Review of Agricultural Economics Volume 24, Number 2 Pages Unbiasedness and Market Efficiency Tests of the U.S. Rice Futures Market

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1 Review of Agricultural Economics Volume 24, Number 2 Pages Unbiasedness and Market Efficiency Tests of the U.S. Rice Futures Market Andrew M. McKenzie, Bingrong Jiang, Harjanto Djunaidi, Linwood A. Hoffman, and Eric J. Wailes This study examines short-run and long-run unbiasedness within the U.S. rice futures market. Standard OLS, cointegration, and error-correction models are used to determine unbiasedness. In addition, the forecasting performance of the rice futures market is analyzed and compared to out-of-sample forecasts derived from an additive ARIMA model and the error-correction model. The results of our unbiasedness tests and the forecasting performance of the rice futures market provide supporting evidence that the U.S. long-grain rough rice futures market is efficient. The results have important price risk management and price discovery implications for Arkansas and U.S. rice industry participants. The importance of commodity futures markets as price risk management and forecasting tools has become a central issue in the aftermath of The Federal Agriculture Improvement and Reform Act of 1996 (FAIR Act). The shift toward less government intervention under the FAIR Act has pushed the U.S. agricultural sector towards a more market-oriented system. An understanding of price risk management alternatives other than government programs is important for market participants. One of the key alternative strategies is the use of futures Andrew M. McKenzie is an Assistant Professor in the Department of Agricultural Economics and Agribusiness, University of Arkansas. Bingrong Jiang is a former research associate in the Department of Agricultural Economics and Agribusiness, University of Arkansas. Harjanto Djunaidi is a research associate in the Department of Agricultural Economics and Agribusiness, University of Arkansas. Linwood Hoffman is an Agricultural Economist, Economic Research Service, U.S. Department of Agriculture. Eric J. Wailes is a Professor in the Department of Agricultural Economics and Agribusiness, University of Arkansas.

2 Unbiasedness and the U.S. Rice Futures Market 475 markets. Price volatility is especially important in the rice sector. Harwood et al. compared price volatility in 24 major grain, oilseed, livestock, and fruit and vegetable markets. They found that rice ranked fifth in price volatility in , before the FAIR Act was passed. The relative hedging effectiveness of a futures market is dependent upon two related concepts: the relative comovement over time of the respective cash and futures prices of a commodity and the intertemporal ability of contemporaneous futures prices to provide unbiased forecasts of subsequent cash prices at contract maturity. This paper focuses on the second of these two issues, otherwise referred to in the literature as the unbiasedness hypothesis, with respect to the U.S. rough rice futures market. The Johansen cointegration procedure (Johansen and Juselius) is used to test for long-run market unbiasedness and the short-run price dynamics are analyzed using an error-correction model (ECM). The ECM framework recognizes that although futures markets may be unbiased forecasters of subsequent cash prices in the long run, the potential exists for futures markets to exhibit short-run pricing inefficiencies and be biased in the short run. More specifically, if futures prices fail to reflect the information contained in historical prices, they violate the weak-form version of the Efficient Markets Hypothesis (EMH) defined by Fama. While our ECM allows us to detect potential short-run inefficiencies in a weakform sense, we would argue that further testing is required to determine if such potential inefficiencies translate into actual pricing biases in a forecasting sense. With this question in mind, we draw upon various econometric and time-series techniques to test the comparative forecasting performance of the rice futures market. The ability of the rice futures market to efficiently perform a price discovery and forecasting function is particularly relevant to the U.S. rice market, which has traditionally been characterized by a high degree of cooperative pool pricing, and which has no central cash market from which to observe price movements. The rough rice futures market is relatively new compared to the well-established futures markets for corn, wheat, and soybeans. Rough rice futures trading was introduced at the Chicago Rice and Cotton Exchange (CRCE) in August In 1994, futures trading was moved to the Chicago Board of Trade (CBOT) and a new rice options contract was introduced. 