rational speculative bubbles and commodities markets - SSRN

RATIONAL SPECULATIVE BUBBLES AND COMMODITIES MARKETS: APPLICATION OF THE DURATION DEPENDENCE TEST

by

Riza Emekter Associate Professor of Finance Robert Morris University Massey Hall 214 6001 University Boulevard Moon Township, PA 15108 +1.412.397.5458 [email protected]

Benjamas Jirasakuldech 1 Associate Professor of Finance Slippery Rock University of Pennsylvania 1 Morrow Way Slippery Rock, PA 16057 +1.724.738.4370 [email protected]

Peter Went Senior Researcher GARP Research Center 111 Town Square Place, Suite 1215 Jersey City, NJ 07310 + 1.201.719.7230 [email protected]

Abstract The presence of rational speculative bubbles in 28 commodities is investigated using the duration dependence test on the stochastic interest-adjusted basis. 11 of 28 commodities experienced some episodes of rational speculative bubble. These commodities are WTI crude oil, coffee, corn, soybean No.2, soybean meal and oil, wheat No.2 soft red and hard winter wheat, lean hogs, gold and platinum. Additionally, natural gas, propane, live cattle, and pork bellies exhibit mean-reversion in the interest-adjusted basis.

August 24, 2011

JEL Classification:

C14, C22, C32, F31, G12, and G15

Keywords:

commodities markets, commodities futures, duration dependence, non-parametric methods, convenience yield, speculative bubble.

1

Corresponding author

Electronic copy available at: http://ssrn.com/abstract=1916345

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RATIONAL SPECULATIVE BUBBLES AND COMMODITIES MARKETS: APPLICATION OF THE DURATION DEPENDENCE TEST

1.

Introduction

This study examines the potential presence of rational speculative bubbles in commodity markets. In a rational speculative bubble, investors realize that traded assets are overvalued, but they are unwilling to liquidate their positions, because they believe that higher future prices will provide adequate compensation for the increased risks of the bubble deflating. The substantial price appreciation and subsequent collapse of crude oil in 2008 serves as a perfect example for a speculative bubble. At the beginning of 2008, one barrel of crude oil – West Texas Intermediate or WTI – was trading around $100. On July 3, 2008, the price of crude oil appreciated to hit $145 per barrel and reached $147.27 eight days later. Yet only three months later, on October 11, 2008, the price of crude oil fell to $77 per barrel and traded at a recorded low of $35 per barrel at the end of the year. Similarly, the substantive price appreciation of natural gas during 2006 came to a halt when Amaranth Advisors LLC, a $9 billion hedge fund collapsed as a result of speculation in natural gas futures. That fund lost $6 billion when its speculative trades went the wrong way. Amaranth’s actions caused “significant price movements in the natural gas market [that] demonstrate[s] that excessive speculation distorts prices, increases volatility, and increases costs and risks for natural gas consumers, such as utilities, who ultimately pass on inflated costs to their customers” (U.S. Senate, 2007) These recent episodes in commodity prices fueled the debate whether the cause of the increase is due to speculative bubble or changing market fundamentals. There are

Electronic copy available at: http://ssrn.com/abstract=1916345

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supporters of both arguments. Those advocating for speculative bubbles emphasize that recent increases in commodity prices are consistent with typical bubble characteristics, which they attribute to the entry of index funds into the markets (Gilbert, 2010; Masters and White, 2008; Wray, 2008; and Robles, 2009). On the other hand, others believed that bubbles do not explain the current rise in commodity prices (Krugman, 2008). Index funds entering the commodity futures markets do not affect the cash prices and do not contribute to bubbles in commodity markets (Irwin et.al, 2009). This study investigates the presence of rational speculative bubbles in commodity futures and uses the interest adjusted basis (IAB) as a proxy measure of fundamental value. It seeks to resolve the ongoing argument whether the recent increases in commodity prices are caused by an increasing demand for commodities in the real sector or artificial demand driven by the financial sector through index funds taking large long position in commodity index. We investigate whether the price movements in 28 widely traded commodities in energy, foodstuffs and industrials, grains and oilseeds, livestock and meats, and metals sectors, were induced by rational speculative bubble-like behavior. McQueen and Thorley (1994) characterized a rational speculative bubble by its negative duration dependence: the probability that negative excess returns replace positive excess returns is a decreasing function of the length of positive excess returns. In other words, the longer the bubble grows, the less likely it will pop. Prior studies employed the duration dependence methodology to identify rational speculative bubbles in equities (Chan, McQueen and Thorley, 1998), currencies, (Jirasakuldech et. al, 2006) and real estate (Lavin and Zorn, 2001). In this study, the

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duration dependence technique is applied on the IAB, a measure of excess return that is believed to capture the fundamental value relationship in both financial and physical commodities markets (Fama and French, 1988; Ludkovski and Carmona, 2004; and Casassus and Collin-Dufresne, 2005). The study offers two main contributions. First, the study expands the literature on rational speculative bubbles in commodities markets, adding to the few studies in the field. Bertus and Stanhouse (2001) found evidence of rational speculative bubbles in gold markets using a state-space model. Gilbert (2010) showed evidence of bubbles in copper and soybeans using first order augmented Dickey-Fuller regression. Employing Granger causality tests, Robles (2009) reported that speculative trading could potentially explain the recent surge in wheat, maize, soybean, and rice prices. Using the same technique, Stevans and Sessions (2008) found evidence of speculative effects in the futures contract with maturities exceeding 6 months. Irwin et.al. (2009) expanded the literature by broadening the number of commodities and found no relationship between the growth of commodity index investing and commodity prices. Our study uses a modified cost of carry relationship as its proxy of fundamental value, IAB, and seeks to answer the question whether a real increase in the demand of actual commodities or the demand fueled from investments by index funds are the potential causes of bubbles. Second, the study addresses some methodological shortcomings in empirical tests previously used to detect speculative behavior in asset markets. Prior studies on speculation in commodities markets used techniques that not only tested for the possible presence of bubbles, but also for the correct specification of the fundamental value

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relationships. The duration dependence test used here differs. First, it does not compare the time series behavior of the fundamental factors determining asset values with asset prices, (Chowdhury, 1991; Krehbiel and Adkins, 1993; Ma and Soenen, 1988; Heaney, 2002; Franses and Kofman, 1991; Kocagil, 1997; and Chang et.al., 1990). Second, it does not require a correct identification of the underlying fundamental pricing model, such as a dividend or cash flow discount model. Prior studies of commodity price behavior employed variance bound tests and cointegration technique as an indirect approach to identify bubbles. Both techniques compare the underlying asset price with fundamental value (Brenner and Kroner, 1995; Barrett and Kolb, 1995; Bryant and Haigh, 2004; and Brorsen, 1989). However, critics argued that these methods contain specification error. The duration dependence test avoids the joint testing of the null hypothesis of no bubbles and no model misspecification. Third, the non-parametric technique does not require normally distributed returns, and returns to both physical commodities and commodities futures exhibit skewness and leptokurtosis (Geman, 2005). Finally, the method accommodates non-linearities in returns, which is one of the characteristics of bubbles. Other studies testing for speculation in commodities markets using variance inequality tests (Wahab, 1995), or autocorrelation, skewness, and kurtosis of returns assume linearity, (Chaudry and Christie-David, 1998). The remainder of the study unfolds as follows. Section II discusses the creation of the IAB series and the duration dependence test. Section III describes our data and section IV summarizes our empirical results, with section V concluding.

