Oil prices in the real economy

Transcription

1 Oil prices in the real economy Haicheng Shu and Peter Spencer This draft: July 2016 Department of Economics and Related Studies The University of York, York, U.K., YO10 5DD We are grateful to Carol Alexander, Laura Coroneo, Adam Golinski, Anh Le, Peter Smith, Mike Wickens and participants in the July 2016 Asset Pricing Workshop, University of York for helpful comments. 1

2 Abstract This paper presents a macro- nance model of the economy and the oil markets that allows us to study interactions between the convenience yield, the spot and futures markets, monetary policy and macroeconomic variables. We use the Kalman lter to represent latent variables that handle the e ect of exogenous shocks to in ation, the oil price and convenience yield and to deal with missing observations. Traditional models use latent variables with little economic meaning to explain commodity futures, while this model also makes the e ect of macroeconomic variables explicit. We nd signi cant interaction between the economy and the oil markets, including an important link in the monetary transmission mechanism running from the policy interest rate to the convenience yield, oil price and in ation. 2

3 1 Introduction This paper presents a macro- nance model of the US economy and the crude oil market. It uses both macroeconomic and oil market data to study interactions between the spot and futures markets, monetary policy and the macro-economy. There is an extensive macro-econometric literature, pioneered by Hamilton (1983) that studies the e ect of oil prices on the economy. He argued that most post-war recessions in the US were caused by oil price shocks. Hamilton and his colleagues have followed this paper up with many studies documenting the adverse e ect of oil price changes on real output and in ation over the last few decades (Hamilton (1985), Hamilton (2003), Herrera and Hamilton (2004), Hamilton (2008), Hamilton and Wu (2014)). Many other authors have studied this e ect (Raymond and Rich (1997), Finn (2000), Hooker (1996) and Hooker (2002).. These papers focus on the e ect of spot oil prices on the economy, neglecting the information in futures, oil inventory and its convenience yield. However, Kilian (2006), Kilian (2008)) attributes a much greater role to demand side pressures, arguing that the e ect of supply side shocks depends critically upon the tightness of the oil market in the run up to the shock, as re ected in inventories. As its name suggests, the convenience yield re ects the convenience and security of holding physical stocks of oil rather than an equivalent position in the oil futures market. If inventories are excessive, the convenience yield can turn negative, re ecting insurance and other holding costs. The relationship between the spot and short forward price is governed by an arbitrage relationship that depends upon the spot convenience yield and the spot 3

4 interest rate. Longer maturity futures prices depend upon the evolution of oil prices and hence the expected interest rate and convenience yield under the risk neutral measure, which makes an allowance for the risk of holding positions in the futures market. The literature on commodity futures dates back to the 1930s and the papers by Hotelling (1931), Working (1943), (1949) and others. Empirical pricing models date back to the early 1980s and the papers of Schwartz (1982), Brennan and Schwartz (1985), Gibson and Schwartz (1990), Brennan (1991), Cortazar and Schwartz (1994), Schwartz (1997), Casassus and Collin-Dufresne (2003)). These empirical models suggest that the term structure of commodity futures prices is similar to the term structure of interest rates in the sense that most of the variation in the cross-section can be explained by three latent factors. However, both the models of futures and interest rates are silent about the nature of the latent variables and their links with the macro-economy. Interest in the structural relationship between the yield curve and the economy has led to the development of macro- nance models, following the pioneering research of Ang and Piazzesi (2003), who successfully introduced macroeconomic factors into the term structure model. They found that although these indicators provide a good description of the behaviour of short rates, it was necessary to retain latent variables to model long term rates. Subsequent macro- nance research has used the semi-structural central bank model (CBM) developed by Svensson (1999), Smets (1999), Kozicki and Tinsley (2005) and others, suggests that this latent variable represents exogenous shocks to the central bank in ation target or underlying rate of in ation (see Kozicki and Tinsley (2005), Dewachter and Lyrio 4

5 (2006), Dewachter, Lyrio and Maes (2006)). The CBM represents the behaviour of the macro economy in terms of the output gap (g t ), in ation ( t ) and the short term interest rate (r t ). There is no role for the oil price in the basic model, despite the evidence of the e ect of oil price shocks on the macro-economy. In this paper, we develop a macro- nance model that introduces the oil price and the convenience yield of inventory into the CBM. We use this model to study the interaction between the oil market and the macro economy and to model the term structure of oil futures prices. We follow the semi-structural strand of the macro- nance literature in modelling the underlying in ation rate as a latent variable using the Kalman lter. We use two additional latent variables to handle exogenous supply side and other shocks to the oil price and convenience yield, which both play a vital role in this model. We also follow the semi-structural approach in using a long sample of macroeconomic data. This begins in 1964 and thus includes the period of the 1970s oil shocks, which helps identify the links between the oil market and the economy. The Kalman lter plays a key role here since it handles missing observations, which other latent variable techniques (such as principal components), nd it very di cult to handle. This allows us to combine a relatively long data set for macro variables and spot oil prices with a relatively short data set (beginning in 1984) for oil futures prices. Observations of the convenience yield are backed out from futures prices. Thus they are only available as hard data from the beginning of 1984, but the Kalman lter provides useful and arguably realistic estimates for earlier years. These estimates are inferred from the macro variables and the way they interact with the oil 5

6 market in the post-1984 period. The empirical results are consistent with our theoretical priors. They are also consistent with the existing macro- nance literature in underlining the importance of both macroeconomic variables and latent factors such as the underlying in ation rate, which re ects salient historical episodes such as the Volker disin ation. Our latent oil market variables also re ect the impact of exogenous shocks such as the two Gulf wars, the internet bubble at the turn of the millennium and the 2008 nancial crisis. We nd strong evidence of the interaction between the oil market and the US economy. For example, re ecting the work of Kilian (2005) and his colleagues, the strength of the economy in the run up to oil price shocks of the 1970s helps explain why these were persistent, and why, because the economy was weak in the run up to the rst Gulf war in 1991, that shock had only a temporary e ect on the oil price. The convenience yield naturally plays a key role in pricing futures, but, more of a surprise, we nd it also plays an important role in the macro-model. It can be viewed as a proxy for oil inventory, which serves as a bu er, damping the e ect of oil and economic shocks on the real economy. We nd that it also plays a key role in the monetary transmission mechanism, providing an important channel through which policy interest rates a ect oil prices, activity and in ation. This is of relevance to Central Bank policy makers. Naturally, when evaluating the e ect of their policies on the economy, they focus on the money and bond markets. However, the oil spot and futures markets are also important and central bank researchers have been studying them recently (see Chin and Liu (2015), Millard and 6