1 Rice industry participants have referred to the rice futures market as a thinly traded futures market. This was certainly the case during the infancy of the contract, which was characterized by both low levels of volume and open interest. Gray noted that a thinly traded market is more likely to be unbalanced and biased than a more heavily traded market. Inefficiency in the rough rice futures market is one argument given by some members of the rice industry for low participation in the market. Figure 1 shows the daily levels of volume and open interest in rough rice futures from August 1986 through November These levels pale in comparison to other established grain futures markets such as soybeans and corn, which frequently register daily volume and open interest in excess of 50,000 and 200,000 contracts, respectively. Although there has been a strong upward trend towards greater participation in the contract from 1986 through 1998, interest declined during Daily open interest averaged between 4,000 and 7,000 contracts during 1999, while the volume of contracts traded averaged between 100 and 1,500 contracts per day. Most previous research on commodity futures market

3 476 Review of Agricultural Economics Figure 1. Daily volume and open interest for all rough rice futures contracts, efficiency has examined highly liquid markets. The results of our paper are of interest in determining whether a relatively thinly traded futures market exhibits inefficiencies. In addition to the relatively low level of trading in the rice futures market, potential price discovery is made difficult by the cooperative marketing system that dominates the U.S. rice industry. As Taylor and Taylor et al. point out, the cooperative system has created an informational void with the predominant use of cooperative pool prices and forward contracting in place of competitive auctiontype markets. In light of the structural nature of the U.S. rice industry, we focused attention on the ability of the rice futures market to provide unbiased and efficient forecasts of Arkansas elevator cash prices at par delivery locations. Arkansas cash prices were chosen because Arkansas dominates other U.S. states in terms of production, accounting for 46% of U.S. production. In addition, Arkansas contains 12 par delivery locations for the rough rice futures contract. An unbiased futures market could be used as a risk management tool for all market participants, including producers, elevator operators, millers, and processors. Unbiasedness also has important implications for derivation of optimal hedge ratios. Benninga, Eldor, and Zilcha note that the minimum variance hedge ratio, which is often calculated in empirical hedging models, is only optimal if futures markets are unbiased. In addition, Zulauf et al. point out that a biased futures market implies avoidable social loss resulting from nonoptimal resource allocation. From a hedging perspective, a downward (upward) biased futures

4 Unbiasedness and the U.S. Rice Futures Market 477 market implies that short (long) hedgers must pay a premium to use futures as a price risk management tool. If futures prices are downward biased, the current futures price is lower than the realized futures price at contract maturity, and futures prices should increase over the life of the contract. In this case, short-hedgers will incur trading losses, which may be considered a form of risk premium payment, and represent the cost of transferring price risk to speculators with long futures positions. Conversely, if futures prices are upward biased, the current futures price is higher than the realized futures price at contract maturity and futures prices should decrease over the life of the contract. In this case, long-hedgers will incur trading losses, which may be considered a form of risk premium payment, and represent the cost of transferring price risk to speculators with short futures positions. Selected past studies on the performance of the rice futures market, including Hoffman; Taylor; and Taylor et al., have provided mixed evidence regarding market efficiency and price discovery. Hoffman detected potential inefficiency in the March 1998 contract using an AR(2) model. Taylor used econometric and time-series models, and concluded that rice futures failed both the necessary and sufficient conditions for semistrong form efficiency, as defined in Garcia et al. However, Taylor s analysis used a monthly average cash price series based on prices received by farmers across all states and for all types of rice. Given that the CBOT rice futures contract specifies the delivery of long-grain rice to any of 12 eastern Arkansas par delivery locations, Taylor s study contains a bias towards the rejection of market efficiency. In this paper, we avoid such data problems by using long-grain Arkansas cash prices. More recently, Taylor et al. used cointegration and error-correction techniques, and presented empirical evidence that a long-run equilibrium relationship exists between Thai and Texas rice cash prices and rough rice futures prices. Their results suggest that rough rice futures prices play an important price discovery role in international rice markets. None of the above studies