whereas i is beer o be long in he 6-9 monh conracs in he low spread regime. The fac ha here is a posiive reurn o a long posiion in he 3-5 monh conracs in he high spread regime suggess ha here is demand for shor posiions in fuures. This could mean ha in he high spread regime, commercial producers are rying o hedge heir shor posiions in naural gas by selling 3-5 monh fuures conracs. Moreover, in he high spread regime here is a posiive reurn o a shor posiion in he 6-9 monh conracs, which could indicae demand for long posiions in hese conracs. In urn, his could indicae ha commercial users are rying o hedge heir long posiions in naural gas by buying 6-9 monh conracs. Similarly, in he low spread regime here is a posiive reurn o a long posiion in he 6-9 monh conracs, which could be a resul of commercial producers rying o hedge heir shor posiions by selling 6-9 monh conracs. Moreover, in he low spread regime here is a posiive reurn o a shor posiion in he 3-5 monh conracs, which could be a resul of commercial users rying o hedge heir long posiions by buying 3-5 monh conracs. The expeced log reurn E P rx (n 1 +1 is relaed o he risk premium invesors demand 14

15 for holding a fuures conrac mauring in n monhs for 1 monh. The expressions for he expeced log reurns condiional on he regime as a funcion of model parameers are derived in Appendix B. E P rx (n 1 +1 s j π jk A (n 1 k log ( π jk e A(n 1 k + B (n 1 λ 0,j + B (n 1 λ 1 X 1 B(n 1 σ j The price of risk λ 1 is saisically significan and negaive. This implies ha an increase in he level of fuures prices decreases he expeced log reurns on he 3-9 monh conracs. The expeced log reurn loading for he n-monh conrac is B (n λ 1. According o my esimaes, a posiive one sandard deviaion shock o he level facor reduces he expeced log reurn on he 3-9 monh conracs by abou 0.88%. The esimaes of he prices of risk λ 0,1 and λ 0, are no saisically significan. Moreover, using a Wald es I find ha λ 0,1 λ 0, is no saisically significanly differen from 0. Thus, I do no find considerable differences in he marke pricing of risk in he wo regimes. 6 Conclusion In his paper I have proposed a new mehod for esimaing regime-swiching affine erm srucure models. My approach allows for compuaionally fas esimaion and avoids he numerical difficulies ha are common when using oher maximum likelihood based mehods in he lieraure. I use my approach o esimae a regime-swiching affine erm srucure model for naural gas fuures prices from 1994 o 013, and find very srong evidence ha here are regimes in he daa. I find ha one regime corresponds o a higher difference beween he longer erm and he shorer erm conracs (higher spread while he oher regime corresponds o a lower difference beween he longer erm and he shorer erm conracs (lower spread. My resuls show ha he marke acs as if regimes are more persisen han hey really are. This could be a resul of hedging pressure or mispricing. I find evidence ha commercial users and commercial producers use naural gas fuures conracs for purposes of hedging. In he high spread regime, commercial producers may be rying o hedge heir shor posiions in naural gas by selling 3-5 monh conracs, while in he low spread regime, commercial producers may be rying o hedge heir shor posiions by selling 6-9 monh conracs. I obain analogous implicaions for commercial users. Moreover, I find ha an increase in he level of fuures prices decreases he expeced reurns on he 15

16 3-9 monh conracs. According o my esimaes, a posiive one sandard deviaion shock o he level facor, which represens a change in prices ha is essenially consan across mauriies, reduces he expeced reurn on he 3-9 monh conracs by abou 0.88%. 16

