Dynamic relationships between spot and futures prices. The case of energy and gold commodities

Resources Policy 45 (2015) 130–143

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Resources Policy journal homepage: www.elsevier.com/locate/resourpol

Dynamic relationships between spot and futures prices. The case of energy and gold commodities Mihaela Nicolau a,n, Giulio Palomba b a b

Danubius University, B-dul Galaţi 3, 800654, Galaţi Romania Università Politecnica delle Marche, Piazzale Martelli 8, 60121 Ancona, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 17 September 2014 Received in revised form 16 April 2015 Accepted 16 April 2015 Available online 15 May 2015

According to the most common financial theories, the price of a futures contract is always influenced by the spot price of its underlying asset (the cost-of-carry model) or by the expected future spot price conditional on information set (the asset-pricing theory). The aim of this paper is to analyze the dynamic relationship between spot and futures prices, and to establish if there is the possibility of a valid “period by period” prediction of the futures price conditional on the prediction of the spot price, and vice-versa. The empirical analysis is conducted on the two most important energy commodities, crude oil and natural gas, and on gold, the most important commodity used for risk hedging and investment during financial turmoil, paying particular attention to the exogeneity issue. We estimate a battery of recursive bivariate VAR models over a sample of daily spot and futures prices, ranging from January 1997 to May 2014. Our results show that some interactions between spot and futures prices clearly exist and they mainly depend on commodity type and futures contracts maturity. Thus, a strong exogeneity operates in the case of the natural gas, while this is not the case for the crude oil, where the exogeneity generally is weak and depends on the contract maturity. On the gold market the results show no possibility of a valid forecasting between spot and futures prices. & 2015 Elsevier Ltd. All rights reserved.

JEL classification: G13 C32 C58 Keywords: Energy commodities Spot and futures prices Recursive estimation Exogeneity

Introduction The price discovery role of futures contracts and the possibility they offer to reduce particular risks increase the importance of studying the futures markets and the relationship between spot and futures prices. The futures contract prices, particularly in commodity markets, transmit information to all economic agents. Producers may base their supply decisions on the futures contract prices, while physical traders might use futures contracts as a reference to price their commodities. Thus, there may be assumed that futures markets dominate spot commodity markets. There are two main financial theories about the spot-futures prices interaction, the non-arbitrage theory (cost-of-carry model) and the asset pricing theory. According to the former approach, the futures price must hold the following condition in order to avoid arbitrage opportunities: F t;τ ¼ ð1 þ r τ ÞSt  ðct;τ  kτ Þ;

ð1Þ

where F t;τ denotes the futures price of a commodity at time t for delivery at t þ τ, St is the spot price, r τ is the risk-free τ-period n

Corresponding author. Tel.: þ 40 745 803201; fax: þ 39 342 3059118. E-mail addresses: [email protected] (M. Nicolau), [email protected] (G. Palomba). http://dx.doi.org/10.1016/j.resourpol.2015.04.004 0301-4207/& 2015 Elsevier Ltd. All rights reserved.

interest rate, Pt represents the spot price at time t, ct;τ is the capitalized flow of marginal convenience yield, and kτ denotes the per-unit cost of physical storage. Cornell and French (1983) introduced the model for stocks. Later, various textbooks extended the theory also to different types of underlying assets such as commodities, non-dividend paying stocks, or currency (see, for example, Hull, 1991, 1993). The second approach, namely the asset pricing theory, establishes a relationship between the futures price and the expected future spot price conditional on an information set It, Et ðSt þ τ Þ. In this case, the futures price is a biased estimate of the future spot price, and it is given by F t;τ ¼ Et ðSt þ τ Þ  ðRτ  r τ ÞSτ ;

ð2Þ

where Rτ denotes the risk-adjusted discount rate, and ðRτ r τ Þ represents the risk premium. Both approaches are used in numerous studies in order to analyze commodity markets; among them the most representative are those presented by Fama and French (1987) or Pindyck (2001). The common way to value a futures contract is by using the cost-of-carry model, which underlines that the futures price should depend upon the current spot price and the cost of carrying or storing the underlying good from now until the delivery. Nevertheless, even if the above equations denote an explicit relationship between spot and futures, no information about the direction of

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causality between these prices is offered. This lacking is covered by different studies that use models as co-integration test, Granger causality test, Error Correction Model (ECM) or causality analysis in the frequency domain; see, for example, Bekiros and Diks (2008), or, more recently Joseph et al. (2014), Alzahrani et al. (2014), or Ding et al. (2014). The aim of this paper is to verify if there are dynamic connections between spot and futures prices of crude oil, natural gas and gold commodities, as statued by the cost-of-carry model, and to identify the direction of causality. Thus, the paper contributes to the existing literature by empirically investigating the direction of information flows between spot and futures through a recursive analysis, in which we focus on the weak exogeneity and the Granger-causality (hereafter WE and GC) issues. It is well known that the former concept is required to obtain efficient inference in conditional models, while the latter represents a test for the impact of lagged explanatory variables onto the dependent variable, easy to compute. Given that the conjunction of weak exogeneity and Granger non-causality defines the strong exogeneity between two time series (see, for instance, Engle et al., 1983 or Urbain, 1992), our analysis aims to establish if it is possible a valid “period by period” prediction of spot prices conditional on prediction of futures prices, and vice-versa. In other words, if the strong exogeneity is found, a valid conditional forecasting between spot and futures prices is possible, since one variable does not depend on the other, and thus it can be treated as “fixed”. Consequently, it can be considered that a joint analysis is required because the prediction of one variable is necessary to predict the other. The results have implications for producers, policymakers, hedgers and speculators. As mentioned at the beginning of this section, producers may base their supply decisions on the commodity futures contract prices. Producers concerned about the risk of price decreases could hedge this risk by taking a short position in futures, while consumers worried about the risk of price increases could hedge the risk by taking a long futures position (Pindyck, 2001). Identifying the direction of information flows between spot and futures prices will help both producers and consumers to establish the most appropriate hedging strategy to be adopted. Our results are important for policymakers too, as long as many scholars emphasize the role of commodity prices in improving the macroeconomic policies. Thus, energy commodities are recognized as leading indicators of inflation (see, for instance, Pecchenino, 1992, Moosa, 1998, or Browne and Cronin, 2010), but also for their role in formulating monetary policy (Awokuse and Yang, 2003), while the role of gold is to hedge against inflation and exchange rate movements, and to diversify the risk in investment portfolios (Oxford Economics, 2011). Other contributions also suggest that price changes in commodity markets precede speculators' position changes, and not vice-versa (see, for instance, Büyükşahin and Harris, 2011 for crude oil case), and thus our results could be significant for speculators, too. The remainder of this paper proceeds as follows. The section “Literature review” contains the literature review about the interactions between spot and futures prices, while the section “Empirical analysis” is dedicated to the data description, the applied methodology and the results of our analysis. Finally, the section “Concluding remarks” concludes.

Literature review The literature examining spot and futures interactions in commodity markets is fairly extensive. Many studies provide evidence for efficient futures markets and equally large number contradicts an unbiased futures price prediction interpretation

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(Serletis, 1991; Bopp and Lady, 1991; Moosa and Al-Loughani, 1994). Some of recent works are closer to our research in terms of studying a similar set of commodities (crude oil, natural gas and gold). Thus, Chinn and Coibion (2014) analyze four groups of commodities that also include crude oil, natural gas and gold. The aim of their research is to examine whether futures prices are unbiased and/or accurate predictors of subsequent prices, and results mainly show that precious metals are poor predictors of subsequent prices changes, while energy futures fair much better. They also emphasize a broad decline in the predictive content of commodity futures prices since the early 2000s. Arouri et al. (2013) also investigate the efficiency of energy and precious metal markets, by employing four linear and non-linear models. Their findings confirm that futures prices do not constitute unbiased predictors of future spot prices, although futures prices are found to be cointegrated with spot prices. Other recent works that focus on the ability of futures price to predict future changes in spot price are made by Alquist and Kilian (2010), and Alquist et al. (2013).