explicitly test for market efficiency with respect to the hypothesis of futures market unbiasedness. Given the relative paucity of recent studies and the current policy framework, this paper addresses both a timely and important issue. The second section of this paper reviews some of the conceptual and technical issues of testing for unbiasedness and market efficiency. The third section discusses the analytical framework and data considerations used in this paper to test for unbiasedness and weak-form market efficiency. The fourth section presents the empirical results, and a final section summarizes the main results. Theoretical and Statistical Issues Relating to Unbiasedness and Efficiency Tests Fama defines three categories of market efficiency tests: weak form, semistrong form and strong form. The weak-form test examines whether current prices fully reflect the information contained in historical prices. The semistrong form test examines how quickly prices reflect the announcement of public information. The strong-form test examines whether investors have private information that is not fully reflected in the market prices. The concept of unbiasedness is a more

5 478 Review of Agricultural Economics restrictive version of Fama s weak-form efficiency. Unbiasedness implies that the current futures price of a commodity should equal the expected cash price of the same commodity at contract maturity. A common approach for the unbiasedness test is to regress cash prices S t on the futures price F t 1 sometime prior to contract maturity and test the null hypothesis that a = 0 and b = 1. (1) S t = a + bf t 1 + e t, where e t is a rational expectations error with the classical properties of a zero mean and constant variance. Previous studies that have utilized this approach with respect to agricultural commodity futures markets include Tomek and Gray s study on corn, soybeans, and potatoes; Kofi s study on wheat, potatoes, soybeans, corn, cocoa, and coffee; and Kahl and Tomek s study on corn, soybeans, and live cattle. Despite the apparent statistical simplicity of testing for unbiasedness as outlined above, a number of theoretical and technical issues have arisen from empirical research in this area. As a result, futures market efficiency has proven to be an elusive concept for empirical researchers. Beck notes that from a theoretical point of view, the unbiasedness test performed on equation (1) is in fact a joint test of both market efficiency and the absence of a risk premium. Rejection of the null hypothesis can therefore be interpreted as either (1) the futures market is inefficient, or (2) a nonzero risk premium exists. If a futures market is weak-form efficient and agents are risk neutral, the market is said to be unbiased. From a hedging perspective, the importance of futures market unbiasedness is of paramount concern and is the concept we initially test in this paper. A second issue relates to the appropriate data construction of the price series being tested. As Kahl and Tomek point out, efficiency tests can be conducted on either different maturity months separately, or by pooling the data for all contracts in one equation. On the one hand, different contract months may vary in terms of relative efficiency so that testing each month separately has merit. However, if a fixed forecast horizon is chosen prior to contract maturity, only one observation is available for analysis each year. As a result, many years must be included in the analysis to gain sufficient degrees of freedom, which in turn causes the potential problem of regime shifts during the sample period. The use of pooled data series has frequently been employed to circumvent the degrees of freedom problem whereby various forecast horizons are estimated using contract months that are trading simultaneously for the same commodity. Kahl and Tomek note that the prices for the various contract months for a particular market are correlated so pooling the data carries the danger of biasing test results towards finding inefficient markets. This problem, referred to as the overlapping observations problem, was first noted by Hansen and Hodrick. In fact, the problem only occurs if the forecast horizon is greater than the sampling interval. In this case, the error structure of the model will follow a moving average process of order j t, where j is the forecast horizon and t is the sampling interval. It has become commonplace to correct for the serially correlated error structure by using Newey and West type methods to obtain consistent estimates of the associated asymptotic covariance matrix. However, Kahl and Tomek note that even after taking into account appropriate statistical corrections, data aggregation of