17 APPENDIX A Relaion beween P-dynamics and Q-dynamics By no arbirage, an asse wih payoff g(x +1 has a price in regime j equal o P (X E M +1 g(x +1 s j E Q g(x +1 s j P (X E M +1 g(x +1 s j ( E exp 1 λ,s λ,s σs 1 v +1 g(x +1 s j ( exp 1 ( λ,j E exp λ,s σs 1 (X +1 µ s ΦX g(x +1 s j ( exp 1 λ,j g(x +1 exp ( λ,j σ 1 j (X +1 µ j ΦX (π 1/ σ 1 j ( exp 1 (X σj +1 µ j ΦX dx +1 (π 1/ σ 1 j + λ,j σ 1 j (π 1/ σ 1 j (π 1/ σ 1 j (π 1/ σ 1 j (π 1/ σ 1 j (π 1/ σ 1 j ( g(x +1 exp 1 1 σ j (X +1 µ j ΦX + (X +1 µ j ΦX + λ,j dx +1 ( g(x +1 exp 1 1 (X +1 µ j ΦX + λ,j σ j ( g(x +1 exp 1 X+1 µ j ΦX + σ j λ,j E Q (g(x +1 s j Therefore, under he Q-measure, ( g(x +1 exp 1 σ j ( g(x +1 exp 1 σ j ( g(x +1 exp 1 σj σ j dx +1 dx +1 X +1 µ j ΦX + σ j λ,j dx +1 X +1 µ j ΦX + λ 0,j + λ 1 X dx +1 X +1 (µ j λ 0,j (Φ λ 1 X dx +1 X +1 s j Q N((µ j λ 0,j + (Φ λ 1 X, σ j (A-1 17

18 or, equivalenly, X +1 s j Q N(µ Q j + ΦQ X, σ j (A- where µ Q j µ j λ 0,j (A-3 and Φ Q Φ λ 1 (A-4 Hence, under he Q-measure, X +1 follows he dynamics X +1 µ Q j + ΦQ X + v Q +1 (A-5 where v Q +1 s j Q N(0, σ j under he Q-measure. B Calculaing expeced reurns E P rx (n 1 +1 s j E P f (n 1 +1 f (n s j E P A (n 1 E P s +1 + B (n 1 X +1 A (n s B (n X s j A (n 1 s +1 + B (n 1 (µ s + ΦX + v +1 A s B (n X s j π P j1 A (n π P j A (n 1 + B (n 1 µ j + (B (n 1 Φ B (n X A (n j (A-6 The fuures price is F (nj e A(n j +B (n X 18

19 Therefore, f (nj A (n j f (nj log F (nj log E Q ( log log log log log log ( ( ( ( ( + B (n X log π Qjk E Q π Qjk e A(n 1 k π Qjk e A(n 1 k π Qjk e A(n 1 k π Qjk e A(n 1 k π Qjk e A(n 1 k ( F (n 1 +1 s j F (n 1k +1 s j E Q + log E Q + log E Q π Q jk e A(n 1 k The above equaion implies he following recursions: or equivalenly A (n j A (n j log log ( ( π Q jk e A(n 1 k π Q jk e A(n 1 k e B(n 1 X +1 s j e B(n 1 X +1 s j e B(n 1 (µ Q j +ΦQ X +v +1 s j + log e B(n 1 (µ Q j +ΦQ X + 1 B(n 1 σj + B (n 1 (µ Q j + ΦQ X + 1 B(n 1 σ j + B (n 1 µ Q j + 1 B(n 1 σ j + B (n 1 Φ Q X + B (n 1 µ Q j + 1 B(n 1 σ j + B (n 1 (µ j λ 0,j + 1 B(n 1 σ j (A-7 and or equivalenly B (n B (n 1 Φ Q B (n B (n 1 (Φ λ 1 (A-8 (A-9 19

20 E P f (n 1 +1 s j π P jk E f (n 1k +1 s j π P jk E A (n 1 k + B (n 1 X +1 s j π P jk π P jk π P jk ( A (n 1 k + B (n 1 E X +1 s j ( A (n 1 k + B (n 1 E µ s + ΦX s j ( A (n 1 k + B (n 1 (µ j + ΦX π P jk A (n 1 k + ( π P jk π P jk A (n 1 k + B (n 1 (µ j + ΦX B (n 1 (µ j + ΦX E P rx (n 1 +1 s j E P f (n 1 +1 f (n π P jk A (n 1 k s j E P f (n 1 +1 s j + B (n 1 (µ j + ΦX log ( f (nj π Q jk e A(n 1 k B (n 1 (µ Q j + ΦQ X 1 B(n 1 σ j log ( π P jk A (n 1 k 1 B(n 1 σ j π Q jk e A(n 1 k π P jk A (n 1 k + B (n 1 (µ j µ Q j + B(n 1 (Φ Φ Q X log 1 B(n 1 σ j ( π Q jk e A(n 1 k + B (n 1 λ 0,j + B (n 1 λ 1 X 0