Crude oil Researches regarding the interactions between crude oil's spot and futures prices are numerous. Many of them analyze the price of West Texas Intermediate (WTI), while others study the price of Brent and Dubai crude oil. Studies with similar set of crude oil spot and futures prices are Bekiros and Diks (2008), Lee and Zeng (2011), Liu and Wan (2011) or Nicolau (2012). Bekiros and Diks (2008) investigate the causal linkages between daily spot and futures prices for maturities of one, two, three and four months of WTI crude oil over two periods: October 1991–October 1999 and November 1999–October 2007. The conventional linear GC test and a new nonparametric test for nonlinear causality after controlling for cointegration are applied. Their results show that the linear causal relationships disappear after cointegration filtering while, in some cases, causal relationships are found in both periods after using a GARCH filtering. This indicates that spot and futures returns may exhibit asymmetric GARCH effects and statistically significant higher order conditional moments. The results imply that, if nonlinear effects are accounted for, neither market leads or lags the other consistently. Lee and Zeng (2011) use the same time series over a period ranging from January 2, 1986 to July 6, 2009. They test causalities between spot and futures oil prices with linear, nonlinear, and quantile methods. Linear and nonlinear methods present bidirectional Granger causalities, while the quantile method shows different results. In this case, spot oil prices indeed cause futures oil prices, but the causality goes in the opposite direction only for one-month futures. The same approach on newer weekly data regarding net long positions and the spot prices of WTI is used by Ding et al. (2014). Their results show that excessive financial speculation in the crude oil futures market destabilizes the spot price of crude oil, and the futures price can help explain the behavior of the spot price oil. The analysis conducted by Nicolau (2012) covers the period 1999/01/01–2012/07/29. In this study a GC test on full sample and two subsamples (before and during the actual crisis) is carried to determine the direction of information flows between spot and futures oil markets. The paper concludes that the financial literature according to which futures prices predict spot prices does not apply for all types of WTI futures contracts. Only futures with shorter delivery dates influence West Texas Intermediate (WTI) spot prices.

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Lean et al. (2010) analyze the market efficiency of crude oil spot and futures, by using two approaches, mean-variance (MV) and stochastic dominance (SD). This contribution uses WTI crude oil data from 1989 to 2008 and the results show no evidence of any MV and SD relationships between crude oil spot and futures indices. Thus, they prove that there is no arbitrage opportunity between crude oil spot and crude oil futures markets, spot and futures do not nominate one another, investors are indifferent to investing spot or futures, and the spot and futures crude oil markets are efficient and rational. More recent studies use GARCH models as methodology (Arouri et al., 2012; Chang et al., 2011; Bu, 2011), or wavelet based tests (Alzahrani et al., 2014). Natural gas market The interactions between natural gas spot and futures prices are analyzed in a less extensive number of studies, and rarely the analysis is conducted exclusively in the natural gas market. In this field of investigation, Herbert (1993) is one of the first who examines the relationship between the spot price for natural gas for a delivery month and the futures contract price for the same delivery month by estimating a regression equation. The results provide a good summary of the relationship between spot and futures prices for the time period, and it appears that the natural gas futures market is inefficient. This is consistent with the results of Chinn and Coibion (2014) or Arouri et al. (2013). Later, Herbert (1995) points out several distinguishing features of the relatively new, but interesting and very volatile short-term futures and cash markets for natural gas. His results show the importance of trading volumes in influencing price volatility. Modjtahedi and Movassagh (2005) analyze exclusively the natural gas futures, but among their results it is also presented that futures prices are less than expected future spot prices, so that futures are backwardated. Buchananan et al. (2001) realize an attempt to predict the direction of natural gas spot price movements using trader positions. More recent studies regarding the interactions spot-futures in natural gas markets analyze the hedging effectiveness on futures when changes in spot prices are partially predictable (see Ederington and Salas, 2008), or emphasize the economic factors that influence prices of spot and contracts that use natural gas as underlying asset (Cartea and Williams, 2008, or Chiou-Wei et al., 2014). Other researches analyze the correlation between futures prices of natural gas and other commodities (e.g. Tonn and McCarty, 2010, for natural gas–crude oil interactions), or use natural gas futures markets data in order to develop models that could improve hedging and forecasting performance (Root and Lien, 2003; Mu, 2007; Park et al., 2008; Kao and Wan, 2009). Gold market The metal markets are frequently analyzed in the literature, but gold market has received the most attention from academic researches. The dynamic properties of gold spot and futures prices have been investigated, among others, by Ball et al. (1985), Bertus and Stanhouse (2001) and Hammoudeh et al. (2011). Gold is considered an important tool for risk hedging as well as investment avenue, therefore predicting its price has become very important to investors. A range of different and complex methods used in this respect are mentioned in the literature.1 1 Shafiee and Topal (2010), for example, present a modified econometric version of the long-term trend reverting jump and deep diffusion model for

There are many studies regarding the relationship between macroeconomic indicators and gold price dynamics, emphasizing the correlation between inflation and gold price. Thus, Tully and Lucey (2007) investigate macroeconomic influences on gold via an asymmetric power GARCH (APGARCH) model. Using gold cash and futures data over a long period (1983–2003), their results confirm that the US dollar is the main, and in many cases the sole, macroeconomic variable which influences gold. The influence of macroeconomic variables on the volatility of gold market is also met in the financial literature. Batten et al. (2010), for example, model the monthly price volatilities of four precious metals (gold, silver, platinum and palladium) prices and investigate the macroeconomic determinants of these volatilities. According to their results, gold volatility is shown to be explained by monetary variables. The intensity direction and the speed of impact of US macroeconomic news announcements on the return, volatility and trading volume of gold are examined also by Elder et al. (2012). Their results show that announcements regarding an unexpected improvement in the economy tend to have a negative impact on gold price, and the volatility is positively influenced by economic news. As in the case of natural gas, gold spot and futures are frequently used in models designed to improve hedging and forecasting performance (see Chinn and Coibion, 2014; Arouri et al., 2013).

Empirical analysis Data description As long as, according to the non-arbitrage and asset pricing theories, the relationship between spot and futures prices explicitly exists, the aim of our work is to identify the dynamic relationship between spot and futures prices in three important commodity markets, crude oil, natural gas and gold, by employing a recursive analysis. We prefer commodity markets because they are more tied to current economic conditions, in contrast to more traditional financial assets. Investors, speculators, hedgers and policymakers look at the commodity markets as alternative investment instrument for hedging against risk in equity markets. Crude oil, natural gas and gold are particularly desirable asset classes for international portfolio diversification due to their different volatile returns and generally low correlations with stocks (Arouri and Nguyen, 2010; Daskalaki and Skiadopoulos, 2011; Hammoudeh et al., 2013). At the same time, the financial literature has proven the inflationary role of crude oil price and the feature of gold investments as hedging tool against inflation. The natural gas commodity has become an increasingly valuable resource. Due to its low environmental impact and to rapid growth in Liquefied Natural Gas (LNG) use, the consumption of natural gas is expected to increase at global level. The available sample is very large and contains daily observations ranging from 1997/01/07 to 2014/05/30 (T ¼4539). The starting date corresponds to that from which the time series for natural gas spot price are available. All prices are expressed in US dollars. In our analysis, we use the logarithmic transformation, so that the first differences can be interpreted as market returns. (footnote continued) forecasting natural-resource commodity price. Their study validates the model and estimates the gold price for the next 10 years, based on monthly historical data of nominal gold price. Pursuing the same goal, Zhou et al. (2012) propose an improved empirical mode decomposition model, and their experiment results show that the proposed system has a good forecasting performance.