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2.

Methodology

Speculative bubbles assume arbitrage opportunities. To preclude arbitrage between the physical commodity and the financial futures, the value of the long position in the futures between time t and the maturity date of the same future contract at T (f tT ), should be identical to the long position in the physical commodity (s,t), adjusted for the interest foregone between t and T (r tT ), the warehousing costs for the same time (w tT ), and the convenience yield (c tT ), or: (1)

f i,tT = si,t + r f,tT + wtT − ctT

This study uses IAB as its measure the fundamental value of the commodity (Ludkovski and Carmona, 2004; and Casassus and Collin-Dufresne, 2005). As most other empirical studies, we disregard the cost of warehousing when calculating the IAB (Sequeira and McAleer, 2000; Fama and French, 1988; and Heaney, 2002). When estimating the IAB, the convenience yield needs to be approximated. This captures the implicit and explicit benefits from having immediate access to the commodity in inventory, or the benefits stemming from possible alternative uses for the product, or the ability to speculate in the price appreciation of the underlying asset and its futures (Gibson and Schwartz, 1990; Chatrath et.al., 1997; Schwartz, 1997, and Considine and Larson, 2001). Empirical results indicate that speculative profits do exist in commodities markets, creating opportunities to earn excess returns (Ma and Soenen, 1988; and Milonas and Henker, 2001). The possibility to earn excess returns in case of temporary disequilibria from trading in either physical or financial commodity markets is IAB. (2)

IABi, tT = f i, tT − si ,t − rf ,tT + ctT

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2.1

Creating contiguous prices series for futures

Physical commodity spot price series are continuous, while financial futures prices are not and are limited by the lifetime of each futures contract. To derive a fundamental value, the physical spot prices must be matched with the financial futures price series, which requires continuous futures price series. This can be artificially created through splicing, rolling over the nearby futures prices into one continuous series, or weighting all futures prices by their time to maturity. Most studies choose the first alternative, a choice that introduces multiple biases 2. First, the calculated value of each observation in the rolled over price time series depends on the effects of the narrowing time to maturity window. It puts a considerable weight on the behavior of the front month contract and the more actively traded nearby contracts, even though the front month contract is more volatile and less relevant for hedging but is used for pricing and speculation. Second, splicing of time-series assumes a continuous roll of the nearest futures contract around expiration and assumes sufficient liquidity in the marketplace. Third, it also introduces expiry related seasonality, which may be particularly pronounced in commodities that have quarterly or unevenly spaced expiration calendar, and can become significant during increased seasonal demands for hedging, such as during growing season for agricultural commodities (Simon, 2002). Fourth, the differences in the conditional variance of the individual futures actually misestimate the unconditional variance of the spiced series (Fama and French, 1988; and Sequeira and McAleer, 2000).

2

For details, see Milonas (1986), Ng and Pirrong (1994), and Heaney (2002).

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Time to maturity invariant futures price series avoids these biases. Rougier (1996) proposed a convex combination of all existing and traded contracts to create an optimal contiguous futures price series, F*. The series include each existing and traded contract: (3)

F* =

n −1

∑ λi (t )Fk +iv i =0

V is the time between contract maturities, λi is the individual contract specific time n −1

weights, and both satisfy the dual necessary conditions of

∑ λ (t ) = 1 and 0 ≤ λ (t ) ≤ 1 i =0

i

i

for all values of i and t. Additionally, the optimal index for n futures contracts must satisfy the following two conditions: n −1

(4)

∑ λ ' (t ) = 0 and i =0

i

n −1

(5)

v ∑ iλi ' (t ) = 1 , i =0

Here, λi ' (t ) is the first derivative of λi (t ) . In the case of n = 2 there is only one optimal time weighting of the components of the price index; for two futures contracts the optimal contiguous time series equals F * =

k −t v − (k − t ) Fk + Fk + v , where Fk is the v v

price of the nearest contract, Fk + v is the price of the next nearest contract, v is the time between expiry of the two adjacent contracts, and k-t represents the time to expiry. With most commodities having multiple existing and traded contracts, several potential optimal combinations of time weights exist as long as they satisfy λi ' (t ) = i=1,…, n-1.

1 iv(n − 1) for

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Our study uses a set of weights that yields a notional time, v, closest to the mean time to expiry of all contracts in the market. This approach reflects all the available trading opportunities in the futures markets and meets all above constraints. Several studies follow the Rougier (1996) approach (e.g., Holmes and Tomsett, 2004; and Lien and Tse, 1998).

2.2

Approximating the convenience yield

As the convenience yield is not observable, it has to be estimated. 3 Heaney (2002) computed the convenience yield refining a stochastic non-arbitrage model for infrequently traded asset previously developed by Longstaff (1995). In this model, a trader initiates a long position in the physical commodity at t, and has perfect foresight from t to T, when the futures contract matures. When commodity price reaches its maximum level between t and T, the trader sells the physical commodity at the maximum price, earns the maximum profit, and realizes the maximum convenience yield. This profit, TS(S i,tT ), can be approximated by the value of an American option to sell the physical asset when its price rises sufficiently between t and T. Assuming a Wiener process in both of physical commodity and financial futures price, the profit from this trading strategy, under continuous compounding, is:

(6)

3

  σ i ,tT 2 ts (s i, tT ) =   2 + 2   

  σ i ,tT 2 N   2  



σ 2 − i , tT   + σ i ,tT e  8  2π 

2

   

  .  

Both Gibson and Schwartz (1990) and Schwartz (1997) developed stochastic convenience yield models. The practical benefit of these stochastic convenience yield models lies in developing term structures of commodities future prices. They do not, however, describe the endogenous factors determining the convenience yield.