7 Shakir (2013), Elekdag et al (2007), Bodenstein M, Guerrieri L and Kilian L (2012)). We use these macro and oil market variables as factors in an a ne model of the term structure of oil futures prices. This is speci ed in nominal terms, with the spot price as the baseline, combining the US general price level (the GDP de ator) and the real price from the macro model. Thus the spot oil price has a unit e ect at the short end of the futures curve, but this in uence fades with maturity. The GDP de ator is non-stationary and has a unit e ect on nominal spot and futures prices, which is permanent. The spot interest rate and convenience yield determine the slope of the short-maturity yield curve. In ation also has an important e ect, but the e ect of the output gap is negligible. As noted, futures prices incorporate risk premiums. The model suggests that these increase with maturity but are strongly cross-correlated. Oil prices play the major role, increasing the premium when the spot price is below the underlying price, as in the mid 1970s. The contribution of the underlying in ation rate is also very important, pushing up the premium until in ation began to decline following the Volker experiment. As in any macro- nance model, the latent variables are updated in line with surprises in both macro and nancial variables. For example, the Kalman variable representing the underlying oil price normally follows the price with a lag, re ecting the delayed e ect of exogenous supply-side surprises. However, it also re ects the e ect of surprises in macro variables that a ect the demand for oil. For example, the negative macro surprises associated with the recent nancial crisis and recession pushed the underlying oil price well below the actual price and largely anticipated the sharp fall seen since mid A latent factor model 7

8 of the oil market that only updated these factors in line with surprises to spot and futures prices would miss this e ect. The paper is organized as follows. The next section speci es the macro- nance dynamic term structure model (DTSM). This speci es the state dynamics under the real-world and risk-neutral probability measures and the risk premium. Section 3 sets out the empirical methodology and econometric model, describes the data that we use, and discusses the empirical ndings and results. Section 4 concludes. 1.1 The real-world dynamics Our research strategy is to follow the semi-structural macro- nance literature, which models the dynamics under both measures in terms of observable variables like the interest rate, convenience yield and oil price, assuming that they are measured without error. Because we use the spot oil price as a factor, this allows us to switch from the nominal prices used in the futures market to the real oil prices relevant for the real economy dividing by the GDP de ator. We will show that this allows future prices to be modelled by specifying both the risk-neutral dynamics of oil prices and the macro economy using real variables. An alternative approach, pioneered by Joslin, Priebsch and Zu (20013) and followed by Heath (2016) would be to assume the cross-section of oil futures prices is a ne in a small number of latent factors (like their own principal components). However, unlike bond prices, which are the relative prices of money in di erent periods, futures prices tend to increase over time due to the e ect of in ation, which it is di cult to allow for without using the spot price as an explicit factor as we do. Moreover, there is an important arbitrage identity linking the 8

9 spot oil price, the convenience yield and interest rate that would be di cult to incorporate into a model that did not identify these explicitly The spot oil market variables The relationship between the spot (S t ) and one period future (F 1;t ) oil price de nes the cost of carrying inventory (r t t ), which can be decomposed into the spot convenience yield of holding physical oil inventories t and the spot interest rate r t : F 1;t = S t e (rt t) ; (1) This shows that if the cost of carry is negative the curve is downward sloping. The market is then said to be in backwardation, as it was in 2015 for example. If the cost of carry is positive, the forward curve is upward sloping and is said to be in contango, as it was in 2012 for example. Many commodity market models, like Heath (2016), treat the cost of carry as a single variable. However, we follow Casasus and Dufresne (2005) in distinguishing the convenience yield and the spot interest rate. Additionally we assume that the latter is observed without error. We could measure the former as: t = r t (f 3;t s t );.where lower case letters denote logarithms of futures prices.. However in practice we nd it better to measure this as: t = r t 3(f 2;t f 1;t ): (2) This centres the value on the mid-month of the quarter but uses simultaneously-recorded 9

10 futures data, avoiding the e ect of any timing or other mis-match problems in observing the spot price. We will see that the decomposition of the cost of carry plays an important role in our model, consistent with the theory of storage (Working (1933), Kaldor (1939), Working (1949), Brennan (1958), Weymar (1968) ). This suggests that the convenience yield is closely related to the level of the commodity stored in inventory. It states that when working inventories (those available in the market for commercial rather than strategic purposes) are tight, the convenience yield will be high; the cost of carry negative and the futures curve in backwardation. On the other hand, when oil inventories are abundant, the convenience yield will be negative, adding to the interest cost of carry and pushing the futures curve into contango. Our model allows the convenience yield and the interest cost to interact, so that a tightening of monetary policy tends to reduce inventories and push up the convenience yield to compensate The other macro variables The model of the economy is naturally speci ed in terms of the real rather than the nominal oil price. The log nominal price is denoted by s t = log S t and the log real price by t = s t p t ; where p t is the log implicit GDP de ator. The e ect of the business cycle is represented by g t ; the US output gap based on the constant price US GDP series (see Section 3). In ation is represented by t ; the annual US in ation rate, calculated using the US GDP price de ator and r t is represented by the US Federal Funds rate. We model the state vector, comprising 0 the convenience yield and the four macro variables, m t = t t g t t r t ; assuming that 10

11 they are measured without error. We argue that this vector is in uenced by three latent variables. These are respectively: a factor denoted by t representing non-transient shocks to oil inventories and the associated convenience yield; the underlying real spot oil price, denoted by t ; and a long run in ation asymptote or policy target, denoted by t. We estimate the real-world parameters of the model using a Kalman Vector Autoregressive (KVAR) The latent factor dynamics The long term convenience yield factor ( t ), underlying real spot price trend ( t ), and in ation asymptote ( t ) are assumed to be mean independent. While t is assumed to be an I(0) variable, t and t are integrated of order one (I(1)): t = + t 1 + ;t ;t N(0; 2 ); (3) t = + t 1 + ;t ;t N(0; 2 ); (4) t = + t 1 + ;t ;t N(0; 2 ): (5) Stacking equations (3), (4) and (5) together, and putting the latent variables in the vector 0 z t = t t we can write the latent factor dynamics compactly as: t z t = A z + B z z t 1 + u z;t u z;t (0; D z ); (6) 11