6 Unbiasedness and the U.S. Rice Futures Market 479 this nature may exaggerate the importance of exceptional observations caused by factors external to the market because the introduction of new information into a market influences the entire constellation of prices. A third technical issue relates to the distributional assumptions of the data. Engle and Granger show if the data series under consideration are nonstationary, then standard statistical hypothesis tests (F and t tests) based on equation (1) will give biased results. Elam and Dixon note this problem with respect to the unbiasedness hypothesis. Voluminous literature has developed in recent years in which researchers have tested for nonstationarity in both futures and cash price series. Typically, prices are found to contain a single unit root and cointegration techniques have subsequently been used to test for market efficiency. For example, Hakkio and Rush, and Kroner and Sultan find cointegration between foreign currency spot and futures rates. Beck, and Antoniou and Foster, find various commodity futures and cash prices to be cointegrated. Finally, a fourth issue of concern when testing for unbiasedness relates to the time dimension under analysis. Theoretically, a futures market that is unbiased in the long run should, on average, provide the best available forecasts of subsequent cash prices at contract maturity. However, even if futures markets are efficient in the long run and provide long-run unbiased forecasts, short-run market inefficiencies may still exist because markets take time to adjust to new information. As we will see in the following section, ECMs present us with a dynamic modeling approach to analyzing futures market unbiasedness, and allow us to distinguish between the short and long run. However, it is difficult to draw conclusive evidence as to weak-form, short-run efficiency through the empirical estimation of an ECM alone. Additional support for short-run efficiency may be provided by comparing the out-of-sample forecasting performance of the ECM to futures prices. The following section of the paper addresses each of the technical issues related to data construction and appropriate statistical estimation procedures used to test for unbiasedness. Analytical Framework and Data Considerations The data used in this study consist of daily long-grain rough rice futures prices traded at the Chicago Board of Trade, which are sampled on a monthly basis, and corresponding Arkansas cash prices for September 1986 to November The cash price series are average long-grain cash price bids collected from representative elevators located at each of the 12 Arkansas par delivery locations for the rough rice futures contract. These prices were obtained from a CBOT rice trader and are available from the authors upon request. The basis is negligible at contract maturity and hence is not explicitly considered in the subsequent empirical estimation of our market efficiency models. 2 There are six futures contracts for January, March, May, July (which began in 1989), September, and November. Three separate data series were constructed, two of which were used to analyze unbiasedness and the third to analyze weak-form market efficiency. Following Kahl and Tomek, unbiasedness tests were conducted for each of the six individual contract months using a single observation a year corresponding to forecast horizons ranging from one to six months prior to contract maturity. Cash price observations were taken during the first week (every Tuesday) of each contract

7 480 Review of Agricultural Economics maturity month. 3 One- to six-month-ahead futures prices were taken as daily closing prices for the first trading day of the calendar month one to six months prior to contract maturity. This provided us with 36 different equations to estimate, specified in equation (1). Each equation had 13 observations, except for July, which had only 11 observations. The one- and two-month-ahead November contract equations had 14 observations. This approach allowed us to determine the degree to which efficiency differed across contract months. This first type of data series provided us with insufficient observations to conduct informative unit root tests, so the equations were estimated in levels using OLS. A pooled-price series was constructed for a two-month forecast horizon across contract months. This series was used to empirically test for long-run and shortrun unbiasedness using the cointegration and ECMs outlined in the following section. A two-month horizon was chosen for two reasons: (1) preliminary analysis of the data indicated that in terms of volume and open interest, each contract was most actively traded two months prior to expiration; (2) a two-month-ahead futures price represented the nearby futures contract, which is the most relevant contract to grain merchandisers and producers utilizing a storage hedge. To avoid the problem of introducing a moving average process into the residuals, cash price observations were taken on a bimonthly basis to match the futures contract time intervals. Cash price observations were again taken weekly (every Tuesday) during the first week of each contract maturity month. Two-monthahead futures prices were taken as daily