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N (*) is the cumulative normal distribution. This symmetrical trading strategy is available to both a trader with a long position in the commodity and a trader with a long position in the financial futures. Further generalization for both long and short positions leads to the following relationship: (7)

(

)

ci ,tT = tsi ,t (si ,tT ) − tsi ,t f * i ,tT . This approximation of the convenience yield requires only two inputs: the time to

maturity of the contracts and variance of the two price series, offering a distinct modeling advantage over alternative approaches 4. Substituting (7) into (2), we estimate the IAB for all the 28 commodities in the study: (8)

IABi, tT = f i, tT − si ,t − r f ,tT + ci ,tT = f i, tT − si ,t − r f ,tT − {ts i ,t (si ,tT ) − ts i ,t ( f i ,tT )}.

2.3

Duration dependence test

Based on these IAB series for each commodity in equation (8), the duration dependence test is performed to examine the relationship between the length of the positive excess returns and the probability that it will end. This test focuses on the hazard rate, h i , for runs of positive excess returns in the IAB. The hazard rate is defined as the probability of obtaining a negative excess return given a sequence of i prior positive excess returns, or h i = Prob(ε t < 0ε t-1 > 0, ε t-2 >0……,ε t-i > 0, ε t-i-1 < 0). Specifically, if a bubble exists, the hazard rates for runs of positive return decreases with i, h i+1 < h i for all i. 5 To apply the duration dependence test, each IAB series is transformed into run lengths of positive and negative excess returns. The numbers of runs of positive or 4

Normally, convenience yield models incorporate the effects of reduced marketability due to the daily mark-to-market and the possibly incorrect pricing of the contract in relation to the underlying due to maturity, storage and convenience effects, (Brenner and Kroner, 1995; Milonas and Hencker, 2001; Ludkovski and Carmona 2004; and Casassus and Collin-Dufresne, 2005). 5 For full derivation of this equation, see McQueen and Thorley (1994).

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negative excess returns of particular length i are then counted. The optimal hazard rate for length i is computed as hˆi = N i /( M i + N i ) by maximizing the log-likelihood function of the hazard function with respect to h i : α

(9)

L(θS T ) =

∑

N i Ln h i + M i Ln(1-h i ) + Q i Ln(1-h i ),

i =1

where N i is the number of completed runs of length i in the sample, and M i and Q i are the numbers of completed and partial runs with length greater than i, respectively. To test the null hypothesis, the functional form of the hazard function is defined as: (10)

hi =

1 1+ e

− (α + β Lni )

.

The test is performed by substituting equation (10) into (9) and maximizing the log likelihood function with respect to α and β. Using a logit regression to estimate the parameters of hazard function, the independent variable is the log of current run length. Here, the dependent variable is 1 if the run ends in the next period and 0 if it does not. The null hypothesis of no rational expectations bubble implies that the probability of a positive run’s ending is unrelated to prior returns, or the hazard rate should be constant, or H 0 : β = 0. The alternative hypothesis of a bubble suggests that the probability of a positive run’s ending should decrease with the length of the run, i.e., decreasing hazard rate or negative duration dependence, or H 1 : β < 0. Under the null hypothesis of no bubble, the likelihood ratio test (LRT) is distributed asymptotically as χ2 with one degree of freedom. L UR is the log-likelihood function using the maximum likelihood estimate (MLE) of the unrestricted parameters and L R is the log-likelihood function using the MLE of the restricted parameters.

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(11)

LRT = 2[L UR – L R ] ∼ χ 12 .

3.

Data

This study considers all 28 commodities with both spot price on the physical commodity and the financial futures prices that are available from the Commodity Research Bureau (CRB database). The commodities cover the following five different commodity groups: 1. energy: Brent crude oil (CB), WTI crude oil (CL), gasoline (HU), heating oil (HO), natural gas (NG), and propane (PN) 2. foodstuffs and industrials: butter (02), coffee (KC), cotton (CT), and sugar (SB) 3. grains and oilseeds: barley (WA), corn (C-), flaxseed (WF), soybean/No. 2 (S-), soybean meal (SM), soybean oil (BO), wheat/No. 2 soft red (W-), wheat domestic feed (WW), and wheat/No. 2 hard winter (KW) 4.

livestock and meats: cattle feeder (FC), cattle live (CL), lean hogs (LH), and pork bellies (PB)

5. metals: copper (HG), gold (GC), palladium (PA), platinum (PL), and silver (SI). After collecting end-of-month prices for the physical commodity and all existing and traded financial future contracts associated with each commodity, the F* series were created. These series together with the end-of-month spot price, the prevailing 13 week T-bill yield from the FRED database, provide the inputs to the monthly cost-of-carry calculations. The convenience yield was approximated using the variance of the monthly prices for the preceding 12 months. PLACE TABLE 1 HERE

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Table 1 reports the detailed descriptive statistics on the IAB series for 28 commodities. Each commodity series runs for different time horizons and in several different exchanges. Butter has the shortest number of observations since the series only go back to 1998, while soybean oil has the longest series dated back to 1960 6. For 12 of the 28 commodities, the mean of IAB is positive. Three in the energy sector - Brent crude oil, unleaded gasoline, and natural gas; three in the foodstuffs and industrials sector - butter, cotton, and sugar; four in the grains and oilseeds sector; barley, soybean No. 2, wheat No. 2 soft red, and domestic wheat; and two in the livestock and meats sector - lean hogs and pork bellies. The highest monthly IAB is that of hogs, 31.91%, a highly seasonal commodity that also exhibits cyclical changes, and the lowest is of high-grade copper, -16.97%, which is a cyclical commodity. 7 The volatility of the monthly IAB ranges from the high of 64.31% for unleaded gasoline to the low of 1.09% for gold. Apart from the high volatility of copper, commodities in metals sector show low overall volatility as do the grains and oilseeds sector. On average, the highest volatility sector is the energy sector. Skewness and kurtosis offer some indication of the possible presence of rational bubbles. Overall, these statistics show significant skewness and excess kurtosis, with the notable exception of butter, barley, soybean oil, wheat No.2, cattle feeder, and cattle live. Positive return autocorrelation is one of the characteristics consistent with a speculative bubble. The autocorrelation coefficients are positive for lags 1 to 4, 6, and 12 for the Brent crude, coffee, cotton, corn, soybean No. 2, soybean meal, soybean oil, wheat soft red, wheat hard winter, cattle feeder, copper, and palladium. Moreover, the 6

Since each series is analyzed individually, using all the data available is better than cutting the most data to match the shortest series in the group. 7 The IAB basis estimates are comparable to those derived by Heaney (2002).