12 1.1.4 The macro dynamics The dynamics of the convenience yield ( t ) depend upon the three latent variables and the lagged values of the oil market variables: t = + ; t + ; t + ; t + ; t 1 + ; t 1 + ;g g t 1 + ; t 1 + ;r r t 1 + ;t (7) The other macro variables depend upon each other (with lag) as well as the latent variables: t = + ; t + ; t + ; t + ; t 1 + ; t 1 + ;g g t 1 + ; t 1 + ;r r t 1 + ;t (8) g t = g + g; t + g; t + g; t 1 + g;g g t 1 + g; t 1 + g;r r t 1 + g;t (9) t = + ; t + ; t + ; t 1 + ;g g t 1 + ; t 1 + ;r r t 1 + ;t (10) r t = r + r; t + r; t + r; t 1 + r;g g t 1 + r; t 1 + r;r r t 1 + r;t (11) Stacking equations (7) to (11), the observed variables m t follow the dynamic system under the measure P : m t = A m + z t + B m m t 1 + m;t m;t (0; m ): (12) The matrix m in this representation plays an important role in the Kalman Filter (see Appendix). It can be factorized with m;t using the standard LDL decomposition as: m = CD 2 mc 0 :: and m;t = CD m u m;t where: D m = Diagfd d d g d d r g; u 0 t = 12

13 u ;t u ;t u g;t u ;t u r;t ; 1 and C is de ned in the Appendix. To use the Kalman Filter (see appendix) z t is contemporary with m t in this representation. To specify the transistion equation we substitute equation (6) into this representation in order to lag z t : m t = A m + (A z + B z z t 1 + u z;t ) + B m m t 1 + m;t = A m;z + B m;z z t 1 + B m m t 1 + m;t ; (13) where B m ; z = B z, A m ; z = A m + A z, and m ; t = CD m u m ; t + u z ; t. Following Dewachter, Lyrio & Maes (2006) and Spencer (2008), we normalize the latent factors by aligning them with the asymptotic values of the observed variables. Dropping the time subscripts from the variables in (13) and setting their error terms to zero gives their asymptotic values: m =A m;z + B m;z z + B m m; =' + Rz; (14) where: ' = (I B m ) 1 A m;z and R = (I B m ) 1 B m;z. Inverting these relationships allows us to specify the dynamic parameters A m;z and B m;z in equation (13) (and hence in equation (15)) as: A m;z = (I B m )' and B m;z = (I B m )R, where ' 0 = ' ' ' g ' ' r is an unrestricted vector and R is a suitably restricted matrix de ned in the Appendix. 1 Diagfag represents a matrix with a vector a along its main diagonal and zeros elsewhere. 13

14 1.2 The companion form Finally, stacking equation (6) and (13) gives the companion form or transition equation for the state vector X 0 t = (z 0 t; m 0 t): 0 z t m t 1 0 A z 1 C A = B A + B z 0 3;5 B C (I B m )' (I B m )R B m or more compactly: z t 1 m t C A + D z 0 3;5 B C (I B m )R CD m u z;t u m;t 1 C A : (15) X t = A + BX t 1 + LDU t U t N(0; I) (16) = A + BX t 1 + W t W t (0; ) 2 The futures prices and the risk-neutral dynamics This macro-econometric model can be estimated as a stand-alone KVAR that models the interaction between t ; t and r t and the other macro variables. However, we now respecify the model dynamics under the risk-neutral measure in order to model the behaviour of oil futures prices. We follow the mainstream macro- nance literature in adopting the same macroeconomic structure, including the real rather than the nominal oil price, but allowing the deterministic parameters of this structure to re ect the change of measure. This has the advantage of allowing information in futures prices to inform the latent variables t ; t and t employed in the macro-model, which remains our primary focus of interest. 14

15 We use the essentially a ne model of Du ee (2002) to change the probability measure. This rede nes the deterministic and stochastic parts of the VAR under measure P, in a way which ensures that the expectation of W Q under the Q measure is zero. This implies a system that is congruent with the companion form (16): X t = A Q + B Q X t 1 + W Q W Q N(0; ); (17) A Q is an unrestricted vector. Thus the rst and fourth columns and rst three rows of B Q are restricted to take account of the exclusion restrictions in (3) to (5). Similarly the fourth row takes account of those in (7). The fth row of B Q imposes the arbitrage identity discussed in the appendix (4), which we can write as: t+1 = t B Q X t + r t t ;t+1 : (18) where B Q X t is the model-implied expectation of t+1 : B Q is speci ed formally in the Appendix. In contrast, Heath (2016) assumes that the nominal spot oil price and cost of carry follow a simple autoregressive scheme under Q; independently of the macro variables. The cross section of futures prices is then a ne in these two factors. The real-world time series dynamics is modelled using a VAR with a state vector that includes these factors alongside macroeconomic variables, which are thus unspanned the sense that they only have a lagged or dynamic e ect on the futures curve, not a contemporaneous one. 15

16 However, we would suggest that there are several problems with this speci cation. First, as our Appendix (4) shows, the parameters of the spot price equation are determined by an arbitrage identity (see equation (39), which is the nominal counterpart of (18)) under the risk neutral measure and should not be freely estimated. Second, his model is speci ed in terms of the nominal spot price, without allowing for the e ect of in ation, which imparts a strong secular uptrend. Our model is speci ed in real terms, to remove the e ect of in ation on nominal prices. We also allow macro shocks to have a contemporaneous e ect on this structure. Since we also split the cost of carry into the spot interest rate and convenience yield, monetary policy and other macro shocks can have an additional indirect e ect on the convenience yield and spot oil price, working through the interest rate. In practice we nd that the e ect of these macro shocks is relatively small, but typically well-de ned given the tiny measurement errors found in cross-section estimates of nancial prices. The appendix shows that A Q and B Q in equation (17) are related to A P and B P in equation (16) by: A Q =A LDD 0 1 (19) B Q =B L 2 (20) where 1 is a 8 1 vector. 2 is a 8 8 matrix of parameter determining the risk premium, 16

17 composed of 1;z, 1;m and 2;z, 2;m, 2;z;m as: 0 1 = 1;z 1;m C A 2;z 0 3;5 2 = B A : (21) 2;z;m 2;m The term structure of futures prices The state dynamics under the risk-neutral measure Q (eq (17)) determine the cross-sectional loadings. First, we adopt the a ne trial solution for the log futures prices: f ;t = + X t + p; p t : (22) The initial condition, is implied by the special case when = 0, that f 0;t = t + p t, which gives the starting values for the rst latent factor ( t ) as: ;0 = 1; p;0 = 1; (23) ;0 = ;0 = g;0 = ;0 = r;0 = 0: (24) This makes the futures prices exponentially a ne in the factors and thus log normal. To verify the trial solution (22) and nd its parameters we take logs of equation (33), using the formula for the expectation of a log normal variable, to get: f ;t = lne t (F 1;t+1 ) = E t (f 1;t+1 ) V ar(f 1;t+1): (25) 17