21 E P rx (n 1 +1 F j1 j1 E P rx (n 1 +1 s j P (s j F π P jk A (n 1 k P (s j F log ( π Q jk e A(n 1 k + B (n 1 λ 0,j + B (n 1 λ 1 X 1 B(n 1 σ j We can also derive an expression for (n 1 EF +1 s j F (nj. F (nj E Q F (n 1 +1 s j π Q jk E Q F (n 1k +1 s j π Q jk E Q e A(n 1 k +B (n 1 X +1 s j π Q jk e A(n 1 k π Q jk e A(n 1 k π Q jk e A(n 1 k E Q E Q e B(n 1 X +1 s j e B(n 1 (µ Q s +Φ Q X +v +1 s j e B(n 1 (µ Q j +ΦQ X + 1 B(n 1 σ j (A-10 E P F (n 1 +1 s j π P jk E P F (n 1k +1 s j π P jk E P e A(n 1 k +B (n 1 X +1 s j π P jk e A(n 1 k π P jk e A(n 1 k π P jk e A(n 1 k E P E P e B(n 1 X +1 s j e B(n 1 (µ P s +Φ P X +v +1 s j e B(n 1 (µ P j +ΦP X + 1 B(n 1 σ j (A-11 1

22 Then E F (n 1 +1 s j F (nj πp jk e A (n 1 k e B(n 1 (µ P j +ΦP X + 1 B(n 1 σj πq jk e A (n 1 k e B(n 1 (µ Q j +ΦQ X + 1 B(n 1 σj πp jk e A (n 1 k e B(n 1 (µ P j µq j +B(n 1 (Φ P Φ Q X πq jk e A (n 1 k πp jk e A (n 1 k e B(n 1 (λ 0,j +λ 1 X πq jk e A (n 1 k πp jk e A (n 1 k πq jk e A (n 1 k e B(n 1 σ j λ,j (A-1 C EM algorihm for firs sage esimaion In he firs sage I esimae he sysem of equaions (14 and (17. I is known ha in he absence of regime-swiching, maximum likelihood esimaion of his sysem is equivalen o OLS esimaion equaion by equaion. Condiional on parameers, he inference abou he regime P r(s j F T 4 can be obained using he Hamilon filering and smoohing algorihm. This suggess esimaion via he EM algorihm. Le θ denoe he vecor of parameers o be esimaed, θ {vec(a, vec(b, Φ, Ω, {µ j, σ j } j1} where vec(a and vec(b are as defined in Secion 4.. Firs, I iniialize he algorihm wih an iniial guess for he vecor of parameers, and compue he corresponding smoohed probabiliies. Then each ieraion l of he algorihm proceeds as follows. Firs, I updae inference for he regression parameers equaion by equaion. An updaed esimae ˆθ (l is derived as a soluion o he firsorder condiions for maximizaion of he likelihood funcion, where he condiional regime probabiliies P r(s Y, θ are replaced wih he smoohed probabiliies P r(s j Y, θ (l 1 compued in he previous ieraion, for Y {Y 1, Y, X } T 1 defined below. Condiional on knowing he smoohed probabiliies, a closed form soluion for he regression parameers of each regime-swiching equaion can be obained by linear regression in which he observaions are weighed by he smoohed probabiliy ha hey came from he corresponding regime. Deails are shown below. Nex, I updae inference abou he smoohed probabiliies P r(s j Y, θ (l, where I am condiioning on he parameer vecor esimae obained in he curren ieraion insead of he unknown parameer vecor θ. 4 F T represens informaion available up o ime T