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The spot prices used in the analysis are the daily London PM Fix for gold, and the closing spot prices of West Texas Intermediate (WTI) crude oil and Henry Hub natural gas. The sources for these data are the Energy Information Administration (EIA) for WTI and Henry Hub, and the Thomson Reuters Datastream for gold. The futures prices of crude oil and natural gas are the official closing prices at 2:30 p.m. from the trading floor of the NYMEX. We use four types of time series regarding these futures contracts, denoted F1, F2, F3 and F4. F1 represents a futures contract specifying the earliest delivery date, while F2, F3 and F4 represent the successive delivery months. For crude oil, each futures contract expires on the third business day prior to the 25th calendar day of the month preceding the delivery month, while natural gas contract expires three business days prior to the first calendar day of the delivery month.2 The source of data for crude oil and natural gas futures contracts is the Energy Information Administration. With regard to the gold futures price, we use the price of the nearby contract quoted on COMEX, as offered by the Thomson Reuters Datastream. Fig. 1 illustrates the available time series plots. Generally two patterns emerge. First, all three graphics reveal that the time series appear to be cointegrated. Second, all futures markets exhibit strong contango condition during the entire sample period, especially for F3 and F4 contracts. The contango position is normal for our futures contracts due to their underlying assets: crude oil, natural gas and gold are non-perishable commodities, so they have a cost-of-carry Fig. 1 (a-c). The continuing decline in the crude oil prices after the mid of 2008 year reflects the decelerating economies across the globe and an attendant slump in oil demand. Although the OPEC have cut its official production starting september 2008, the decline continued until the second quarter of 2009, when the prices rose moderately due to more favorable economic news (EIA, 2008; EIA, 2009). The natural gas prices usually follow almost the same pattern of crude oil prices. This confirms the economic theory according to which the prices of crude oil and natural gas are related because these commodities are substitutes in consumption and also complements in production. It should be noted, however, the evolution of natural gas time series before the crisis when, for very short periods of time, both spot and futures prices reached high levels. The most important is the spike in spot price from 2003/02/25. According to the EIA reports, the high price level is due to an Arctic blast of cold, combined with strong space-heating demand and limited supply alternative,3 being well known that natural gas prices are highly sensitive to weather swings and supply disruptions (Mastrangelo, 2007). Spikes in natural gas spot prices are also noticed in February and March of 2014. These price risings coincide with the socio-political events that took place in Ukraine,4 and with the Henry Hub natural gas demand variation caused by waves of unexpected cold weather.5 The continuous increasing in gold price, starting the end of 2007, confirms the importance of gold as hedging tool during economic turmoil.

2 Sources: http://www.eia.gov/dnav/pet/TblDefs/pet_pri_fut_tbldef2.asp and http://www.eia.gov/dnav/ng/TblDefs/ng_pri_fut_tbldef2.asp. 3 Source: http://www.eia.gov/naturalgas/weekly/archive/2003/02_27/ngup date.asp. 4 Due to its geographic position, Ukraine is an important transition country for natural gas and petroleum liquid provided by Russia. According to the EIA, approximately 3.0 trillion cubic feet of natural gas flowed through Ukraine in 2013 to European Union's countries, Moldavia and Turkey. Source: http://www.eia. gov/countries/country-data.cfm?fips=up. 5 According to the EIA Natural Gas weekly updates. Source: http://www.eia. gov/naturalgas/weekly/archive/2014/02_06/index.cfm and http://www.eia.gov/nat uralgas/weekly/archive/2014/03_06/index.cfm.

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a

b

c

Fig. 1. Time series plots (in logarithms). (a) Crude oil. (b) Natural gas. (c) Gold.

Methodology In order to study the dynamic relationships between spot and futures prices, we use a recursive analysis to detect possible structural breaks. In the financial literature, a common technique to assess the model stability over time is to compute parameter estimates over a rolling window of a fixed size through the sample (see, for example, Hernandez and Torero, 2010). Although, due to the large size of our available sample, we opt for the recursive estimation for the following reasons. First, with this method there is no need to choose the optimal window size.6 Second, the rolling window has always a fixed size in each estimation period, because the subsample is updated by

6 It is well known that, in the rolling estimation, the use of different window sizes may lead to different empirical results. The determination of the optimal window size in a rolling estimation has been widely discussed in literature. See, for example, Granger and Newbold (1986), West and McCracken (1998), Pesaran and Timmermann (2007), Inoue and Rossi (2010) or Giacomini and Rossi (2013).

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dropping the first observation, while another observation is added at the end. This mechanism of excluding the first observations determines a loss of information. Therefore, we decide to use the recursive estimation that preserves such information by enlarging the sample size at each iteration, though the estimation performs better. We consider this method more useful to progressively account for the effects of the actual financial crisis. In this paper we estimate a battery of bivariate VAR(q) models in which the first variable is the logarithm of the spot price (St) and the second is the logarithm of a futures price (Ft). Practically, we carry out 9 sequences of estimations, 4 for crude oil and natural gas commodities (spot versus F1, F2, F3 and F4), and a unique sequence spot-F1 gold. The general setup is AðLÞxt ¼ dt þ εt

)

xi;t ¼ di;t þ x~ 0i;t  1 aii;t þ x~ 0j;t  1 aij;t þ εi;t ;

ð3Þ

where dt is a time-trend polynomial, xt ¼ ½St F t 0 , i ¼ 1  j is a selection index (i¼ 1,0) for which St ¼ x0;t and F t ¼ x1;t , the vector x~ i;t  1 ¼ ½xi;t  1 … xi;t  q 0 contains q lagged variables and εt is a bivariate white noise process. It is worth noting that the model parameters aii;t and aij;t are time-varying, since they are recursively computed; then, we carry out the recursive WE and GC analyses, and we repeat the procedure throughout the sample. Let T be the total amount of our sample observations, for t ¼ τ; τ þ1; τ þ 2; …; T, where τ represents the starting window; all the estimations are computed starting from the initial window τ ¼ 3005, which corresponds to the period that goes from 1997/01/ 07 to 2008/07/14. The choice of the initial window period is motivated by the fact that on July 14, 2008 the financial authorities stepped into assist America's two largest lenders, Fannie Mae and Freddie Mac, owners or guarantors of about 5 trillion worth of mortgages and home loans.7 We suspect that this information caused significant reactions on financial markets. Fig. 1(a) and (b) clearly shows that the natural gas and crude oil spot and futures prices are rapidly declining after that date, respectively. The whole procedure runs a series of T  τ þ 1 ¼ 1535 iterations and, at each iteration, it displays the WE and GC test statistics and the related critical values taken from the conventional χ 2q distribution. Since it is well known that in sequential tests the conventional critical values lead to size distortions by rejecting the null hypothesis with asymptotic probability close to one, at each estimation period we calculate the Inoue and Rossi (2005) critical values, which provide a convenient solution to this problem. Once a confidence level α is given, we reject H0 when the test statistic is greater than such critical values. Before carrying out the above mentioned tests, it is necessary to specify an appropriate model at each iteration. Since in our analysis we have to carry out T  τ þ 1 estimations for each bivariate VAR, we need to specify a total amount of 9  1351 ¼ 13; 815 models. Therefore, an automatic procedure is required. Accordingly, we use an algorithm that, for each couple St–Ft and for all estimation periods, proceeds as follows: 1. initially, we determine the optimal VAR lag-length (q). The procedure consists of minimizing the Akaike (AIC), the Bayesian (BIC) and the Hannan–Quinn (HQC) information criteria for each bivariate VAR in levels. Even if, according to the Efficient Markets Theory, excess returns cannot be predicted by using strategies based on historical prices, we try to improve the statistical accuracy of our models by allowing the current prices to depend on the past information.8 Accordingly, we assume that qmax ¼ 5 is the maximum value for lags. This value 7