18 Increasing t and reducing in equ (22) by one, substituting p t+1 = t+1 + p t and then equ (17): E t (f 1;t+1 ) = 1 + 1E t (X t+1 ) + p; 1 E t (p t+1 ) = 1 + 1(A Q + B Q X t ) + p; 1 (B Q X t + p t ); (26) V ar(f 1;t+1 ) = 1 0 1: (27) Substituting these into equation (25) using the starting values (23) and (24) veri es the trial solution in equation (22) provided that: = 1B Q + B Q ; (28) p; = p; 1 = 1; (29) = 1 + 1A Q ; (30) where is constant. Equation (29) shows that all the loadings p are equal to one, simplifying (22) to: h ;t = f ;t ip t = + 0 X t ; (31) Stacking these equations gives the A ne Term Stucture Model (ATSM): h t = + zz t + mm t (32) 18

19 2.0.2 The Kalman Filter and the likelihood function To complete the dynamic term structure model we now describe the maximum likelihood approach used to estimate the Kalman lter and the model parameters. The lter uses surprises in forecasting the macro variables and the futures prices to update the estimates of the latent variables. Appendix 4.1 sets out the revisions algebra as eq (52), derives the log likelihood function eq (53) for this model. We use the Kalman Filter instead of the principal components to capture the latent variables, for several reasons. This avoids assuming that any yields or combination of yields are measured without error. It allows us to align the latent variables with the asymptotes,, and. But most important, the Kalman lter nicely resolves the missing variable problem (see Appendix 4.1). 3 The empirical model The empirical model consists of a heteroscedastic VAR describing the three latent variables and ve macroeconomic variables (16) and the auxiliary equations describing the representative futures prices (32). It is estimated by maximum likelihood and the Kalman lter, which gives optimal linear estimates of the latent variables in this situation. The likelihood function is derived in appendix 4.1. This section describes the data and the empirical results. 3.1 Data sources and description We estimate the model using quarterly time series of the macro variables and crude oil futures. All data are downloaded from Thompson Reuters DataStream. Summary statistics are presented in Table 1. Figure 8 shows the WTI oil futures prices and Figure 9 shows the 19

20 four observed macro variables and the implied convenience yield. We use data for the US output gap, US in ation and US Fed Funds rate, from Q to Q This allows us to analyse the e ect of the oil shocks of the 1970s in our research. The Fed Funds rate is speci ed as a quarterly decimal fraction (the annual rate as % divided by 400). We generate the US output gap by applying the HP lter to log US GDP. US in ation is the log di erence of the US implicit price de ater. The spot oil price is a composite series. The West Texas Intermediate (WTI) spot price, which matches the futures data, is available from Q1 1983, while the Brent price, which gives the price of a similar grade, is available from Q However, the crude oil price was xed close to $2.25 per barrel between 1964 and Q So this value is used until then; the Brent price from Q to Q and WTI thereafter. To represent the term structure of oil futures, we use the prices of WTI light crude oil futures traded on New York Mercantile Exchange (NYMEX) starting from the year 1984, when these oil futures contracts started trading. We study oil futures contracts with 1, 2, 3, 6, 9, 12, 18 and 24 month maturities. The series for the prices of oil futures with 1, 2, 3, 6, 9 and 12 months maturities are available from Q1 1984, the 18 month contract from Q3 1989; and the 24 month contract. Finally, we generate the implied convenience yield using equation (2). This is the quarterly convenience yield, expressed as decimal fraction 2 Brent and WTI spot oil price series only diverge signi cantly in recent years, when the latter went to a discount because of export controls and the development of the US shale hydrocarbon industry. 3 Before the 1970s, the oil market was monopolized by the major western oil companies, and the oil price at that time was described by the phase : take the price used by Exxon, add it to that used by Shell and divide the sum by two (Carollo (2012)). 20

21 3.2 The behaviour of the macro and spot oil market variables Figure 10 shows the estimates of the state variables alongside their observed values. Table 2 shows their root mean squared errors (RMSEs). The long term in ation asymptote ( ) in gure (10) picks up the secular trends in in ation and interest rates. This variable resembles the in ation target identi ed by Ireland (2007), which largely accommodates the oil price hikes in the 1970s but then falls back sharply after the Volker de ation in the early 1980s. The top panel of Figure 13 plots the real spot price against the latent variable ( ) representing the underlying real spot oil price. The underlying price normally follows the price with a lag, re ecting the lagged e ect of oil price surprises. However ( ) also re ects the e ect of surprises in macro variables and in particular those associated by the nancial crisis and the ensuing recession, which pushed well below the price, anticipating the sharp fall seen since mid The second panel of Figure 13 shows how oil prices interact with the output gap. The output gap was very high before both of the oil shocks of the 1970s, re ecting the strength of the US economy. This helped tighten the oil market, causing the underlying oil price to trend upward. These price hikes were quite persistent, provoking a sharp fall in the output gap as the economy moved into recession. In contrast, the economy was not as strong prior to the oil price spike seen at the time of the rst Gulf war in This was much less persistent, and was not followed by a serious slowdown. These episodes re ect the arguments of Kilian (2005), suggesting that oil shocks caused by political or other exogenous events only have a big impact on the real economy if the economy is strong and the underlying oil price trend is upward. The US economy was also strong as the oil price 21

22 peaked in 2008, but the ensuing recession was arguably due to the nancial crisis rather than the high oil price. These shocks are also re ected in the convenience yield (). Recall that this is backed out from the one and two months futures data using eq (2). The top panel of Figure 14 shows the model estimates of the convenience yield (^) alongside the data available since Model estimates in earlier periods are inferred by the Kalman lter used to deal with such missing variables (see appendix 4.1). It does this by using estimates of the e ect of oil price and other macro shocks on the convenience yield post-1985 to estimate what e ect they would have had in the earlier period. This estimate picks up the tightness of the market in 1974 and 1980 quite nicely. It is close to the data when this is available for the post-1984 period except in 2008 Q4 when there is a large negative residual associated with the Lehman crisis, and again in the winter of 2014/15 when the price fell sharply. The bottom panel of gure 14 shows the inverse correlation between the estimated convenience yield from the model and the US oil inventory (excluding the Strategic Petroleum Reserve (SPR)), which is the oil inventory that is available to the market. Although inventory is not part of the model, this panel shows that short-run swings in the convenience yield and the oil inventory tend to be inversely related. However there are some notable spikes in the estimated convenience yield () which are not re ected in inventory. For example there is a sharp spike in Q2 1974, which arguably re ects rationing and other e ects designed to conserve oil stocks and help shield the economy from the Arab oil embargo. 4. Figure 15 4 For example the US Congress passed the Emergency Highway Energy Conservation Act to impose national maximum speed limit of 55 mph (about 88 km/h) in 1974, with similar restrictions imposed in European countries. In the UK, petrol coupons were issued in preparation for the possibility of petrol 22