23 The regime-swiching sysem I esimae is of he form Y 1 µ s + Φ X + ε (1 1 (1 1 (1 1 (1 1 (1 1 Y A s + B (7 1 (7 1 X (7 1(1 1 + u (7 1 ε s N(0, σ s u s N(0, Ω (A-13 (A-14 Equaion(14 when sacked across all mauriies n 3,..., 9 is of he form of he above equaion (A-14 wih Y (f (3, f (4, f (5, f (6, f (7, f (8, f (9, while equaion (17 is of he form of equaion (A-13 wih Y 1 X +1. I esimae he vecor sysem using a parially resriced algorihm equaion by equaion. The algorihm for a single equaion is described in Appendix D. Suppose a he previous ieraion of he algorihm I have esimaes θ (l and Pr(s j θ (l, Y for Y {Y 1, Y, X } T 1. Ieraion (l + 1 of he algorihm works as follows. Sep 1. Updae inference for Y regression parameers. 1a Taking each n 1,..., 7 one a a ime saring wih n 1, consruc ω (l n row n, col. n elemen of Ω (l λ (l n1 λ (l n Y (n P r(s 1 Y, θ (l ω n (l P r(s Y, θ (l ω (l n n h elemen of Y A (l in nh elemen of A (l i B (l n n h elemen of B (l For 1,..., T, define and ỹ (l n λ (l n1y (n x (l n λ (l n1x z (l n1 λ (l n1 z (l n 0 ỹ (l n,t + λ(l ny (n x (l n,t + λ(l nx 3

24 z (l n1,t + 0 z (l n,t + λ(l n Condiional on knowing λ (l n1 and λ (l n, a closed-form soluion for (Â(n 1, Â(n, ˆB (n can be found by performing an OLS regression on an arificial sample of size T, ỹ (l n A (n 1 z (l n1 + A (n z (l (l n + B (n x n + ũ n, 1,,..., T Consruc û (l+1 n1 û (l+1 n Y (n Y (n Â(l+1 1n Â(l+1 n (l+1 ˆB n X (l+1 ˆB n X 1b For each n 1,..., 7 calculae ω (l+1 n { 1 T ( T 1 û (l+1 n1 P r(s 1 Y, θ (l + T 1 û (l+1 n P r(s Y, θ (l } 1/ Sep. Updae he inference for he Y 1 parameers. This involves he analogous seps o hose above using he parially resriced algorihm for a single equaion as described in Appendix D. The facor variance is updaed as σ (l+1 1 T 1 P r(s 1 Y, θ (l (Y 1 µ (l+1 1 Φ (l+1 X T 1 P r(s 1 Y, θ (l σ (l+1 T 1 P r(s Y, θ (l (Y 1 µ (l+1 Φ (l+1 X T 1 P r(s Y, θ (l Sep 3. Updae he inference abou he ransiion probabiliies. The ransiion probabiliies are updaed as ˆπ P ij(l+1 T P r(s j, s 1 i Y T, θ (l T P r(s 1 i Y T, θ (l Specifically, ˆπ P 11(l+1 T πp 11(l P r(s 1 Y T,θ (l P r(s 1 1 Y 1,θ (l P r(s 1 Y 1,θ (l T P r(s 1 1 Y T, θ (l ˆπ P 1(l+1 T πp 1(l P r(s 1 Y T,θ (l P r(s 1 Y 1,θ (l P r(s 1 Y 1,θ (l T P r(s 1 Y T, θ (l 4