closing prices for the first trading day of the calendar month two months prior to contract maturity. For example, the cash price for the first week of January represented the two-month-ahead expected price corresponding to the January futures price taken on the first trading day of November. In addition, a four-month-ahead futures price series corresponding to the same contract months as the two-month-ahead futures prices was also sampled over every two-month period. Empirical estimation of ECMs requires taking first differences of the price series. To avoid inducing unnatural price jumps in our futures price series, first differences were taken between the two-month-ahead and the four-month-ahead futures prices series, both of which were observed for the same contract maturity months. 4 Constructing the data in this manner provided us with price series consisting of 78 observations. 5 If cash and futures prices are both nonstationary and require differencing to make them stationary, then, in general, most linear combinations of the two series will also be nonstationary. However, a cointegrating vector may exist, which makes a specific linear combination of the two series stationary. For example, if equation (2) below is a stationary series, a and b are the cointegrating terms and the regression S t = a + bf t 1 + e t as in equation (1) above is the cointegrating or equilibrium regression. (2) e t = S t a bf t 1. This result implies that S t and F t 1 cannot move too far apart from each other despite the fact that they are both nonstationary. Cointegration between the two series is a necessary but not sufficient condition for market efficiency. Cash and futures prices are determined by the same fundamentals, and so efficiency implies

8 Unbiasedness and the U.S. Rice Futures Market 481 that they cannot move too far apart. However, cointegration does not rule out short-run market inefficiencies, whereby past information can improve futures market forecasts of future cash prices. The Granger representation theorem states that a time-series model of a cointegrated series can be rewritten in an error-correction form. Such a transformation renders the series stationary, and allows for normal hypothesis testing. The ECM, which incorporates the long-run relationship between the two price series while allowing for short-run dynamics, is specified as follows: (3) m S t = λ + ρ e t 1 + β F t 1 + β i F t i + i=2 k ψ j S t j + v t, j=1 where is defined as the change or difference in a variable from one period to the next, e t 1 is the error-correction term, and v t is a stationary series. Long-run unbiasedness implies that a = 0 and b = 1 in the cointegrating regression specified in equation (1). The assumption that a = 0 and b = 1 can be tested using the Johansen multivariate cointegration procedure. This approach estimates Likelihood Ratio tests for restrictions on the parameters of the cointegrating regression. The Engle Granger two-step cointegration procedure cannot be used to test these restrictions, as the test procedure does not have well-defined limiting distributions. Rejection of the hypothesis, a = 0 and b = 1, would imply either market inefficiency or the possible presence of a risk premium in the futures market. Futures markets containing a risk premium would be biased, but could still be efficient. In other words, such markets would impound information efficiently while simultaneously taking into account the risk premium. If a futures market is deemed to be unbiased in the long run, the concepts of short-run unbiasedness and market efficiency are synonymous. If the long-run unbiasedness assumptions that a = 0 and b = 1 are correct, short-run unbiasedness requires the restrictions ρ = 1,β = 1, and β i = ψ i = 0 from equation (3) to hold. This can be seen by rewriting equation (3) as S t = λ + (1 + ρ) S t 1 + β F t 1 (ρb + β)f t 2 ρa (4) m + β i F t i + i=2 k ψ j S t j + v t, j=1 where (S t 1 a bf t 2 ) is substituted for e t 1 from equation (2). If the nonlinear restrictions ρ = 1, β= 1, β i = ψ j = 0 hold along with the long-run unbiasedness restrictions, a = 0 and b = 1, equation (4), with the additional constant term λ included, is consistent with our long-run cointegrating relationship in equation (2). Thus, short-run unbiasedness can be determined by estimating the ECM specified in (3) using OLS and testing the above restrictions. If the above restrictions do not hold, then past future and cash prices would contain relevant information not completely incorporated into current future prices, which could be used to predict S t. This would violate the weak form of the efficient markets hypothesis, which states that all past information should already be incorporated into the