Source: http://www.economist.com/node/16640249. See, for example, Chapter 2 of Campbell et al. (1997) for a detailed discussion on this topic. 8

corresponds to the workweek, therefore it can be reasonable for daily financial data; on the other hand, we consider spurious all the correlations of higher order. As it is widely known, the three information criteria can return the same result or not. When they provide the same result, the corresponding lag is automatically selected; otherwise, the selection of lags is made by carrying out the Ljung and Box (1978) test residuals for autocorrelation (hereafter AC) from order 1 to order 5. The procedure is simple: for each lag a bivariate model is estimated and it is accompanied by 10 values of the AC tests (5 tests for each couple of variables). The selected lag is that of the model with the minimum number of rejections. If the maximum number of rejections is obtained for two different lag specifications, the minimum lag order is selected in order to reach the most appropriate model parsimony; 2. the optimum lag selected in the previous step is crucial for carrying out the Johansen (1991, 1996) test for cointegration. Checking for cointegration is extremely important in establishing which VAR to estimate. The cointegration rank (r) is crucial. If r¼0, the spot and futures time series are two independent Ið1Þ processes (integrated of order one), so we need to estimate a bivariate VAR in differences. The cointegration rank r¼2 indicates that there are two Ið0Þ processes (stationary or integrated of order zero), thus a VAR in levels is required. In the case of our analysis, the cointegration rank r¼ 1 seems to be the most appropriate, as long as the time series concerning the same commodity follow common trends (see Fig. 1). Under this condition, the following stationary VECM model

ΘðLÞΔxt ¼ dt þ γ t β0t xt  1 þ εt ) Δxi;t ¼ di;t þ γ i;t ðxi;t  1 þ δt xj;t  1 Þ þ Δx~ 0i;t  1 θii;t þ Δx~ 0j;t  1 θij;t þ εi;t ð4Þ is derived from Eq. (3). For convenience, the cointegrating vector 0 at time t may be normalized as βt ¼ ½1 δt  without implicating any loss of generality, and γt is the corresponding vector of loading factors; Δx~ i;t  1 and Δx~ j;t  1 are (q  1)-dimensional vectors containing the first differences, while θii and θij are the vectors of time-varying estimated coefficients. Using Eq. (4), the tests for WE and GC can easily be performed. On the one hand, testing recursively the hypothesis H 0 : γ i;t ¼ 0 makes it possible to assess the spot-futures causality issue focusing on the direction of error correction. As statued by Johansen (1991) and Urbain (1992), among others, in the ECM models this condition is sufficient for weak exogeneity of xi;t for βi;t . On the other hand, the recursive GC procedure applies to sample observations in the window ½1; t by using the Toda and Yamamoto (1995) (see also Dolado and Lütkepohl, 1996) method. This technique consists of estimating an augmented bivariate VAR(q þ 1) in levels, then testing the GC simply via Eq. ð1Þ ðqÞ (5) by imposing H 0 : θj;t ¼ … ¼ θj;t ¼ 0, where the superscript indicates the single element of vector θj;t . In practice, this consists of the recursive Wald test statistic 0 1 W j;t ¼ θ^ j;t Σ^ j;t θ^ j;t

with j ¼ 0; 1;

ð5Þ

where, at each iteration, the estimated covariance matrix of θj;t is

Σ^ j;t ¼

t 1X x~ x~ 0 : t k ¼ 1 j;k j;k

ð6Þ

The vector x~ j;k is already defined in Eq. (3). Eq. (6) can be easily computed by multiplying by ðt  2q 1Þ=t the covariance obtained via the OLS. For each t, the null hypothesis is H 0 : θj;t ¼ 0. The deterministic terms we adopt in the recursive estimation depend on the commodity. From a graphical inspection of Fig. 1

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the “restricted constant” case seems the more appropriate for natural gas, while the “unrestricted constant” is used for crude oil and gold.9 3. finally, for each recursively estimated bivariate VAR, two test statistics for WE and two test statistics for GC are provided. At this step, two different critical values are calculated: the first is taken from the standard χ 2q distribution and the second, suggested by Inoue and Rossi (2005), is   t ; ð7Þ κ q;t ¼ c2q;α þ q ln

τ

where α is the desired significance level and q is the number of restrictions imposed by H0. The test for WE always imposes one parameter restriction, therefore q¼ 1. The values of cq;α are taken from Table 1 of Inoue and Rossi (2005). Moreover, at each iteration we calculate some measures of forecast accuracy to assess the predictive properties of our recursive model. Accordingly, we use the RMSE, the Theil (1966)'s U statistic and the Diebold and Mariano (1995) test. Preliminary analysis Before carrying out the method previously described, we make a preliminary analysis to confirm the necessity of a recursive method. Tests for structural instability It is well known that the recursive analysis may be redundant when there is no evidence of structural instability. In order to avoid this issue, we test for the presence of some beaks in our sample by using two separate univariate ECM equations for Δxi;t , with i ¼0,1 and εi;t  Nð0; σ 2i Þ. Several approaches have been tried on this issue and the results are provided in the Appendix. In Table 2 some traditional tests are carried out. After setting 2008/07/14 as the split date, the Chow (1960) test always rejects the null, while the QLR test for a structural break at an unknown point in time (Quandt, 1960) documents that the change in regime often occurs after that date. On the other hand, the CUSUM test (Harvey and Collier, 1977) provides no evidence in favor of some parameter instability. An alternative approach is the Bai and Perron (2003) model. In this context, Eq. (4) is used as pure structural change model, therefore all the parameters can vary over m þ 1 different subsamples, where m is the number of estimated breaks. The lag specification is obtained by using the BIC, HQC and LWZ (Liu et al., 1997) information criteria,10 while the amount of breaks is determined via a sequential application of the supF t ðℓ þ 1j ℓÞ test. Table 3 provides the estimated dates of break by using two separate columns for breaks occurring before and after 2008/07/14. Excluding F1, F2 and F3 for the natural gas, at least one structural change seems to occur for all the other time series starting from the summer of 2008; this result is also confirmed by the tests UDmax, WDmax and sup F T ðkÞ which are always significant. Finally, the parameter stability test introduced by Hansen (1992) and the cointegration stability test proposed by Hansen and Johansen (1999) have been tried. The results are in Table 4. The former test is presented in both forms, the individual parameter stability test (Lr with r ¼ 1; 2; …; 2 þ 2q) and the joint parameter stability test (Lc), while the latter consists of the fluctuation tests on the individual eigenvalues supτðtÞ T and the 9 More precisely, both restricted/unrestricted constant cases have been tried and then compared for all the commodities. The preliminary estimates are available upon request from the authors. 10 Bai and Perron (2003) suggest to use the BIC and the LWZ, since AIC performs very badly.

135 0

fluctuation of beta coefficient (β-stability test, with β ¼ ½1 δ). The results show that some instability occurs for all time series.