25 Sep 4. Updae he inference abou smoohed probabiliies. This sep calculaes he smoohed probabiliies P (s j Y, θ (l+1 using he Hamilon filering and smoohing algorihms, which are described in Appendix E. The iniial probabiliy vecor ρ is updaed as ρ (l+1 j P r(s 1 j Y T, θ (l D EM algorihm for scalar regression Here I presen he general form of he EM algorihm I use for esimaion of each equaion from my regime-swiching vecor sysem. Suppose he variances and some bu no all of he parameers change wih he regime, ha is y x β + z c s + σ s v for y a scalar, x an (m 1 vecor, z an (r 1 vecor, and v N(0, 1. Thus η 1 πσ 1 exp 1 exp πσ { (y x β z c 1 σ1 { (y x β z c σ } } log η σ 1 log η σ log η β log η c 1 log η c (y x β z c 1x σ 1 (y x β z c x σ (y x β z c 1z σ 1 1 σ 1 1 σ The MLE for θ (β, c 1, c, σ 1, σ saisfies 0 0 (y x β z c z σ + (y x β z c 1 σ (y x β z c σ 4 T ( log η ˆξ T 0.. (A-15 1 θ 5

26 Define λ 1 λ σ1 1 σ 1 Pr(s 1 Y Pr(s Y for Y {y, x, z } T 1 he full se of observed daa. of β (using equaion (A-15 can be wrien Then he FOC associaed wih choice ( T x y λ T x y λ 1 ( T T x x λ 1 + x x λ β 1 ( T 1 ( T + 1 x z λ 1 c x z λ c. Take he analogous FOC for choice of c 1 and c and sack he hree equaions ogeher: ( T ( T 1 x y λ 1 + T 1 x y λ ( T 1 z y λ 1 ( T 1 z y λ 1 x x λ 1 + ( T 1 x T ( x λ 1 1 x T z λ 1 1 x z λ ( T ( 1 z T β x λ 1 1 z z λ 1 0 c ( T ( 1 z T 1. x λ 0 1 z z λ c (A-16 Condiional on knowing λ 1 and λ, a closed-form soluion for ( ˆβ, ĉ 1, ĉ can be found by performing a single OLS regression on an arificial sample of size T, ỹ x β + z 1c 1 + z c + ṽ 1,,..., T, where for 1,,..., T I have defined ỹ y λ 1 x x λ 1 z 1 z λ 1 z 0 6

27 whereas he nex T observaions (denoed T + for 1,..., T are from ỹ T + y λ x T + x λ z T +,1 0 z T +, z λ. The OLS coefficiens from his arificial sysem are given by ˆβ ĉ 1 ĉ 1 x x T 1 x z 1 T 1 x z 1 T 1 1 z 1 x T 1 z 1 z 1 T 1 z 1 z x ỹ T 1 1 z x T 1 z z 1ỹ z 1 T 1 z z T 1 z ỹ 1 x x λ 1 + ( T 1 x T ( x λ 1 x T z λ 1 1 x z λ ( T ( 1 z T x λ 1 1 z z λ 1 0 ( T ( 1 z T x λ 0 1 z z λ 1 x y λ 1 + T 1 x y λ ( T 1 z y λ 1 T T T ( T ( T ( T 1 z y λ 1 which will be recognized as a closed-form soluion o he FOC for he MLE as given in equaion (A-16. Thus an EM algorihm would work as follows. A he previous sep I have calculaed esimaes ˆσ 1, ˆσ, ˆξ T, from which I can consruc λ 1 and λ. I hen use hese o consruc {ỹ, x, z 1, z } T 1 and do an OLS regression of ỹ on x, z 1, z o ge new esimaes of β, c 1, c. Taking firs order condiions for σ 1 and σ resuls in he following expressions for he nex sep esimaes: ˆσ 1 T 1 (y x ˆβ z ĉ 1 Pr(s 1 Y T 1 Pr(s 1 Y ˆσ T 1 (y x ˆβ z ĉ Pr(s Y T 1 Pr(s. Y 7