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Ljung-Box Q statistics strongly reject the null hypothesis of no autocorrelation up to lags 6 and 12 at the traditional significance level for all commodities, with the exception of some energy commodities – crude oil WTI, unleaded gasoline, heating oil, propane and platinum. We note that the return behavior of Brent crude and WTI exhibit different patterns and the observed persistence of positive autocorrelation in the IAB series strengthens the possibility that bubbles could exist in some commodities. 8 Yet, interpreting these results warrants some caution as other factors could contribute to the persistent autocorrelation in excess return series, particularly in highly seasonal commodities (Geman, 2005). The autocorrelation coefficients for ρ 1 to ρ 3 , show a pattern of persistent short-term autocorrelation for foodstuffs and industrials and grains and oilseeds, reflecting the effects of the summer growing season. In the case of natural gas, during the shoulder season in the U.S. (between April and October) inventories are built up and supply of natural gas exceeds actual demand. During the heating season (between November and March), demand for heat and weather-related demand shocks may exceed actual supply, depleting storage. Moreover, empirical evidence suggests that both skewness and kurtosis reflect changing economic or political fundamentals rather than the emergence of speculative bubbles (Antonshin and Samiei, 2006). Asymmetric fundamentals and information asymmetry affecting publicly information could cause skewness in returns and batched arrival of information could cause leptokurtosis. Additionally, positive autocorrelation could result from time-varying and deterministic risk premium (Fama and French, 1988),

8

To conserve some space, the autocorrelation coefficients and Ljung-Box Q-statistics for each commodity are not reported here but are available from authors upon request.

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emergence of fads (Porterba and Summers, 1988), non-synchronous trading (Lo and MacKinely, 1990), or pure psychological effects (Westerhoff, 2003).

4.

Empirical results

4.1

Energy

In Table 2, the logit regression estimate of β for WTI crude oil is negative and significant (-1.0252). 9 The likelihood ratio test (LRT) of the null hypothesis of no duration dependence, or constant hazard rate, is rejected at the 5% significance level with the LRT = 6.0872. Clearly, WTI has experienced some rational speculative bubbles during 1985 and 2005. However, there is no evidence that Brent crude oil, gasoline, and healing oil are affected by rational speculative bubble. For two commodities as closely linked as Brent and WTI are, this finding is surprising and counters previously reported empirical evidence, reported by Antoniu and Foster (1992); and Milonas and Henker (2001). We offer two alternative explanations. One relates to the differences in volatility estimates between the two IAB estimates: IAB’s volatility estimate for the WTI is 2.8719 and for Brent 9.5180. The other relates to the available length of the series. Our available price information on Brent futures starts in 1991, while WTI starts in 1985. For natural gas and propane, the null hypothesis of β=0 is rejected in favor of β>0 with the LRT = 5.7768 at the 5% significance level for natural gas and LRT = 3.6161 at the 10% significance level for propane. A significant positive β implies that the excess

9

For Brent crude oil, there are exactly 4 runs that last 4 months and 13 runs that last at least 4 months. The hazard rate states that if a positive run persists for 4 consecutive months, there is a 30.80% probability that the bubble will burst in the next month. For the WTI, there is a 25.00% probability that positive run lasting for 4 months will revert to a negative return in the next period. The hazard rate of other commodities can be interpreted in the same way.

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return series of both commodities exhibit mean-reversion, where the likelihood of positive excess return reverting to negative return in the next period increases as the established length of positive excess return increases. The mean reversion could possibly be explained by the pronounced seasonal effects.

4.2

Foodstuffs and Industrials PLACE TABLE 3 HERE

The logit regression estimates of β for all four commodities including butter, coffee, cotton, and sugar as shown in Table 3 are negative, but are not significant except for coffee. The null hypothesis of β=0 is rejected in favor of β= -0.6364 for coffee at the 5% significance level with the LRT = 5.0536. Therefore, out of the four commodities in foodstuffs and industrial sector, rational speculative bubble only existed in coffee sometime between April 1974 and April 2005. The presence of rational speculative bubbles in the coffee market can be attributed to changes in the international coffee market. The market for coffee has undergone significant changes with the collapse of the International Coffee Agreement in 1989, an agreement that kept global coffee prices artificially low between 1975 and 1989. After 1989, coffee prices as well as price volatilities increased. As a result of this structural change and higher prices, most of the new entrants into the international coffee markets produce Robusta coffee, which has a higher yield and lower production cost than Arabica coffee (Adrangi and Chatrath, 2003). Both the spot coffee price and the NYBOT/ICE futures contract used to calculate the IAB basis is based on Robusta coffee, which is widely produced in Brazil, one of the largest coffee producers and exporters in the world. By 1998, coffee prices started a gradual decline due to global overproduction of coffee.

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4.3

Grains and oilseeds PLACE TABLE 4 HERE

Table 4 reports the duration dependence test results for nine commodities in the grains and oilseed sector. Six commodities exhibit a characteristic that is consistent with a rational speculative bubble: corn, three different representations of soybean (bean#2, meal, and oil), and two representations of wheat (soft red and hard red). Logit regression estimates of β range from -0.4365 for corn, -0.4501 for soybean #2, -0.6640 for soybean meal, -0.9758 for soybean oil -0.2808 for wheat #2 soft red, and -0.7316 for wheat #2 hard winter. The null hypothesis of no bubble in these six commodities is rejected at the traditional significant level. The existence of speculative bubbles in corn is likely a spillover between two types of animal feed: soybean meal and corn. As the entire soybean spectrum – soybean #2, soybean meal, and soybean oil – exhibits rational speculative bubbles, confirms that there must have existed speculative pressures in these commodities. These pressures have impacted both soybeans and its derivatives: soybean meal and oil. Global production of soybeans has increased dramatically in the late 1980s with the entrance of Brazil as one of the largest exporter of soybeans. The U.S. remains one of the largest consumers of soybeans meal, a direct competitor for corn, an animal feed. As corn exhibited a rational speculative bubble, the complex relationship between the corn and the soybean is most likely explained by the increasing production of soybeans and feed corn to meet the demand from cattle growers during the 1990s. The existence of speculative bubble in the wheat complex – soft red and hard winter – may reflect

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seasonal weather related patterns in the U.S. The harvest period for wheat stretches from June to September. Only half of the wheat production can be stored, the rest must be sold immediately, and there may be shorter periods of seasonal oversupply, particularly during the summer growing period. The stacking of expiration calendar around the growing season in this commodity futures complex also supports this contention. Since this seasonal price pattern is regular, speculative positions can be profitable (Dutt et al, 1997).

4.4

Livestock and meats PLACE TABLE 5 HERE

Table 5 shows that the duration dependence test rejects the null hypothesis of no rational speculative bubble in lean hogs at the 1% level with LRT =12.93. The β estimate is 0.6825 and is statistically significant at the 1% level. Thus, lean hogs, a very liquid market, experienced some episodes of rational speculative bubbles during April 1968 to April 2005. The market for hogs in the U.S. has a history of longer cyclical price patterns, with peaks occurring every 4 – 6 years. On the other hand, both live cattle and pork bellies show positive and significant β estimates of 0.6479 and 0.3243, respectively. The null hypothesis of β =0 is rejected favor of β>0 at the 1% level for cattle and at the 10% level for pork bellies, which indicates that the excess return series of these two commodities exhibit mean reversion, suggesting that these markets are different.