9 482 Review of Agricultural Economics current futures price, and therefore it should have no effect on the future cash price. It should be noted that various other stationary-inducing transformations of equation (1) have been used to test the unbiasedness hypothesis. A good review of these alternative methods can be found in Brenner and Kroner. For example, a frequently used test is the first differences model used by Hakkio and by Baillie, Lippens, and McMahon: (5) S t = A + B F t 1 + u t. The unbiasedness hypothesis is a test of the joint restrictions, A = 0 and B = 1. However, such simple stationary-inducing transformations are misspecified if the futures and cash prices are cointegrated. If the two series are cointegrated, an ECM should be estimated to take into account the long-run equilibrium relationship between the two series. Another popular transformation of equation (1) which has often been applied to unbiasedness tests of commodity futures regresses cash price changes on the basis. Fama and French, and more recently, Zulauf et al. used this approach: (6) S t = A + B( F t 1 S t 1 ) + u t. Once more the unbiasedness hypothesis is a test of the joint restrictions, A = 0 and B = 1. This model specification is only consistent with the ECM specification in (3) if the additional restrictions, ρ = 1, β= 1, β i = ψ j = 0, ρ = 1, β= 1, β i = ψ j = 0, along with the long-run unbiasedness restrictions, a = 0 and b = 1, from equation (1) are also imposed. The ECM specified in equation (3) nests the basis regression specification in equation (6). In the presence of cointegration, the ECM is preferable to the basis regression, which restricts the short-run dynamics to zero and is inconsistent with tests of short-run market unbiasedness. After conducting tests for long- and short-run unbiasedness, we proceeded with more elaborate tests of short-run, weak-form market efficiency by comparing the relative forecasting performance of the futures market against a standard timeseries model and the ECM specified above. Leuthold and Hartmann; Rausser and Carter; and Garcia et al. are examples of previous studies that have compared the forecasting performance of futures markets to time-series and econometric models. Rausser and Carter compare the forecasting performance of futures markets within the soybean complex to a structural time-series model. Leuthold and Hartmann compare the forecasting performance of the live hog futures market to an econometric forecasting model. Garcia et al. use both time-series and econometric models to analyze the forecasting performance of the live cattle futures market. Leuthold and Hartmann, and Garcia et al., which incorporate publicly available information within an econometric framework, may be considered tests of semistrong form market efficiency. On the other hand, forecasting tests using pure time-series models, as in this paper, should be considered weak-form efficiency tests. The intuition behind all of these forecasting tests rests on the premise that econometric and time-series models, which provide more accurate forecasts than futures prices, are indicative of an inefficient futures market. However, as argued by Garcia et al., obtaining a model with more accurate forecasts than futures

10 Unbiasedness and the U.S. Rice Futures Market 483 prices is only a necessary condition for inefficiency. A sufficient condition also requires justifying the costs associated with a forecasting model. In other words, for a futures market to be deemed inefficient compared to a forecasting model, both the relative accuracy of the forecast and the informational costs of obtaining the forecast should be taken into account. A third data set was constructed to perform the forecasting tests. Weekly (every Tuesday) Arkansas long-grain rough rice cash prices, sampled on the first week of each month, were used to estimate a time-series model for September 1986 through August This yielded 108 in-sample observations. The 51 observations for September 1995 through November 1999 were set aside for out-of-sample, oneto six-month-ahead forecasts. A factored autoregressive integrated moving-average (ARIMA (2, 1, 1)) model was identified based on standard Box Jenkins time series methods and using the in-sample data. Residual autocorrelation was found to be significant at lags 6 and 7, suggesting that, in addition to monthly price changes lagged one month, the monthly changes in the cash price are also affected by price information lagged six to seven months. This half-year phenomenon may be related to the fact that Southern Hemisphere rice markets typically have their main harvest six months after the U.S. rice harvest. The resulting change in world rice production has a significant impact on the Arkansas long-grain cash price. Incorporating both the six- and seven-month lags into the model, we obtained the following model: (7) (1 B)Z t = (1 θ 1 B)(1 θ 6 B 6 )(1 θ 7 B 7 ) a (1 φ 1 B φ 2 B 2 t, ) where Z t is Arkansas long-grain rough rice cash prices, B is the backshift operator, θ i B i is the moving-average operator, φ i B i is the autoregressive operator, and a t is the independent disturbance term. All the coefficients were found to be significant at the 5% significance level with the exception of the