‘One shot’ analysis The analysis carried out in this section consists of two ‘one shot’ estimations, using the starting sample (the so-called pre-crisis period) and the entire sample. These scenarios correspond to the first and the last iterations of the algorithm proposed in the section “Methodology”. The results are presented in Table 1 in the Appendix. The estimation results reveal generally that, in the case of crude oil and natural gas, Ft is weak exogenous for the cointegrating vector β for each bivariate VAR, independent of futures maturity or estimation period. As pointed out by Urbain (1992), this means that no useful information is lost conditioning the model on futures variables without specifying their generating process. The only exception appears for the long-run relationship spot-F1 for crude oil in the pre-crisis analysis that rejects the weak exogeneity. The weak exogeneity is always rejected in the case of futures gold market, for both periods, and in the case of the spot time series for all commodities. The ‘one-shot’ estimations also highlight that the three examined commodity markets have different dynamics. In crude oil market, for example, the Granger-causality goes always both-ways, independent of the sample length. So, the price of futures contracts conveys information about expectations of market participants concerning the spot price at the maturity date, and, vice-versa, the spot price conveys informations about the futures price. This allows us to conclude about the absence of any strong exogeneity between spot and futures prices. Thus a valid forecasting between spot and futures prices in crude oil market is not possible since spot variable depends on futures variable and vice-versa. Moreover, this result is not surprising giving the price discovery role of futures market, and as such, its influences on spot oil price. With regard to the natural gas market, the weak exogeneity of the futures prices is always accompanied by the Granger non-causality from spot to futures, independent of the maturity and the estimation period. In this commodity market the cointegrating vector does not appear in the equation of futures prices, and the additional restriction “St does not Granger-cause Ft” operates, consequently the strong exogeneity seems to hold. From the economic perspective this result suggests that the multi-step ahead predictions of spot prices conditional on futures prices are valid, since Ft is not dependent on any feedback of St. In other words, the conditional variable Ft accounts only for its own lags, therefore it can be treated as “fixed” for predictions of spot prices. As mentioned before, in the gold market the tests for weak exogeneity always reject the null and the Granger-causality is bidirectional. In this context, the conditions for strong exogeneity are never satisfied. The conclusion is straightforward: the prediction of one variable is necessary in order to have a conditional forecasting about the other, therefore a joint analysis is always required. As in the case of crude oil market, there is no possibility in gold market of a valid forecasting between spot and futures prices. The results presented in Table 1 are consistent with the statistical property according to which Granger-causality must exist in at least one direction when two variables are cointegrated (see, among others, Miller and Russek, 1990). Moreover, the sample size is sufficiently large to overcome the problem of low power that generally affects the results provided by the Toda and Yamamoto (1995) approach. Anyway, these are preliminary results obtained via a ‘one shot’ estimation with conventional χ 2q critical values, therefore they can suffer from some data contaminations that could lead to a misleading inference. Specifically, when the available time series are possibly affected by some outliers or breaks, the OLS method returns biased estimates. In this situation, the necessity of estimating time series parameters more

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robustly arises and this is the reason why we use the recursive analysis in the next section.

Recursive analysis As already claimed, we carry out a recursive analysis using the the algorithm proposed in the section “Methodology” when the number of observations augments from τ ¼ 2008=07=14 to T ¼ 2014=05=30 with a one-day periodicity. Figs. 2–4 show the results of our analysis. The values of the recursive WE and GC are plotted in bold, the horizontal grey line indicates the χ 2q critical values, and the increasing black line corresponds to the Inoue and Rossi (2005) critical values κ q;t calculated according to Eq. (7). All the results are sustained by good predictive ability of the estimated models. Thus, at each iteration t ¼ τ þ 1; τ þ 2; …; T, the RMSE of all bivariate VAR models exhibits very small values and the Theil (1966)'s U statistic is always less than one. Moreover, the Diebold and Mariano (1995) test carries a similar message by accepting the null hypothesis of a good forecast accuracy.11 In general, the plots confirm all the results already obtained in “One shot analysis” presented in the section “‘One shot’ analysis”. The conclusions made exploiting the full sample are substantially the same as those obtained step-by-step via the recursive analysis, hence the estimated relationships in terms of WE and GC appear stable against time. Fig. 2 shows that the sequences of recursive WE and GC analysis for the crude oil market evidences instability at the beginning of the analyzed period. This issue does not depend on the use of the recursive method, but it is caused by the economic environment. On the one hand, the instability in financial markets played a key role. In fact, the announcement regarding Fannie Mae and Freddie Mac on 14th of July 2008 coincides with the starting point of the continuously decline of crude oil price. On the other hand, oil price is subject of external influences such as economic growth, OPEC behavior, geo-political events and hurricanes. The WTI crude oil price was quite unsettled during July 2008, registering both a new record high and the largest one-month decline in several years. According to the EIA, the declines were primarily driven by demand and supply issues due to the previous very high prices that affected the consumption of oil products. Although the demand for crude oil decreases continuously by the end of 2009 (according to IEA, 2010), a further increase in oil price is recorded at the end of September 2008, during one week. This growth, unjustified by the decreasing demand, could be explained in the context of crude oil seen as alternative investment instrument.12 The economic recovery, although anemic, has driven to stability from 2010 onwards, as also seen in Fig. 2. The recursive analysis reveals accurately the crude oil market's behavior under the influence of economic environment described above. Thus, the weak exogeneity test for F1 rejects the null before the end of September 2008 (more precisely: 2008/09/23), and accepts it always after this date. Consistently with the estimated break dates provided in Table 3, the test statistics plotted in Fig. 2 seem to follow two separate regimes before and after the beginning of 2009 year. This evidence is documented by some visible spikes in the recursive analysis regarding the weak exogeneity of St, and by a sort of jump illustrated in the GC plots. All the test results 11

These estimates are available upon request from the authors. The new spike in crude oil prices from September 2008 overlaps to the announcement that Lehman Brothers filed for the Chapter 11 bankruptcy protection and sold the core business to Barclays (15–22 September, 2008). After the announcement, the main stock indexes (e.g. Dow Jones industrial average, or S&P's 500-stock index) registered significant drops, so it is very possible that market participants looked for hedging on commodity markets, in our case the crude oil market. 12

become stable from 2010 onwards, which coincides with the beginning with economic recovery. The extent to which the initial instability affects the results in terms of WE and GC depends on which quantiles are used to analyze the results, the conventional χ 2q quantiles or the Inoue and Rossi critical values. Thus, when the χ 2q quantiles are used, the initial instability does not affect the results in terms of WE and GC, except in some isolated cases. Conversely, when Inoue and Rossi critical values are used, different conclusions arise with regard to the couples spot-F1 and spot-F4. In the former case the strong exogeneity of F1 is substantially not rejected over the period 2009/03/11–2012/10/10. From the economic perspective, this confirms the possibility of a valid multistep ahead prediction of crude oil spot price conditional on one month maturity crude oil futures price. When we use the Inoue and Rossi critical values, three scenarios arise: (a) from the starting date to 2009/01/06 the weakly exogeneity of spot price is not rejected, but F4 clearly Granger-causes spot; (b) in the subperiod ranging from 2009/01/07 to 2009/07/22 the analysis reveals the strong exogeneity of St; (c) from 2009/07/23 onwards the spot price dynamics are not affected by the past of F4, but they depend on the cointegration relation. Thus, in this case the dynamic relationships between spot and futures prices in crude oil markets vary over time and depend on contract maturity. However, as we stressed in section “Methodology”, the use of Inoue and Rossi critical values in recursive analysis is more appropriate than the use of standard χ 2q , therefore we consider the later results more significant. Regarding the natural gas market (see Fig. 3), the results are stable over time. According to our analysis, the financial crisis and the geopolitical instability from the East of Europe (Russia-Ukraine conflict) do not seem to produce significant effects on the spotfutures dynamic relationships. Moreover, from the statistical perspective, the model lag order is constant (q ¼4), except for the couple spot-F1 for which it augments from 4 to 5 during the period 2010/09/22–2014/02/11. In this case, the strong exogeneity holds between spot and futures prices because the conjunction of weak exogeneity and Granger non-causality is always met for Ft, independent of contract maturity and estimation period. Focusing on the gold commodity, the entire sequence of estimations confirms the ‘one shot’ results: endogeneity between spot and futures prices holds over time. Fig. 4 clearly shows that, with a size of α ¼ 0:05, the direction of causality goes from futures to spot and vice-versa, independent of the critical value we use. Moreover, the statistical features of the recursive VAR models remain almost unchanged over the estimation periods. Specifically, the AIC, BIC and HQC jointly suggest the lag order q¼ 2 until the beginning of october 2013. From this period onwards, the information criteria come into clash and often the best selected lag is q¼ 1. This sort of turbulence in lag specification produces some effects only when the weak exogeneity of the futures price is investigated by using the Inoue and Rossi (2005) critical values.