4.5

Metals PLACE TABLE 6 HERE

The duration dependence test results in Table 6 show evidence of rational speculative bubbles in gold between April 1976 and February 2005 and platinum between October

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1986 and March 2005. Both metals can be considered as investible assets (Cai et al., 2001). The null hypothesis of β =0 is rejected in favor of β = -1.2602 at the 1% significance level for gold; for platinum the β = -0.2515 at the 10% significance level. These results are consistent with previously published studies regarding the behavior of gold prices (Berthus and Stanhouse, 2001). Our empirical evidence does not support the existence of rational speculative bubbles among high grade copper, palladium, and silver. Although we failed to identify bubbles in some commodities, we believe that this is caused by the unique features of these commodities, such as low trading volume which attracts less attention from hedgers, speculators and investors (e.g., cattle feed, barley, and butter). The large body of literature about the positive relationship between trading volume and volatility should serve as an explanation (see for example, Wang and Yau, 2000). Due to relatively lower trading volume and volatility, it is less likely that bubblelike behavior in these commodities could be observed. In other markets with distinct history of heavy institutional participation, such as the market for metals, the structure of market participation and infrastructure could serves as a candidate explanation. Even though there are examples of speculative attempts in the market for copper (Sumitomo’s failed attempt to corner the copper market in the mid 1990s) and silver (the Hunt Brothers’ likewise failed attempt to corner the silver market in the early 1980s), we failed to indentify rational speculative bubbles. A final explanation that bubbles might not have emerged in certain sectors could be attributed to the commodity itself. For commodities where hedging is essential and the number of speculators is limited, long-term deviations from a fundamental value relationship may not persist. For instance, having an unorthodox price view in the market

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for butter could potentially generate substantial short-term excess returns, yet cornering that market for a perishable good with high warehousing costs and ready alternative substitutes (such as “I Can’t Believe It’s Not Butter”) may not be practically feasible.

5. Conclusions This study investigates the existence of rational speculative bubbles in commodities by using interest adjusted basis as the fundamentals value. It helps to fill the gap in literature whether the recent increase in commodity prices is due to an increasing demand of commodities causing bubbles or artificial (financial) demand caused by index funds taking large long positions in commodity index futures. We apply the duration dependence test on the monthly interest-adjusted basis, a measure of potential excess returns earned on commodities Our empirical findings suggest that there is evidence of rational speculative bubbles in 11 of the 28 commodities. Notably, rational speculative bubbles exist in WTI crude oil in the energy sector but not in Brent crude oil. We attribute the results to time differences in the available price information. In the foodstuffs and industrials sector coffee exhibits evidence for speculation, which we attribute to changes in the global market for coffee. The wide agricultural sector of commodities also shows evidence for speculative bubbles. In the grains and oilseeds sector, there is evidence of speculative bubbles in corn, the entire soybean complex, and large part of the wheat complex. The existence of speculative bubbles in the wheat complex is likely caused by seasonal price patterns. In the livestock and meats sector lean hogs have contained rational speculative bubbles. Among ferrous metals gold and platinum show speculative bubbles; both these metals are considered as

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investible assets. Additionally, the duration dependence procedure also allows for the identification of mean reversion in natural gas, propane, live cattle and pork bellies. With no methodological consensus in the academic literature on identifying bubbles in financial assets and financial markets, alternative interpretation of approaches and results are possible. Yet, all agree that predicting the bubbles ex ante is almost impossible. Some even contend that bubbles may not be known and identified ex post 10. As information asymmetry and excessive risk taking among speculators are often blamed for as the cause of bubbles in asset markets, this debate is likely to continue (Cochrane, 2000; and Herring and Wachter, 1999). Consequently, policymakers should actively take actions to improve transparency in order to mitigate information asymmetry between traders and to restrict excessive risk taking among speculators. Our results demonstrate that information asymmetry and speculative trading may in fact have an impact on grains and oilseeds futures prices. Regulatory objectives should prioritize the transparency in commodity by requiring traders to disclose more and better information. Disclosure fuels transparency and fosters market discipline. Making the hedgers and investors aware of the possible asymmetries in the markets would aid in reducing informational gap. In testimony to the U.S. congress, Meghnad Desai, George Soros, Michael Masters, and Senator Joe Lieberman (2008) called for increasing regulations in commodity future markets and trading. Yet, by imposing more regulations, the effects can be detrimental. Increasing position reporting and more transparent market structure as well as the changes in the post Dodd-Frank world should facilitate the discussion on

10

Some authors even dispute 1929 crash as bubble. See McGrattan and Prescott (2001) for example.

- 21 -

how to identify optimal regulatory policies that may inhibit speculative financial bubbles in the future.

- 22 -

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- 26 -

Table 1 - Descriptive Statistics of Monthly Interest Adjusted Basis (IAB) Commodity Energy Crude Oil, Brent/Global Spot (CB)

Exchange IPE

Period

N

1991:02-2005:04

171

Mean 2.5939

Std. Dev. 9.5180

Skewness ***

Kurtosis ***

1.2952

5.4614

***

***

Crude Oil WTI/Global Spot (CL)

NYMEX

1985:04-2005:05

242

-0.6179

2.8719

6.7024

83.3574

Gasoline, Unleaded/Regular (HU)

NYMEX

1986:02-2005:05

232

2.8581

64.3186

2.9790

89.8816

Heating Oil#2/Fuel Oil (HO)

NYMEX

1980:01-2005:05

305

-0.8456

13.7762

-3.8758

30.5078

Natural Gas, Henry Hub (NG)

NYMEX

1995:01-2005:07

127

0.9762

13.1534

-1.2933

4.4119

Propane (PN)

NYMEX

1988:11-2005:05

199

-0.8456

13.7762

-3.8758

30.5078

Foodstuffs and industrials Butter, Aa (02) Coffee (KC)

CME NYBOT

1998:03-2005:04 1974:04-2005:04

86 373

0.4384 -3.7353

11.8742 6.5044

-0.4784* *** -2.2358

Cotton/1-1/16" (CT)

NYBOT

1980:04-2005:06

303

10.2821

20.3649

-2.8023

***

*** *** ***

***

***

Sugar#11/World Raw (SB)

NYBOT

1962:03-2005:05

519

0.4965

16.7706

Grains and Oilseeds Barley, Western/No. 1 (WA)

WCE

1993:04-2005:04

145

1.4501

8.2423

0.1136

4.5408

***

Corn/No. 2 Yellow (C-)

CBOT

1961:02-2005:02

529

-1.2371

1.8328

*** *** ***

***

0.3271 *** 9.8668 ***

40.7668

***

13.1018

***

6.9135

***

-1.1495

6.9126

***

***

Flaxseed/No. 1 (WF)