23 shows model estimates of the latent variable ( ) used to represent the exogenous oil supply shocks. These are not very informative until 1985 when the crude oil futures prices become available. This factor is reasonably persistent under the risk neutral measure but acts like a white noise shock under P, as we explain in the next section. It exhibits a series of spikes corresponding to exogenous supply shocks such as the two Gulf wars as well as demand e ects like the Dot-com boom and the Asian nancial crisis. Table 4 reports the estimates of the parameters obtained from the KVAR under the measure P The estimates of the key parameters conform to economic priors and are generally signi cant statistically. The estimates of ;g, ;, and ;r are signi cant at the 99% con dence level, capturing the impact e ect of the real economy on the real oil price, while the signi cance of ;, ;r captures the e ect on the convenience yield, in line with the results of Casassus and Collin-Dufresne (2003). As we would expect, the estimate of ; indicates that the real oil price has a signi cant e ect on in ation. Impulse response functions The dynamics of these interaction e ects can be seen from Figures 17 and 18, which depict the impulse response functions. These show the dynamic e ects of innovations in the macroeconomic variables. Because these innovations are correlated empirically, we work with orthogonalized innovations using the triangular factorization de ned in section (1.1.4). The orthogonalized impulse responses show the e ect on the macroeconomic system of increasing each of these innovations by one percentage point for just one period using the Wald represenrationing, although they were never used. 23

24 tation of the system. Each column shows the e ect of a unit shock to a macro variable, while the rows show their e ects The relationships between the output gap, in ation and interest rates are in line with economic priors and similar to those seen in previous macro- nance models. In particular, the use of Kalman lters to pick up the e ect of unobservable expectational in uences helps to solve the notorious price puzzle - the tendency (noted originally by Sims (1992)) for increases in policy interest rates to anticipate in ationary developments and apparently cause in ation. As we will see, the introduction of the oil market variables also enhances the e ect of interest rates on in ation. The results re ect the Taylor rule, which suggests that the central bank adjusts the policy interest rate in order to maintain low of in ation. The nal row of Figure 17 shows that the US Fed increases interest rates in response to in ation and excessive economic expansion. The nal column in turn shows that output and in ation fall in response to the higher interest rate. The novelty here is the introduction of oil prices and the convenience yield into this macro system. As we would expect, Figure 17 shows that oil price innovations act as supply-side shocks that depress output but increase in ation and interest rates. In turn, the oil price responds positively to the output gap and in ation and negatively to the interest rate. This is part of the monetary transmission mechanism, which policy makers study since they need to understand the way that the policy interest rate policies is transmitted to output and the in ation. Our results suggest that oil inventories and the convenience yield form an important link in this chain. As noted in section 5, we can take the convenience yield as an indicator of inventory e ects, since the cost of carry a ects inventories (negatively) while 24

25 shocks that depress inventory increase the convenience yield. The impulse responses are very revealing in this respect, suggesting that these are important links in the transmission mechanism. The nal column shows that an increase in interest rates reduces the oil price. It also pushes up the convenience yield. A plausible explanation is that interest rates increase the cost of carry and reduce inventories. The fall in inventories increases the convenience yield and depresses the spot price, and hence in ation, as indicated in Figure 17. To illustrate this e ect, Figure 19 shows how this impulse response changes as we set di erent parameters to zero, to shut o di erent parts of this monetary transmission mechanism. Of course, monetary policy also a ects in ation through its e ect on the output gap. Analysis of Variance These real-world dynamics are re ected in Figures 20 and 21. These report the results of the Analysis of Variance (ANOVA) exercise and show the share of the total variance attributable to the innovations at di erent lag lengths. These are also obtained using the Wald representation of the system, as described in Cochrane and Piazzesi (2009). They indicate the contribution each innovation would make to the volatility of each model variable if the error process was suddenly started (having been dormant previously). As such they re ect the variances of the shocks and well as the impulse responses. The rst column of Figure 20 shows the e ects of oil market shocks while the second shows the e ects of macro shocks. The rst two rows show that the variance of the convenience yield and oil price are is dominated by oil market shocks. Although the impulse responses show 25

26 that macro shocks have a signi cant e ect on oil prices, their relatively low variance means that these e ects are dwarfed by the e ect of the high volatility oil shocks. The variance of the oil price is naturally dominated by the its asymptote after 60 months. Nevertheless;, macro shocks account for 10% of the variance in the convenience yield after 24 months and 6% of the variance of the oil price after 12 months. The remaining rows show that oil market shocks have a signi cant e ect on the variance of output and in ation, accounting for nearly 10% of the variance of the former after 24 months and nearly 10% of the variance of the latter after 12 months. The e ects of oil shocks on interest rates are much smaller. The variances of in ation and interest rates are naturally dominated by the in ation asymptote. 3.3 The behavior of the futures market Figure 12 shows the estimates of the futures prices alongside their observed values, where these data are available. The estimates for the missing 18 and 24 month maturity data are extracted from the Kalman lter. Table 3 shows the root mean squared error of the futures prices. Table 5 reports the parameters estimated under the measure Q, and embedded in the a ne model of the term structure of futures prices (eq 32). Again, these are nicely in line with their priors. They are related to those of the KVAR by the prices of risk and variance estimates (eq 21), shown in Table (6). The signi cance of the parameter estimates under the measure Q is generally higher than those of the KVAR under P. Cochrane and Piazzesi (2009) suggest that, Q parameters are precise because they come from the cross section which has tiny measurement errors, while the P parameters come from the KVAR which has large forecast errors. 26