28 E Le Filering and smoohing algorihm 1{s 1} ξ 1{s } (A-17 Le ˆξ τ E(ξ Y τ. Then P r(ξ e 1 Y τ ˆξ τ P r(ξ e Y τ (A-18 where Y τ consiss of informaion available up o ime τ, e 1 (1, 0, e (0, 1. Le y be he vecor of dependen variables of all he equaions. Le η be he vecor of densiies of y condiional on ξ and Y 1 : p(y θ 1, Y 1 p(y ξ e 1, Y 1 η p(y θ, Y 1 p(y ξ e, Y 1 where θ has been dropped on he righ hand side for breviy. In my model, η 1 (π ( (K+N/ Ψ 1 1/ exp η (π ( (K+N/ Ψ 1/ exp where and K 1 and N ( ( Y 1 Y Y 1 Y Ψ j ( ( µ 1 + ΦX Ψ 1 1 A 1 + BX ( ( µ + ΦX Ψ 1 A + BX σ j 0 K N 0 N K Ω j Y 1 Y Y 1 Y (A-19 ( µ 1 + ΦX A 1 + BX ( µ + ΦX A + BX The densiy of y condiional on Y 1 is given by p(y Y 1 η ˆξ 1 1 (η ˆξ 1 where signifies elemen-wise marix muliplicaion. The conemporaneous inference ˆξ abou he unobserved sae vecor ξ is given in marix noaion by he filering recursions ˆξ η ˆξ 1 1 (η ˆξ 1 (A-0 ˆξ +1 P ˆξ (A-1 8

29 where P is he marix of ransiion probabiliies. The recursion is iniialized wih ˆξ 1 0 ρ The smoohed inference abou he unobserved sae vecor ξ is given by ˆξ T ˆξ ( P (ˆξ +1 T ( ˆξ +1 (A- where he sign ( denoes elemen-by-elemen division. The smoohed probabiliies ˆξ T are found by ieraion on equaion (A- backward for T 1, T,, 1. This ieraion is sared wih ˆξ T T, which is obained from equaion (A-0 for T. 9

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32 Figure 1: Log of he observed hree monh naural gas fuures price 3

33 Figure : The spread beween he log of he nine monh fuures price and he log of he hree monh fuures price (in blue vs. smoohed probabiliy of he low spread regime (in red. Shaded areas represen he low spread regime. 33

34 Figure 3: Log of he observed hree monh naural gas fuures price. Shaded areas represen he low spread regime. 34

35 Figure 4: Firs principal componen (in blue vs. smoohed probabiliy of he low spread regime (in red. Shaded areas represen he low spread regime. 35

36 Figure 5: Expeced reurns condiional on he high spread regime (regime 1, in blue and condiional on he low spread regime (regime, in red 36

37 Table 1: Firs sage reduced form parameer esimaes Regime 1 µ ( Φ A ( ( A ( ( A ( ( A ( ( A ( ( A ( ( A ( ( B (3 B (4 B (5 B (6 B (7 B (8 B (9 σ ( Ω (3 Ω (4 Ω ( ( ( ( ( ( ( ( ( ( ( ( Regime ( ( ( ( ( ( ( ( (

38 Table 1: Firs sage reduced form parameer esimaes (coninued Ω (6 Ω (7 Ω (8 Ω (9 π P11 π P ρ 1 Regime ( ( ( ( ( ( Regime logl 71.6 Asympoic sandard errors are in parenheses 38

39 Table : Second sage parameer esimaes Regime 1 µ ( Φ A ( ( B (3 σ ( Ω (3 Ω (4 Ω (5 Ω (6 Ω (7 Ω (8 Ω (9 π P11 π P π Q11 π Q λ ( λ ( ( ( ( ( ( ( ( ( ( ( ( ( ( Regime ( ( ( (

40 Table 3: Wald -saisics H 0 Sandard error Wald -saisic Reduced-form parameers (firs sage A (4,1 A (4, A (5,1 A (5, A (6,1 A (6, A (7,1 A (7, A (8,1 A (8, A (9,1 A (9, Reduced-form parameers (second sage µ 1 µ σ1 σ A (3,1 A (3, Srucural parameers (second sage λ 0,1 λ 0, π P11 π Q π P π Q

41 Table 4: Reduced-form versus model implied values for A (n,j and B (n Firs sage esimaes Model implied values A (3, A (3, A (4, A (4, A (5, A (5, A (6, A (6, A (7, A (7, A (8, A (8, A (9, A (9, B ( B ( B ( B ( B ( B ( B (