WCE

1993:04-2004:05

145

-1.2473

9.5194

-3.1402

21.3280

Soybean/No. 2 Yellow (S-)

CBOT

1961:03-2005:05

531

1.0443

3.5106

-0.4690

5.5396 2.9250 6.3701 9.5778

*** ***

Soybean Meal/48% Protein (SM)

CBOT

1961:04-2005:04

529

-1.9425

6.7891

Soybean Oil/Crude (BO)

CBOT

1960:03-2005:04

528

-1.3765

7.0806

-1.0174 0.2314

Wheat/No. 2 Soft Red (W-)

CBOT

1961:04-2005:04

529

0.7650

7.0631

-0.1030 ***

*** *** *** ***

***

WW Domestic Feed/No. 3 (WW)

WCE

1993:04-2005:04

145

2.4577

10.5103

2.1327

11.8064

Wheat/No. 2 Hard Winter (KW)

WCE

1971:04-2005:04

409

-3.8790

6.4383

-2.1764

9.4340

Livestocks and Meats Cattle, Feeder/Average (FC) Cattle, Live/Choice Average (LC)

CME CME

1978:12-2005:02 1966:01-2005:05

315 473

-7.1285 -0.1075

6.2388 7.6653

Hogs, Lean/Average Iowa/Smi (LH)

CME

1968:04-2005:04

445

31.9076

21.3809

***

-0.1981 0.1678 ***

***

0.1971 *** 3.6859 ***

0.6506

2.7401 3.4395

***

Pork Bellies, Frozen 12-14 lbs (PB)

CME

1964:05-2005:04

492

3.2964

10.0910

-0.5040

Metals Copper/High Grade (HG)

NYMEX

1978:12-2005:02

315

-16.9722

25.3446

-1.0434

*** ***

*** ***

2.1052

***

Gold (GC)

NYMEX

1976:04-2005:02

347

-0.4195

1.0934

0.3658

26.9499

Palladium (PA)

NYMEX

1974:04-2005:04

202

-1.6485

4.9661

-9.2977

108.695

*** ***

***

***

Platinum (PL)

NYMEX

1986:10-2005:03

222

-0.2406

1.8712

9.6208

126.995

Silver (SI)

NYMEX

1976:07-2005:05

443

-0.3943

1.6815

3.6029

61.0970

***

***

Notes: The spot and future prices of each commodity are obtained from Commodity Research Bureau (CRB). The future exchanges are International Petroleum Exchange (IPE), New York Mercantile Exchanges (NYME), Chicago Mercantile Exchange (CME), New York Board of Trade (NYBOT), Winnipeg Commodity Exchange (WCE), and Chicago Board of Trade (CBOT). The descriptive statistics are provided for the interest adjusted basis (IAB). IAB is the difference between costs of carry and convenience yield, and is f i, tT − s i ,t − r f ,tT + ctT . F* denotes the future price which is constructed as the contiguous price index that reflects a time weighted mean of the price of the nearest and next nearest contracts. The study period for each commodity varies depending on the availability of the data. N is the number of quarterly observations. Mean and standard deviations are expressed in percent. Asymptotic standard of skewness is (6/N)1/2. Asymptotic standard errors of coefficient of excess kurtosis is (24/N)1/2. ***, **, and * indicate significance at the 1%, 5%, and 10% level, respectively.

- 27 -

Table 2 – Duration Dependence Test for Rational Speculative Bubbles in Commodities in Energy Sector

Run Length 1 2 3 4 5 6 7 8 9 10 11 Total Positive Run Log-Logistic Test α β LRT of H 0 : β = 0 (p-value)

Crude Oil, Brent/Global Spot (CB) Actual Sample Run Hazard Counts Rate 5 0.1852 8 0.3636 1 0.0714 4 0.3077 3 0.3333 0 0.0000 2 0.3333 0 0.0000 2 0.5000 2 1.0000 0 0.0000 27

Crude Oil WTI/Global Spot (CL) Actual Sample Run Hazard Counts Rate 27 0.7500 5 0.5556 0 0.0000 1 0.2500 2 0.6667 0 0.0000 0 0.0000 1 1.0000 0 0.0000 0 0.0000 0 0.0000 36

***

-1.3071 0.2131 0.4773 (0.4896)

Gasoline, Unleaded/Regular (HU) Actual Sample Run Hazard Counts Rate 20 0.4167 8 0.2857 6 0.3000 5 0.3571 6 0.6667 2 0.6667 1 1.0000 0 0.0000 0 0.0000 0 0.0000 0 0.0000 48

**

0.8579 ** -1.0252 ** 6.0872 (0.0136)

Heating Oil#2/Fuel Oil (HO)

Natural Gas, Henry Hub (NG)

Actual Run Counts 13 1 7 4 6 1 4 1 2 0 1

Actual Run Counts 3 7 3 3 2 5 0 1 0 0 0

Sample Hazard Rate 0.3250 0.0370 0.2692 0.2105 0.4000 0.1111 0.5000 0.2500 0.6667 0.0000 1.0000

40 **

-0.5722 0.1856 0.3753 (0.5401)

Sample Hazard Rate 0.1250 0.3333 0.2143 0.2727 0.2500 0.8333 0.0000 1.0000 0.0000 0.0000 0.0000

Propane (PN)

Actual Run Counts 20 18 5 4 2 0 0 0 0 0 0

24 ***

-1.2468 0.2085 0.6367 (0.4248)

Sample Hazard Rate 0.4082 0.6207 0.4545 0.6667 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

49 ***

-1.8375 ** 0.9551 ** 5.7768 (0.0162)

Notes: The duration dependence test is performed on interest-adjusted basis (IAB). IAB is the difference between costs of carry and convenience yield, and is calculated as f i, tT − s i ,t − r f ,tT + ctT . The sample hazard rate, h i = N i /(M i +N i ), indicates probability that a run ends at length i provided it lasts until i. β is the hazard rate which is estimated using a logit regression where the independent variable is the log of current length of runs and dependent variable is 1 if a run ends and 0 if it does not end in the next period. The likelihood ratio test (LRT) of the null hypothesis of no duration dependence or constant hazard rate (H 0 : β = 0) is asymptotically distributed χ2 with one degree of freedom. The critical values are 6.635, 3.841 and 2.706 for the 1%, 5%, 10% significance levels, respectively. P-value is the marginal significance level--the probability of obtaining the calculated value of LRT or higher under the null hypothesis. ***, **, and * indicate significance at the 1%, 5%, and 10% level, respectively.