27 One of the key di erences between the real world and risk neutral estimates relates to the persistence parameter in the model of the underlying convenience yield. In the real world, is small and insigni cant (Table 4), which is why we call it a shock absorber. Indeed, we could eliminate it from a stand alone KVAR, since it would then add to the error terms in the equations for without disturbing the parameters. However Q is larger and signi cant (Table 6), meaning that is persistent in the risk neutral world, acting as an underlying convenience yield rather than as shock. This di erence is clearly re ected in the associated risk premium in Table (5). The behaviour of the futures curve is dictated by the factor loadings, which depend in turn upon the parameters of the risk-neutral factor dynamics (Section 2.0.1). Empirically, this system has a single a unit root under Q. that is closely associated with the underlying in ation rate (which has a unit root under P by assumption) and means that the loadings of the futures on the factors ( ) increase with maturity (). Dividing these loadings by maturity gives the factor loadings for the annualised cost of carry ( =). The loadings are depicted in Figure 7, as a function of maturity (expressed in quarters). The rst panel shows the loadings on the three latent variables and the second those on the observed variables. Figure (16) presents the model estimates of the risk premium in the futures market. The top panel shows estimates for 6 months, 2, 5 and 10 years. Recall that we only have futures data for the period since 1984, and that the estimates before then depend up inferences about the convenience yield obtained from the Kalman lter. The 5 and 10 year estimates are obtained by extrapolation. This gure indicates that the risk premiums increase with 27

28 maturity but are strongly correlated. Further analysis, based on the factor decomposition of the 2 year risk premium shows that as we would expect, oil prices play the major role. However the e ects of and largely net out, with the rst having a positive and the second a negative e ect. The net e ect is shown in the lower panel: the risk premium is high when the spot price is below the underlying, as in the mid 1970s. The contribution of is also very important, especially before and during the Volker experiment. The contribution of other factors is relatively small, although the output gap in ation have a positive e ect and helps explain some of the spikes in the risk premium. 4 Concluding remarks This paper presents a macro- nance model that includes crude oil prices and makes the crude oil futures exponential-a ne in the state variables. We use the maximum likelihood method based on the Kalman lter algorithm to estimate the model. The Kalman lter solves the severe missing observation problem. The paper also shows how the convenience yield can act as a bu er between oil prices and the real economy. It can also act as a conduit, transmitting monetary signals to the real economy, in uencing output and in ation. This model also allows variations in the oil futures term structure to be explained by latent variables as well as macro-economic indicators. It successfully captures the dynamic interaction between the oil futures market and the macro-economic system. It helps us understanding the Central Banks role in the crude oil futures market and provides a framework for policy makers to evaluate how their monetary policies can in uence the commodities 28

29 market. References Ang, A., and M. Piazzesi (2003): A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables, Journal of Monetary Economics, 50, Bodenstein, M., L. Guerrieri, and L. Kilian (2012): Monetary policy responses to oil price uctuations, IMF Economic Review, 60(4), Brennan, M. J. (1958): The Supply of Storage, The American Economic Review, 48(06), (1991): The price of convenience and the valuation of commodity contingent claims, Stochastic Models and Option Values, 200, Brennan, M. J., and E. S. Schwartz (1985): Evaluating Natural Resource Investments, The Journal of Business, 58(2), Carollo, S. (2011): Evolution of the Price of Crude Oil from the 1960s up to John Wiley & Sons, Ltd. Casassus, J., and P. Collin-Dufresne (2005): Stochastic convenience yield implied from commodity futures and interest rates, The Journal of Finance, 60(5), Chin, M., and Z. Liu (2015): A joint a ne model of commodity futures and US Treasury yields, Bank of England Working Paper. Cochrane, J. H., and M. Piazzesi (2009): Decomposing the Yield Curve, SSRN 29

30 Working Paper Series. Cortazar, G., and E. S. Schwartz (1994): The valuation of commodity contingent claims, The Journal of Derivatives, 1(4), Dewachter, H., and M. Lyrio (2006): Macro factors and the term structure of interest rates, Journal of Money, Credit and Banking, pp Dewachter, H., M. Lyrio, and K. Maes (2006): A joint model for the term structure of interest rates and the macroeconomy, Journal of Applied Econometrics, 21(4), Duffee, G. R. (2002): Term premia and interest rate forecasts in a ne models, The Journal of Finance, 57(1), Elekdag, S., R. Lalonde, D. Laxton, D. Muir, and P. Pesenti (2008): Oil price movements and the global economy: A model-based assessment, Discussion paper, National Bureau of Economic Research. Finn, M. G. (2000): Perfect competition and the e ects of energy price increases on economic activity, Journal of Money, Credit and Banking, pp Gibson, R., and E. S. Schwartz (1990): Stochastic convenience yield and the pricing of oil contingent claims, The Journal of Finance, 45(3), Hamilton, J. D. (1983): Oil and the macroeconomy since World War II, The Journal of Political Economy, pp (1985): Historical causes of postwar oil shocks and recessions, The Energy Journal, 6(1), (2003): What is an oil shock?, Journal of Econometrics, 113(2),

31 (2008): Understanding crude oil prices, Discussion paper, National Bureau of Economic Research. Hamilton, J. D., and J. C. Wu (2014): Risk premia in crude oil futures prices, Journal of International Money and Finance, 42, Heath, D. (2016): Macroeconomic factors in oil futures markets,. Herrera, A. M., and J. D. Hamilton (2001): Oil shocks and aggregate macroeconomic behavior: the role of monetary policy, Department of Economics, UCSD. Hooker, M. A. (1996): What happened to the oil price-macroeconomy relationship?, Journal of Monetary Economics, 38(2), (2002): Are oil shocks in ationary?: Asymmetric and nonlinear speci cations versus changes in regime, Journal of Money, Credit, and Banking, 34(2), Hotelling, H. (1931): The economics of exhaustible resources, The Journal of Political Economy, pp Ireland, P. N. (2007): Changes in the Federal Reserve s in ation target: causes and consequences, Journal of Money, credit and Banking, 39(8), Kaldor, N. (1939): Speculation and economic stability, The Review of Economic Studies, 7(1), Kilian, L. (2006): Not all oil price shocks are alike: Disentangling demand and supply shocks in the crude oil market,. (2008): Exogenous oil supply shocks: how big are they and how much do they matter for the US economy?, The Review of Economics and Statistics, 90(2),

32 Kozicki, S., and P. A. Tinsley (2001): Shifting endpoints in the term structure of interest rates, Journal of Monetary Economics, 47(3), Millard, S., and T. Shakir (2013): Oil shocks and the UK economy: the changing nature of shocks and impact over time, Bank of England Working Paper. Raymond, J. E., and R. W. Rich (1997): Oil and the macroeconomy: A Markov stateswitching approach, Journal of Money, Credit, and Banking, pp Schwartz, E. S. (1982): Option pricing theory and its application, The Journal of Finance, 37(2). (1997): The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging, Journal of Finance, 52(3), Sims, C. A. (1992): Interpreting the macroeconomic time series facts: The e ects of monetary policy, European Economic Review, 36(5), Smets, F. (2002): Output gap uncertainty: does it matter for the Taylor rule?, Empirical Economics, 27(1), Svensson, L. E. (1999): In ation targeting as a monetary policy rule, Journal of Monetary Economics, 43(3), Weymar, F. H. (1968): The dynamics of the world cocoa market., The dynamics of the world cocoa market. Working, H. (1933): Price relations between July and September wheat futures at Chicago since 1885, Wheat Studies, (06). (1949): The Theory of Price of Storage working, The American Economic Review, 32