Concluding remarks The lower diversification benefits from equity investments during financial crises, justifies the interest of investors, hedgers and speculators in considering spot and futures commodity markets as alternative investment instruments for hedging against risk in equity markets. Furthermore, the lately increasing positive correlation between equity and commoditity markets puts emphasis on the spot-futures price dynamics in commodity markets, in order to minimize the risks that could occur also inside of these markets. In this paper we aim to analyze the relationships between spot and futures prices in commodity markets. To do so, we propose a battery of recursive VAR models between spot and futures prices of the crude oil, natural gas and gold commodities, focusing on the

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weak exogeneity and Granger-causality issues. The entire study is conducted over a sample which ranges from January 7, 1997 to May 30, 2014. The first estimation period, or pre-crisis period, consists of all the sample observations until July 14, 2008, the day in which the collapse of Fannie Mae and Freddie Mac was announced. The recursive analysis starts from this event onwards, so that the possible effects produced by the actual crisis are taken into account. We propose an algorithm that, in all estimation periods, determines the more appropriate lag order and the cointegration rank for each model, and then carries out the tests for weak exogeneity and Granger-causality. Besides, some measures of forecast accuracy are also computed in order to assess the predictive properties of all the estimated models. The main result of our approach is that the dynamic interactions between spot and futures prices substantially depend on commodity market. In practice, we do not find an univocal relationship that might be universally valid, because each commodity market and each couple spot-futures we examined present its own peculiarities. Spot and future prices are always cointegrated and this is not surprising, since the time series share a common path. In general, all the statistical features such as the number of the optimal lags or the results of the autocorrelation tests do not vary significantly over time, especially when the sample is sufficiently large. Focusing on weak exogeneity and Granger-causality, we also found that the causal relationships between spot and futures prices are substantially stable over the entire estimation sample. In particular, the recursive estimation results are the same as those obtained via the ‘one shot estimation’ conducted over the entire sample, although a preliminary analysis concluded in favor of some breaks in time series. Only for the oil commodity a sort of instability is visible in the initial estimation period. This is consistent to the uncertainty of financial markets and economic environment during the period July 2008–November 2009. The different results obtained for the crude oil confirm that this market is more sensitive to the behavior of market participants when economic and financial announcements are presented. Consequently, we can assess that, among the commodity markets we analyze, the crude oil is one of the most used for hedging and speculations during financial turmoil, and the market participants are not anymore indifferent in investing spot or future or in choosing the commodity market for further investments. Despite the very high co-movement between natural gas and crude oil prices, the direction of causality between spot and futures in these markets is not the same since weak exogeneity of futures prices is not rejected in both markets. The Granger-causality generally operates in both directions for the crude oil commodity, while a Granger non-causality of spot price on futures prices is found for natural gas. This implies a strong exogeneity (see, for instance, Engle et al., 1983 or Urbain, 1992) independent of contract maturity. Our results are consistent with previous studies regarding the predictive content of futures contract for one commodity, which emphasize the ability of the natural gas futures to predict subsequent prices, while crude oil does it relatively worst (Chinn and Coibion, 2014). However, in our opinion, the natural gas market will become more instable during the next future period, considering the last European events13 and knowing that natural gas markets are globalizing (see, for example, Cronshaw et al., 2008). With regard to the gold market the results of our recursive analysis show that there is no possibility of a valid forecasting between spot and futures prices.

13 We recall here the confrontation between Russia and Ukraine, that has shaken the forecasts for the European natural gas markets, the consequences of the economic constraints imposed to Russia by the EU, as well as the Russian measures with regard to its international trade with energy products, particularly with countries from the Baltic region and south-eastern Europe. This argument is well analyzed in Dickel et al. (2014).

137

Both producers and consumers could consider all our results in setting their hedging strategies. Additionally, our results are important for policymakers. This is due to the role that commodity prices have in formulating monetary policy and in indicating the inflation pressure, as mentioned in Pecchenino (1992), Moosa (1998), Browne and Cronin (2010), and Awokuse and Yang (2003). Since the start of economic downturn, it is often noticed the poor record of commodity futures markets in forecasting the course of price rises (Bernanke, 2008; Chinn and Coibion, 2014). It is also well known that for the surge in spot price of oil during 2003–2008 has to be blamed the influx of financial investors into oil futures market, and not the change in economic environment. Therefore, analyzing the directionality between spot and futures prices, in order to understand if the information offered by commodity futures markets is still useful, is more important than ever. Accordingly, adding some exogenous regressors (for example, daily economic variables or specific dummy variables) could improve our approach. An analysis on the dynamics of spot and futures daily returns taking into account changes in volatility would be also useful. These topics represent fields of investigation for our further research.

Acknowledgements We thank, without implicating them for errors, Allin Cottrell, Peter Summers, Ilaria Traini and all the Gretl Conference 2013 participants for their useful and constructive comments. Appendix See Figs. 2–4 and Tables 1–4.

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Fig. 2. Crude oil.

Fig. 3. Natural gas.

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139

Fig. 4. Gold.

Table 1 ‘One shot’ estimations. Bivariate VAR

Lags

IC

Coint. rank

AC tests (reject)

Weak exogeneity

GC

Ft

St

St -F t

F t -St

Pre-crisis analysis (from 1997/01/07 to 2008/07/14, T ¼3005 observations) Crude oil: spot-F1 1 BIC 1

10/10

14:403

79:377

15:301

86:479

Crude oil: spot-F2

4

BIC

1

10/10

0:0049

16:716

25:675

49:980

Crude oil: spot-F3

4

BIC-HQC

1

10/10

0:0001

9:1089

18:816

32:922

Crude oil: spot-F4

4

BIC-HQC

1

10/10

0:0907

6:3580

18:949

23:648

Nat. gas: spot-F1

4

BIC

1

10/10

0:6503

105:86

3:2161

969:18

Nat. gas: spot-F2

4

BIC

1

10/10

0:0006

74:609

4:5693

846:15

Nat. gas: spot-F3

4

BIC-HQC

1

10/10

0:0004

50:255

6:1766

746:96

Nat. gas: spot-F4

4

AIC-BIC-HQC

1

10/10

0:2268

39:804

3:2990

679:13

Gold: spot-F1

2

AIC-BIC-HQC

1

10/10

26:852

178:64

26:219

657:12

Full sample analysis (from 1997/01/07 to 2014/05/30, T ¼ 4539 observations) Crude oil: spot-F1 4 AIC-HQC 1