-0.3312 ** 0.7884 * 3.6160 (0.0572)

- 28 Table 3 – Duration Dependence Test for Rational Speculative Bubbles in Commodities in Foodstuff and Industrial Sector Butter, Aa (02) Run Length 1 2 3 4 5 6 7 8 9 10 11 12 13……16 17 18 19 20 21 22……31 32 33 34…….51 52 Total Positive Runs Log-Logistic Test

Actual Run Counts 9 1 0 0 2 2 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

Sample Hazard Rate 0.5625 0.1429 0.0000 0.0000 0.3333 0.5000 0.0000 0.5000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

16

Coffee (KC) Actual Run Counts 10 2 2 2 0 0 0 1 2 0 0 1 0 0 0 1 0 0 0 0 0 0 0

Cotton/1-1/16" (CT)

Sample Hazard Rate 0.4762 0.1818 0.2222 0.2857 0.0000 0.0000 0.0000 0.2000 0.5000 0.0000 0.0000 0.5000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

21

Actual Run Counts 42 17 12 6 6 3 2 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Sample Hazard Rate 0.4565 0.3400 0.3636 0.2857 0.4000 0.3333 0.3333 0.2500 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

92

Sugar#11/World Raw (SB) Actual Run Counts 3 0 2 0 1 1 1 1 0 0 1 0 0 2 0 1 1 1 0 1 1 0 1

Sample Hazard Rate 0.1667 0.0000 0.1333 0.0000 0.0769 0.0833 0.0909 0.1000 0.0000 0.0000 0.1111 0.0000 0.0000 0.2500 0.0000 0.1667 0.2000 0.2500 0.0000 0.3333 0.5000 0.0000 1.0000

18

α

-0.2628

-0.3995

-0.2873

-2.3171***

β

-0.5942

-0.6364**

-0.1977

-0.1247

0.9878 (0.3202)

0.2818 (0.5955)

LRT of H 0 : β = 0 (p-value)

2.2745 (0.1315)

**

5.0536 (0.0245)

Notes: The duration dependence test is performed on interest-adjusted basis (IAB). IAB is the difference between costs of carry and convenience yield, and is calculated as f i, tT − si ,t − r f ,tT + ctT . The sample hazard rate, h i = N i /(M i +N i ), indicates probability that a run ends at length i provided it lasts until i. β is the hazard rate which is estimated using a logit regression where the independent variable is the log of current length of runs and dependent variable is 1 if a run ends and 0 if it does not end in the next period. The likelihood ratio test (LRT) of the null hypothesis of no duration dependence or constant hazard rate (H 0 : β = 0) is asymptotically distributed χ2 with one degree of freedom. The critical values are 6.635, 3.841 and 2.706 at the 1%, 5%, 10% significance levels, respectively. P-value is the marginal significance level--the probability of obtaining the calculated value of LRT or higher under the null hypothesis. ***, **, and * Indicate significance at the 1%, 5%, and 10% level, respectively.

- 29 Table 4 – Duration Dependence Test for Rational Speculative Bubbles in Commodities in Grains and Oilseeds Sector Run Length

Barley, Western/No. 1 (WA) Actual Sample Run Hazard Counts Rate

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24……28 29 30 31 32……45. 46 47

6 0 0 1 1 0 1 3 2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

0.3750 0.0000 0.0000 0.1000 0.1111 0.0000 0.1250 0.4285 0.5000 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Corn/No. 2 Yellow (C-) Actual Sample Run Hazard Counts Rate

24 12 9 2 1 2 2 1 2 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0.4138 0.3529 0.4091 0.1538 0.0909 0.2000 0.2500 0.1667 0.4000 0.6667 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Flaxseed/No. 1 (WF) Actual Sample Run Hazard Counts Rate

4 1 1 5 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.2667 0.0909 0.1000 0.5556 0.0000 0.2500 0.3333 0.5000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Soybean/No. 2 Yellow (S-) Actual Sample Run Hazard Counts Rate

12 10 5 5 1 1 0 2 4 6 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1

0.2308 0.2500 0.1667 0.2000 0.0500 0.0526 0.0000 0.1111 0.2500 0.5000 0.0000 0.0000 0.1667 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2000 0.0000 0.0000 0.2500 0.0000 0.0000 0.0000 0.3333 0.0000 0.5000 1.0000

Soybean Meal/48% Protein (SM) Actual Sample Run Hazard Counts Rate

18 3 9 1 2 3 0 0 0 1 0 0 0 0 0 0 0 1 4 0 0 0 0 0 1 0 0 0 0 0

0.4186 0.1200 0.4091 0.0769 0.1667 0.3000 0.0000 0.0000 0.0000 0.1429 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1667 0.8000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Total Positive Run

16

58

15

52

43

Log-Logistic Test α β LRT of H 0 : β = 0 (p-value)

**

-1.3563

-0.3412 **

-0.1752

-0.4365

0.3008

4.9265

(0.5833)

(0.0264)

**

-1.3979

***

-1.0866

**

-0.5153 -0.6640

***

15.1864

0.0634

-0.4501

0.0343

11.0115

(0.8530)

***

***

(0.0009)

***

***

(0.0000)

- 30 Table 4 – Duration Dependence Test for Rational Speculative Bubbles in Commodities in Grains and Oilseeds Sector (Cont.) Run Length

Soybean Oil/Crude (BO) Actual Sample Run Hazard Counts Rate

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22……40 41 42…….44 45

22 7 2 2 1 0 0 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 1 0 0

Total Positive Run

41

Log-Logistic Test α β LRT of H 0 : β = 0 (p-value)

0.5366 0.3684 0.1667 0.2000 0.1250 0.0000 0.0000 0.1429 0.0000 0.0000 0.0000 0.0000 0.1667 0.2000 0.0000 0.0000 0.2500 0.3333 0.0000 0.0000 0.5000 0.0000 1.0000 0.0000 0.0000

Wheat/No. 2 Soft Red (W-) Actual Sample Run Hazard Counts Rate 12 7 7 4 4 3 4 2 0 4 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1

0.2353 0.1795 0.2188 0.1600 0.1905 0.1765 0.2857 0.2000 0.0000 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2500 0.3333 0.0000 0.0000 0.5000 0.0000 0.0000 0.0000 0.0000 1.0000

51 -0.0037 -0.9758*** 32.4804*** (0.0000)

WW Domestic Feed/No. 3 (WW) Actual Sample Run Hazard Counts Rate 5 2 3 2 0 2 2 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0.2632 0.1429 0.2500 0.2222 0.0000 0.2857 0.4000 0.0000 0.0000 0.3333 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

19 -1.1248*** -0.2808* 3.3998* (0.0652)

Wheat/No. 2 Hard Winter (KW) Actual Sample Run Hazard Counts Rate 12 2 2 2 0 0 0 1 2 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0

0.5217 0.1818 0.2222 0.2857 0.0000 0.0000 0.0000 0.2000 0.5000 0.0000 0.0000 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