33 39(06). 33

34 Online Appendix The arbitrage relationship In this section we specify the arbitrage relationships relating the futures prices to the real and nominal spot price, convenience yield and interest rate. We start with the well known property of nominal futures prices: they follow a martingale under the risk-neutral measure Q. F ;t = E Q t (F 1;t+1 ) 1: (33) This is because these contracts do not yield dividends or convenience bene ts. The maturity value of the futures price will always equal the future spot price. So for the special case of = 1: F 0;t+1 = S t+1 : Substituting this into (33): F 1;t = E Q t (S t+1 ): (34) The risk-neutral spot oil price dynamics follow by combining this with the standard arbitrage condition for a forward price. Importantly, for the special case of = 1, the interest rate is known and so this relationship also holds for the futures price: F ;t = S t e (rt s). Taking logs: 34

35 ln E Q t (S t+1 ) = s t + r t t : (35) where: s t = ln S t Finally, suppose that S t+1 is log normal under Q so that taking logs again: s t+1 = s t + t + Q s;t+1 Q s;t+1 N(0; 2 s): (36) where Q s;t+1 is a risk neutral oil price shock and: E Q t (S t+1 ) = S t e ( t s ) : (37) Taking logs and substituting into (35) gives: t = r t t s: (38) Finally, substituting this back into equation (36) gives the dynamic equation for the nominal spot price under probability measure Q: s t+1 = s t + r t t s + Q s;t+1 Q s;t+1 N(0; 2 s) (39) The macro model works with the real oil price s R t+1, which is the nominal price s t+1 less the log price level p t+1. This follows a real arbitrage relationship derived by adjusting (39) for in ation by subtracting p t+1 = p t t+1 from both sides: 35

36 t+1 = t t+1 + r t t s + Q s;t+1: (40) where t+1 = p t+1 p t is in ation: the rst di erence of the log price level. Strictly speaking this is the monthly change, which is di cult to model. so we follow mainstream macro- nance studies in approximating this by a twelfth of the annual rate. This introduces an additional error term into (40), but this is small relative to the volatility of he real oil price. The transition and measurement equations The transition equation 15) is speci ed under the measure P and incorporates the matrices: c ; c g; C = ; R = c ; c ;g 1 0 B A B C 0 c r; c r;g c r; A The matrix R does not restrict, but it rotates the latent variables t ; t and t to align them (up to additive adjustments) with t ; t and t. The nal row of R ensures that the interest rate asymptote also re ects t : Under the probability measure Q, the matrices K Q and Q in eq (17) are de ned as: 36

37 A Q = k Q k Q k Q k Q k Q k Q g k Q k Q r Q Q Q Q B Q ; = Q ; Q ; Q ; Q ; 0 0 Q ;r ; 0 Q ; Q ; 1 1 Q ; Q ;g Q ; 1 Q ;r 0 Q g; Q g; 0 Q g; Q g;g Q g; Q g;r 0 Q ; C B Q ; 0 Q ; Q ;g Q ; Q ;r C A 0 Q r; Q r; 0 Q r; Q r;g Q r; Q r;r The observed macro and futures variables are stacked in the vector yt 0 = (h 0 t; m 0 t) and related to the state vector X t by the measurement equation. The futures prices are determined by (32). The macro variables are observed without error and thus related to X t by identity. Stacking these relationships gives the measurement equation: y t = G + HX t + e t ; (41) e t N(0; Q): where: e 0 t = (& 0 t; 0 5 )G 0 = ( 0 ; 0 5;5 ); H 0 = ( 0 ; I 5 ); Q = Diag( q1 2 q2 2 ::: q 2 ; 0 5 ): 37

38 4.1 The change of probability measure The analogue of (6) under measure Q, is: z t = A Q z + B Q z z t 1 + u Q z;t; (42) Following Du ee (2002), the essential a ne setting, implies: u Q z;t = u z;t + L z D z z;t 1 ; (43) where z;t is a ne in the state variables: z;t = D z 1;z + D 1 z 2;z z t + D 1 z 3;z m t ; (44) where the price of risk parameters are de ned in the main text. We assume that 3;z = 0 3;5 to preserve the mean-independent dynamics of z t and keep the system recursive. Substituting (43) and (44) into (42) and comparing this with (6) gives: A Q z = A z L z D z D 0 z 1;z (45) B Q z = B z L z 2;z ; (46) Similarly the analogue of (13) under measure Q, is: m t = A Q m + B m;z z t 1 + B Q mm t 1 + Q m;t; (47) 38

39 De ning: Q m;t = m;t + L m D m m;t 1 m;t = D m 1;m + D 1 m 2;m m t + D 1 m 3;m z t and substituting these into (47) gives: A Q m = A m L m D m D 0 m 1;m (48) B Q m = B m L m 2;m (49) B m;z = B m;z L m 2;z;m (50) Stacking these relationships gives eqs (19) and (20) of the main text. 39

40 The Kalman lter and maximum likelihood estimation We represent expectations conditional upon the available information at time t with a hat (so that ^z t = E t (z t ); ^z jt = E t (z ); t) and de ne the following conditional covariances: P zz = E t (z t ^z t )(z t ^z t ) 0 = ^V t ; (51) P mm = E t (m t+1 ^m t+1jt )(m t+1 ^m t+1jt ) 0 = ^V t+1jt 0 + m P hh = E t (h t+1jt ^ht+1jt )(h t+1jt ^ht+1jt ) 0 = ( z + m) ^V t+1jt + m m 0 m + Q: P zh = E t (z t+1 ^z t+1jt )(h t+1 ^ht+1jt ) = ( z + m) ^V t+1jt ; P mh = E t (m t+1 ^m t+1jt )(h t+1 ^ht+1jt ) = m 0 m + ^V t+1jt ( 0 z + 0 m 0 ); P mz = E t (m t+1 ^m t+1jt )(z t+1 ^z t+1jt ) = ^V t+1jt ; where ^z t+1jt, ^m t+1jt, and h t+1jt follow from (6), (13) and (32). This allows the expectations 40