10/10

0:0082

115:72

19:350

150:71

Crude oil: spot-F2

4

BIC

1

10/10

0:0000

23:305

41:389

37:459

Crude oil: spot-F3

4

BIC-HQC

1

10/10

0:0415

13:158

35:529

21:078

Crude oil: spot-F4

4

BIC-HQC

1

10/10

0:1937

9:7278

35:257

15:411

Nat. gas: spot-F1

4

BIC-HQC

1

10/10

0:7678

191:03

2:1694

1469:9

Nat. gas: spot-F2

4

BIC-HQC

1

10/10

0:2662

114:83

3:8188

1247:6

Nat. gas: spot-F3

4

BIC-HQC

1

10/10

0:1472

73:161

4:2011

1138:1

Nat. gas: spot-F4

4

AIC-BIC-HQC

1

10/10

0:0585

56:319

2:6258

1072:2

Gold: spot-F1

1

BIC

1

9/100

7:1169

854:96

15:697

970:93

ð0:0001Þ ð0:9444Þ ð0:9910Þ ð0:7632Þ ð0:4200Þ ð0:9801Þ ð0:9850Þ ð0:6339Þ ð0:0000Þ

ð0:9277Þ ð0:9952Þ ð0:8385Þ ð0:6598Þ ð0:3809Þ ð0:6059Þ ð0:7013Þ ð0:8088Þ ð0:0076Þ

Note: Logarithmic transformations of the variables spot and futures are used. The null hypotheses of the WE tests are H0: “Ft is weakly exogenous for β” and H0: “St is weakly exogenous for β”. The null hypotheses of the GC tests are H0: “St does not Granger-cause Ft” and H0: “Ft does not Granger-cause St”. The p-values are those of the conventional χ 2q distribution. The confidence level for each test carried out in the algorithm is α ¼ 0:05. Johansen test with unrestricted constant for crude oil and gold and with restricted constant for natural gas.

ð0:0000Þ ð0:0000Þ ð0:0025Þ ð0:0117Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ

ð0:0000Þ ð0:0000Þ ð0:0003Þ ð0:0018Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ

ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:5223Þ ð0:3344Þ ð0:1863Þ ð0:5091Þ ð0:0000Þ

ð0:0007Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:7046Þ ð0:4311Þ ð0:3795Þ ð0:6222Þ ð0:0000Þ

ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ

ð0:0000Þ ð0:0000Þ ð0:0003Þ ð0:0039Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ ð0:0000Þ

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Table 2 Traditional tests for breaks. Market

Lags (q)

Chow

CUSUM

QLR test

(expl. var. -dep. var.)

max F

estimated break date

Crude oil: F1 -spot

4

4:8467

0:9907

44:2863

2005/05/05

Crude oil: F2 -spot

4

5:3253

 0:3637

52:4941

2008/09/25

Crude oil: F3 -spot

4

5:5421

0:0019

54:3134

2008/09/25

Crude oil: F4 -spot

4

5:4610

0:2855

52:7234

2008/09/25

Crude oil: spot -F1

4

4:4103

1:9443

51:4289

2004/06/30

Crude oil: spot -F2

4

5:6481

1:4418

60:2795

2008/12/24

Crude oil: spot -F3

4

6:0541

1:2884

59:2017

2008/12/23

Crude oil: spot -F4

4

5:8240

1:2456

55:7184

2008/12/19

Nat. gas: F1 -spot

4

7:0449

 0:4953

89:2567

2009/08/27

Nat. gas: F2 -spot

4

4:4807

 1:4151

61:0758

2009/08/28

Nat. gas: F3 -spot

4

3:2925

 2:0399

50:4141

2001/01/01

Nat. gas: F4 -spot

4

2:6907

 1:8890

49:2656

2009/09/08

Nat. gas: spot -F1

4

2:0994

1:8010

55:9206

1999/09/02

Nat. gas: spot -F2

4

2:0634

0:8528

47:8102

2001/01/05

Nat. gas: spot -F3

4

2:6443

0:4062

31:5338

2001/01/05

Nat. gas: spot -F4

4

3:3804

0:3160

33:7939

2008/07/07

Gold: F-spot

1

6:2968

1:0324

43:9869

2002/01/02

Gold: spot -F

1

4:1768

0:0637

28:1175

2005/06/02

ð0:0000Þ

ð0:3219Þ

ð0:0000Þ

ð0:0000Þ

ð0:7161Þ

ð0:0000Þ

ð0:0000Þ

ð0:9985Þ

ð0:0000Þ

ð0:0000Þ

ð0:7753Þ

ð0:0000Þ

ð0:0000Þ

ð0:0519Þ

ð0:0000Þ

ð0:0000Þ

ð0:1494Þ

ð0:0000Þ

ð0:0000Þ

ð0:1977Þ

ð0:0000Þ

ð0:0000Þ

ð0:2130Þ

ð0:0000Þ

ð0:0000Þ

ð0:6204Þ

ð0:0000Þ

ð0:0000Þ

ð0:1571Þ

ð0:0005Þ

ð0:0000Þ

ð0:0414Þ

ð0:0040Þ

ð0:0000Þ

ð0:0590Þ

ð0:0263Þ

ð0:0000Þ

ð0:0718Þ

ð0:0293Þ

ð0:0000Þ

ð0:3938Þ

ð0:0047Þ

ð0:0000Þ

ð0:6846Þ

ð0:0004Þ

ð0:0060Þ

ð0:7520Þ

ð0:0003Þ

ð0:0026Þ

ð0:3019Þ

ð0:0058Þ

ð0:0000Þ

ð0:9492Þ

ð0:0001Þ

Note: The QLR test is performed with 15% trimming at the beginning and end of the sample period.

Table 3 Bai and Perron (2003) regression. Market

Lags (q)

Estimated break dates Before 2008/07/14

Crude oil: F1 -spot

5

2003=03=24

After 2008/07/14 ,

2007=04=20

2009=01=16

½2002=09=24; 2003=09=22 ½2007=01=24; 2007=05=24

Crude oil: F2 - spot 4

2003=03=24

,

½2008=10=10;2009=03=12

2007=04=19

2009=02=11

2002=10=24; 2003=05=28 2007=02=22; 2007=05=30

Crude oil: F3 - spot 5 Crude oil: F4 - spot 4

1998=11=25

,

2003=03=24

2008=10=21; 2009=04=08

,

2007=04=19

2009=02=11

½1998=07=06; 1999=03=02 ½2002=10=11; 2003=06=18 ½2007=03=02; 2007=06=14

1999=02=15

,

2003=03=24

,

½2008=10=23; 2009=04=14

2007=04=19

2009=02=11

½19987=08=26; 1999=05=31 ½20027=11=22; 2003=08=06 ½20077=03=02; 2007=06=27

,

½20087=12=03; 2009=03=17

2010=11=22 ½20107=10=11; 2011=02=10

Crude oil: spot - F1 3 Crude oil: spot - F2 5 Crude oil: spot - F3 5 Crude oil: spot - F4 5

2001=11=14

2008=12=24

½2001=02=08; 2002=10=08

1999=02=15

,

½2007=10=04; 2009=05=21

2002=11=14

,

2007=04=18

2009=01=23

½1998=08=07; 1999=04=30 ½2002=10=09; 2003=03=21 ½2006=12=25; 2007=08=14

2002=02=22

,

½2008=09=05; 2009=04=22

2007=04=20

2009=02=04

½2001=11=22; 2002=05=28 ½2006=12=08; 2007=09=04

2002=02=22

,

½2008=09=18; 2009=05=11

2007=04=20

2009=02=04

½2001=11=26; 2002=05=28 ½2006=11=13; 2007=09=04

,

½2008=11=21; 2009=03=24

2010=11=22 ½2010=09=03; 2011=01=24

Nat. gas: F1 - spot

3

1998=12=03

,

2001=05=24

,

2003=02=24

,

2004=11=25

½1998=09=18; 1999=02=03 ½2001=03=19; 2001=06=29 ½2003=01=31; 2003=03=11 ½2004=11=11; 2005=01=13

2009=08=25

,

½2009=04=22; 2010=01=15

2012=08=01 ½2012=04=09; 2012=10=19

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141

Table 3 (continued ) Market

Lags (q)