23 -1.3345*** -0.0511 0.0301 (0.8622)

Notes: The duration dependence test is performed on interest-adjusted basis (IAB). IAB is the difference between costs of carry and convenience yield and is calculated as f i, tT − si ,t − r f ,tT + ctT . The sample hazard rate, h i = N i /(M i +N i ), indicates probability that a run ends at length i provided it lasts until i. β is the hazard rate which is estimated using a logit regression where the independent variable is the log of current length of runs and dependent variable is 1 if a run ends and 0 if it does not end in the next period. The likelihood ratio test (LRT) of the null hypothesis of no duration dependence or constant hazard rate (H 0 : β = 0) is asymptotically distributed χ2 with one degree of freedom. The critical values are 6.635, 3.841 and 2.706 for the 1%, 5%, 10% significance levels, respectively. P-value is the marginal significance level—the probability of obtaining the calculated value of LRT or higher under the null hypothesis. ***, **, and * indicate significance at the 1%, 5%, and 10% level, respectively.

-0.2314 -0.7316** 7.0593*** (0.0078)

- 31 Table 5 – Duration Dependence Test for Rational Speculative Bubbles in Commodities in Livestock and Meat Sector

Run Length

Cattle, Feeder/Average (FC) Actual Sample Run Hazard Counts Rate

1

10

Cattle, Live/Choice Average (LC) Actual Sample Run Hazard Counts Rate

0.5000

25

0.2976

Hogs, Lean/Average Iowa/Smi (LH) Actual Sample Run Hazard Counts Rate 1

Pork Bellies, Frozen 1214 lbs (PB) Actual Sample Run Hazard Counts Rate

0.1667

13

0.2203

2

5

0.5000

19

0.3220

0

0.0000

3

0.0652

3

2

0.4000

13

0.3250

0

0.0000

7

0.1628

4

1

0.3333

15

0.5556

0

0.0000

10

0.2778

5

1

0.5000

6

0.5000

1

0.2000

3

0.1154

6

1

1.0000

4

0.6667

0

0.0000

1

0.0435

7

0

0.0000

2

1.0000

1

0.2500

3

0.1364

8

0

0.0000

0

0.0000

1

0.3333

7

0.3684

9

0

0.0000

0

0.0000

2

1.0000

5

0.4167

10

0

0.0000

0

0.0000

0

1.0000

7

1.0000

11

0

0.0000

0

0.0000

1

1.0000

3

0.3000

12

0

0.0000

0

0.0000

0

0.0000

2

0.1667

13….15

0

0.0000

0

0.0000

0

0.0000

0

0.0000

0

0.0000

0

0.0000

0

0.0000

1

0.0769

16 Total Positive Runs

20

84

7

65

Log-Logistic Test α

-0.0359

-1.0119***

β

-0.0245

0.6479

LRT of H 0 : β = 0 (p-value)

0.0020 (0.9640)

-1.5524***

-1.9436***

***

-0.6825***

0.3243*

***

***

3.1843*

7.6512

(0.0056)

12.9309

(0.0003)

(0.0743)

Notes: The duration dependence test is performed on interest-adjusted basis (IAB). IAB is the difference between costs of carry and convenience yield and is calculated as f i, tT − si ,t − r f ,tT + ctT . The sample hazard rate, h i = N i /(M i +N i ), indicates probability that a run ends at length i provided it lasts until i. β is the hazard rate which is estimated using a logit regression where the independent variable is the log of current length of runs and dependent variable is 1 if a run ends and 0 if it does not end in the next period. The likelihood ratio test (LRT) of the null hypothesis of no duration dependence or constant hazard rate (H 0 : β = 0) is asymptotically distributed χ2 with one degree of freedom. The critical values are 6.635, 3.841 and 2.706 for the 1%, 5%, 10% significance levels, respectively. P-value is the marginal significance level--the probability of obtaining the calculated value of LRT or higher under the null hypothesis. ***, **, and * indicate significance at the 1%, 5%, and 10% level, respectively.

- 32 Table 6 – Duration Dependence Test for Rational Speculative Bubbles in Commodities in Metals Sector

Run Length

Copper/High Grade (HG) Gold (GC) Actual Sample Actual Sample Run Hazard Run Hazard

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Counts 5

Rate 0.2778

3 3 1 1 1 1 0 1 1 0 0 0 0 1

0.2308 0.3000 0.1429 0.1667 0.2000 0.2500 0.0000 0.3333 0.5000 0.0000 0.0000 0.0000 0.0000 1.0000

Total Positive Runs 18 Log-Logistic Test α β LRT of H 0 : β = 0 (p-value)

Counts

Rate

Counts

Rate

37 3 3 1 1 0 1 0 0 0 0 0 0 0 0

0.8043 0.3333 0.5000 0.3333 0.5000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

15 5 0 0 0 0 0 0 0 0 0 0 0 0 0

0.7500 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

46 **

-1.0339

Palladium (PA) Platinum (PL) Silver (SI) Actual Sample Actual Sample Actual Sample Run Hazard Run Hazard Run Hazard

20 1.2551*** ***

Counts Rate Counts Rate 22 7 4 0 1 1 0 0 0 0 0 0 0 0 0

0.6286 0.5385 0.6667 0.0000 0.5000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

35

42 14 9 1 2 1 0 1 0 0 0 0 0 0 0

0.6000 0.5000 0.6429 0.2000 0.5000 0.5000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

70

1.1630**

0.4912

0.3855*

-0.1221

-1.2602

0.5055

-0.2515

-0.3202

0.1379 (0.7104)

7.4723*** (0.0062)

2.4040 (0.1210)

3.1843* (0.0743)

1.0118 (0.3144)

Notes: The duration dependence test is performed on interest-adjusted basis (IAB). IAB is the difference between costs of carry and convenience yield and is calculated as f i, tT − si ,t − r f ,tT + ctT F denotes the future price which is constructed as the contiguous price . index that reflects a time weighted mean of the price of the nearest and next nearest contracts. The sample hazard rate, h i = N i /(M i +N i ), indicates probability that a run ends at length i provided it lasts until i. β is the hazard rate which is estimated using a logit regression where the independent variable is the log of current length of runs and dependent variable is 1 if a run ends and 0 if it does not end in the next period. The likelihood ratio test (LRT) of the null hypothesis of no duration dependence or constant hazard rate (H 0 : β = 0) is asymptotically distributed χ2 with one degree of freedom. The critical values are 6.635, 3.841 and 2.706 for the 1%, 5%, 10% significance levels, respectively. P-value is the marginal significance level--the probability of obtaining the calculated value of LRT or higher under the null hypothesis. ***, **, and * indicate significance at the 1%, 5%, and 10% level, respectively.