41 to be updated as: ^z t+1 = ^z t+1jt + ^V t+1 = ^V t+1jt 0 0 h t+1 m t+1 P zh P zm 1 C A : ^ht+1jt ^m t+1jt P hh P hm where : = P zh P B C A P mh P mm C A ; (52) 1 and is the Kalman gain matrix. The log likelihood function for period t + 1 is: lnl(h t+1 ) = nt 2 ln(2) 1 2 ln Det P hh P hm B AA P mh P mm P hh P hm 2 h t+1 h t+1jt m t+1 m B C A P mh P mm h t+1 m t+1 ^ht+1jt ^m t+1jt 1 C A (53) The likelihood for the full sample follows by substituting (51), (52), (6), (13) and (32) and summing over t = 0; :::T 1. It is optimized with respect to the parameters of (6), (13) and (32). 41

42 Missing observations Recall that we lack many of the futures data for the pre-1995 period. We also lack observations on the convenience yield in the pre-1985 period. We solve the rst problem by introducing indicator matrices to identify the observed data in the data set, suggested by Tsay (2005), Durbin and Koopman (2012). Let y t be the vector of observed data in period t, excluding any missing observations. Let matrix J t be an indicator matrix with the same number of rows as y t that picks these variables from the full vector y t : y t = J t y t. For the post 1995 period there are no missing observations and J t is an identity matrix. We de ne: G = J t G, H = J t H, e t = J t e t and Q = J t QJt, 0 allowing us to re-write the measurement eq (41) in a compressed form as: y t = G + H X t + e t e t N(0; Q )The likelihood function is compressed in a similar way, allowing us to estimate the model using maximum likelihood estimation with the Kalman lter even when data is missing. For example, during the period between Q to Q2 1989, the oil futures prices for 18 and 24 month maturities are missing, and the compressed measurement equation system excludes the measurement equations for these maturities: 0 h t m t 1 C A = 0 j 0 1 C A + 0 B j C B z t m t 1 C A + e t ; e t N(0; Q ): The Kalman lter automatically provides an estimate of the convenience yield in the pre period.given any estimates of the rst observation and the model parameters. In this 42

43 case the compressed measurement equation system is trivially: m t = m t : 0 m t = 0 0 I z t t m t 1 : C A where m t excludes t. Since h t+1 = ^h t+1jt the updating equations () are supplemented with by: ^z t+1 = ^z t+1jt + ^V t+1 = ^V t+1jt 0 0 h t+1 m t+1 P zh P zm 1 C A : ^ht+1jt ^m t+1jt P hh P hm where : = P zh P B C A P mh P mm C A ; (54) 1 ^ t = ^ tjt 1 + h (h t+1 ^ht+1jt ) + m (m t+1 ^m t+1jt ): where is the It generates this using eq (7) with ;t = 0: This e ectively updates the estimate of the convenience yield using the oil price and other macro shocks. Similarly, given any estimate of the rst observation, it updates ^ t over the pre-1984 period using the rst updating equation in (52): 43

44 ^ t = ^ tjt 1 + h (h t+1 ^ht+1jt ) + m (m t+1 ^m t+1jt ): However, in the absence of surprises in futures prices and the convenience yield, this only depends upon surprises in oil prices, which have a small impact Thus the estimates ^ t of are broadly constant before

45 Figures and tables 45

46 Table 1: Summary statistics Mean S.D Skew. Kurt. ADF Obs WTI futures prices F F F F F F F F Observed state variables o R;o g o o r o These data were supplied by Datasteam, and are discussed in the text. Mean denotes sample arithmetic mean, S.D. standard deviation; Skew. and Kurt. report skewness and excess kurtosis, standard measures of the third and fourth moments. Obs. reports the number of observations. ADF shows the p-statistic for the Augmented Dickey Fuller statistic testing the null hypothesis of non-stationarity. The lag length is determined by the Akaike information criterion. The critical values are -3.47, and for the 1%, 5% and 10% con dence level, based on Mackinnon (1996) one-sided p-values. 46

47 Table 2: The root mean squared errors of the estimates of the macro variables m o t g r The RMSE of the convenience yield (), interest (r) and in ation rates () are reported as annualised basis points. Table 3: The root mean squared errors of the futures prices h ;t 3m 6m 9m 12m 18m 24m The RMSE of the log futures prices (see eq (32)) are reported in basis points. Table 4: Estimates of the parameters of the real world dynamics Parameters Estimates t-stat Parameters Estimates t-stat k s;g k s s; k s;r ' g;s ' s g;g ' g g; ' g;r ' r ;s ; ;g ; ; ;s ;r ;g r;s ; r;g ;r r; s; r;r s;s This table presents the parameters of the dynamic structure eq (16) de ned under the real world measure P; with their asymptotic t-statistics. 47

48 Table 5: Estimates of the parameters of the risk-neutral dynamics Parameters Estimates t-stat Parameters Estimates t-stat k Q Q ; k Q s Q ; k Q Q ;s k Q Q ;g ks Q Q ; kg Q Q ;r k Q Q ;s kr Q Q ; Q ; Q ;s Q s ;s Q ;g Q ; Q ; Q g;s Q ;r Q g; Q r;s Q g;s Q r; Q g;g Q r;s Q g; Q r;g Q g;r Q r; Q ; Q r;r Q ;s This table presents the parameters of the dynamic structure eq (17) de ned under the risk neutral measure Q; with their asymptotic t-statistics. 48

49 Table 6: Risk premium parameter estimates Parameters Estimates t-stat Parameters Estimates t-stat 1; ;g; ; ;g;s ;s ;g;g ; ;g; ;s ;g;r ;g ;;s ; ;; ;r ;;s ; ; ;;g ;s ;s ;; ; ; ;;r ;; ;r;s ;;s ;r; ;; ;r;s ;; ;r;g ;;s ;r; ;;r ;r;r ;g;s These parameters allow for risk, connecting the parameters shown in tables (4) and (5),estimated under the measures P and Q. (See eqs (19) and (20)). Table 7: Volatility parameter estimates Parameters Estimates t-stat Parameters Estimates t-stat c ;s d s c g;s d c ;s d c ;g d s c r;s d g c r;g d c r; d r d These are the parameters speci ed in the LDL decomposition of the covariance matrix (section (1.1.4)). 49

50 Table 8: The term structure of log WTI oil futures contract This gure shows the term structure of WTI light crude oil futures taking natural logarithm, which started trading on NYMEX in The data are from Datastream. The 1, 2, 3, 6, 9 and 12 months maturities are available from Q1 1984, the 18 month contract from Q3 1989; and the 24 month contract from Q