Estimated break dates Before 2008/07/14

Nat. gas: F2 - spot

5

1998=12=03

After 2008/07/14 ,

2001=05=28

,

2003=02=24

,

2004=11=25

,

2009=11=13

½1998=09=11; 1999=02=02 ½2001=03=14; 2001=07=06 ½2003=01=31; 2003=03=13 ½2004=11=11; 2004=12=23

2012=08=24

½2006=08=01; 2006=11=14

Nat. gas: F3 - spot

5

Nat. gas: F4 - spot

5

1998=12=03

½2012=06=07; 2012=10=03

,

2000=12=28

,

2003=02=24

,

2004=11=23

,

2009=09=04

½1998=08=25; 1999=02=26 ½2000=10=17; 2001=02=19 ½2003=01=20; 2003=03=17 ½2004=10=25; 2005=01=06

2007=11=28

2012=07=25 ½2012=04=10; 2012=10=02

,

2000=09=28

,

2003=02=24

,

2004=11=23

,

2009=09=04

½1998=10=14; 1999=01=07 ½1999=06=29; 2000=11=08 ½2003=01=22; 2003=03=14 ½2004=10=28; 2005=01=06

Nat. gas: spot - F1

2

Nat. gas: spot - F2

5

Nat. gas: spot - F3

5

,

½2009=07=29; 2009=10=19

½2007=10=04; 2008=02=28

1998=12=03

,

½2009=09=03; 2010=01=27

2006=08=25

,

½2009=07=16; 2009=10=09

2007=11=28

2012=08=31

½2007=09=28; 2008=03=24

½2012=05=14; 2012=11=15

1999=07=13 ½1999=06=07; 2000=01=28

1999=05=12

,

2001=02=06

½1999=03=25; 1999=06=25 ½2000=11=10; 2001=06=19

1999=11=01

,

2001=08=01

½1999=08=19; 1999=12=31 ½2001=04=10; 2002=01=09

Nat. gas: spot - F4

3

Gold: F - spot

5

2000=02=07

,

2001=11=14

2008=07=03

½1999=11=17; 2000=03=17 ½2001=10=02; 2002=06=12

1999=10=04

,

2001=12=31

,

½2007=01=11; 2009=02=06

2007=01=02

2008=10=15

½1999=07=16; 1999=11=16 ½2000=02=24; 2002=03=12 ½2006=07=24; 2007=04=05

,

½2008=09=01; 2008=12=19

2010=07=26 ½2010=03=12; 2011=01=07

Gold: spot - F

4

2000=02=09

,

2001=12=19

,

2005=08=29

2008=10=22

½1999=10=25; 2000=04=14 ½1999=12=15; 2002=03=07 ½2005=02=08; 2005=11=22

,

½2008=06=25; 2009=05=06

2011=09=26 ½2010=12=29; 2012=06=19

Note: Minimum distance between breaks: h ¼454 observation (ϵ ¼ 0:1). Maximum number of breaks: M ¼ 8. 5% confidence interval for the break dates are in brackets. UDmax and WDmax (absence of breaks against an unknown number of breaks) and sup F T ðkÞ always reject the null (see Bai and Perron, 2003 for details). sup F T ðkÞ (absence of breaks against a fixed number of breaks, k) always reject the null (see Bai and Perron, 2003 for details).

Table 4 Stability tests. Market

Parameter stability test (Hansen, 1992)

(expl. var. -dep. var.) ci

ϕ1

ϕ2

ϕ3

Cointegration stability test (Hansen and Johansen, 1999)

γ1

γ2

γ3

αi

σ2i

Lc

VAR

Crude oil: F1 -spot

0.0984

0.1327

0.2011

0.5546

0.0441

0.1269

0.1124

0.0161

2.5601

4.4279 Crude oil:

Crude oil: F2 -spot Crude oil: F3 -spot

0.1486 0.1189

0.0885 0.0824

0.1864 0.1912

0.4969 0.4870

0.0347 0.0540

0.0640 0.0608

0.0540 0.0863 0.0538 0.0613

2.4788 2.4772

4.3808 spot-F1 4.3226 Crude oil:

Crude oil: F4 -spot Crude oil: spot -F1

0.1095 0.0511

0.0794 0.1372

0.1995 0.0828

0.4967 0.0771

0.0609 0.3271

0.0543 0.1130

0.0619 0.0438 0.3660 0.2373

2.4843 2.0894

4.3281 spot-F2 4.6586 Crude oil:

Crude oil: spot -F2 Crude oil: spot -F3

0.0564 0.0598

0.1605 0.1382

0.1467 0.1490

0.0426 0.0603

0.3349 0.2708

0.1601 0.1747

0.5326 0.0291 0.6017 0.0446

1.8209 1.8088

3.8890 spot-F3 3.6284 Crude oil:

Crude oil: spot -F4 Nat. gas: F1 -spot

0.0626 0.1236

0.0571 0.3401

0.1505 0.1060

0.0722 0.1831

0.1344 0.1968

0.2076 0.0438

0.6100 0.0485 0.3387 0.0286

1.7605 0.4320

3.3303 spot-F4 1.9028 Nat. gas:

Nat. gas: F2 -spot Nat. gas: F3 -spot

0.1069 0.1000

0.2203 0.1770

0.1156 0.1011

0.2054 0.1985

0.0377 0.1338

0.0690 0.0768

0.2513 0.1196

0.0504 0.0392

0.3888 0.4323

1.5024 spot-F1 1.3958 Nat. gas:

Nat. gas: F4 -spot Nat. gas: spot -F1

0.1035 0.0434

0.1325 0.0977

0.0969 0.4885

0.2405 0.2948

0.1681 0.2223

0.1090 0.1317

0.1263 0.0329 0.0852 0.0619

0.4654 2.0967

1.5433 spot-F2 3.3984 Nat. gas:

Nat. gas: spot -F2 Nat. gas: spot -F3

0.0620 0.0790

0.1113 0.1332

0.3642 0.4734

0.1628 0.4337

0.1639 0.1268

0.0914 0.1336

0.0963 0.0568 0.0898 0.0615

1.5979 1.4421

2.7686 spot-F3 2.9770 Nat. gas:

Nat. gas: spot -F4 Gold: F -spot

0.0870 0.4544

0.1889

0.8922

0.6775

0.1426

0.0958

0.1320

0.0569 1.2412

1.3685 6.2181

3.6475 spot-F4 8.1210 Gold:

Gold: spot -F

0.2270

0.8345

4.6061

5.7409 spot-F

test sup τðtÞ T β-stab. supτðtÞ T β-stab. supτðtÞ T β-stab. supτðtÞ T β-stab. supτðtÞ T β-stab. supτðtÞ T β-stab. supτðtÞ T β-stab. supτðtÞ T β-stab. supτðtÞ T β-stab.

r. const

r. trend

1.8327

1.2591

1.5988 6.3233

2.7949 4.9315

0.9249 5.5880

3.2783 4.3227

0.6689 4.3369

1.8352 3.6878

0.7679 3.6494

1.2435 3.1278

1.1882 2.6588

3.1685 2.0594

0.7026 2.7003

2.4702 1.6959

0.3882 2.3581

1.9133 1.4028

0.2576 1.2627

1.5666 1.1994

0.6784

1.0243

Note: Parameters are those of Eq. (4). Critical values (5%): Individual stability test 0.47#Joint stability test for crude oil and natural gas (Lc with q¼ 5) 2.75 #Joint stability test for gold (Lc with q ¼5) 1.01#Fluctuation tests on the individual eigenvalues 1.36#Fluctuation test of beta coefficient (restricted constant) 2.20#Fluctuation test of beta coefficient (restricted trend) 2.28 The null hypothesis of stability is rejected if the test statistic exceeds the critical value.

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