AR (1) term, the inclusion of which was deemed necessary to reduce the residuals to white noise. The model was used to forecast weekly average cash prices for the first week of each calendar month. All forecasts were based on price information available up to the current month. For example, the model was initially used to forecast monthly out-of-sample cash prices for September 1995 through February The data set was subsequently updated a month at a time and the model was reestimated. The revised model estimates were used to forecast prices for the next six months starting in October This procedure was continued until October 1999, when a final forecast for November 1999 was made. Model identification was confirmed by using a tentative order selection test based on the smallest canonical correlation analysis. To achieve consistency with futures price forecasts, which only provide us with forecasts of the cash prices at contract maturity, only the ARIMA forecasts that coincide with periods of futures contract maturity were retained for formal statistical forecasting comparisons. This gave us 26 out-of-sample observations for a one-month forecast horizon, 25 observations for the two- and three-month forecast horizons, 24 observations for the four- and five-month forecast horizons and 23 observations for the six-month

11 484 Review of Agricultural Economics forecast horizon. The ECM was reestimated for the same in-sample period as the ARIMA model, which gave us 52 bimonthly observations. The model was subsequently updated for every two-month out-of-sample forecast horizon, which again yielded 25 observations. Given that the ECM was specified on a bimonthly basis, only forecasts for the two-month horizon were obtained. Our forecasting approaches used out-of-sample observations at forecast horizons that are identical, and thus the three different forecasts were directly comparable to each other. Futures price forecasts were obtained by sampling futures prices on the first calendar day of a month, one through six months prior to contract maturity. Root mean squared error (RMSE) and Theil s inequality coefficient were used as criteria to compare the performance of the models forecasts with futures price forecasts. Under the RMSE criteria, larger forecast errors are penalized more than smaller ones. Theil s inequality coefficient is a unit-free measurement ranging from zero to infinity and a value of unity is equivalent to a random walk forecast. Forecasting accuracy increases with both lower RMSE and Theil inequality coefficient values. Empirical Results The empirical results for the long-run unbiasedness tests using the annualized data series are summarized in table 1. Early research such as Tomek and Gray argues that R 2 is a good indicator of the relationship between cash and futures prices when estimating OLS regressions based on the model specified in equation (1). Kofi uses R 2 as a measure of the forward pricing performance of commodity futures in this context. Futures prices for continuous inventory commodities were found to outperform their discontinuous inventory counterparts, which is consistent with the theory of storage. In addition, Kofi s results indicate that futures prices with longer forecast horizons were associated with lower R 2 values, and hence provided poorer forecasts of future cash prices. Similarly in our study, R 2 estimates are, in general, lower across all contract month regressions as the forecast horizon was increased. One- and two-monthahead futures prices for most of the contract months are able to explain most of the variation in cash prices as reflected in the relatively high R 2 estimates. We also note a consistently large drop in R 2 values for the January, March, and November contracts when moving from postharvest to preharvest forecast horizons. This suggests that unexpected production shocks at harvest time impede the forecasting performance of the futures market to accurately predict harvest time and postharvest time cash prices. 6 In particular, the November futures contract has the poorest forecasting performance in terms of the R 2 criteria. This may be explained by the fact that the November contract for rice represents the expectation of the harvest time cash price for rice, and unrealized rice production at harvest time adds an extra dimension of uncertainty to the forward pricing decision process. This raises a question of the usefulness of the November contract, which is most often used by producers for anticipatory hedges, as a price risk management tool. However, the F test of the joint null hypotheses of (a = 0 and b = 1) is not rejected for any of the November contract regressions, indicating that we cannot reject unbiasedness. In fact, with respect to the OLS regression results for all contracts and all

12 Unbiasedness and the U.S. Rice Futures Market 485 Table 1. OLS regression results of individual contracts Month Lags a a b a R 2 F b a b R 2 F January Contract (n = 13) July Contract c (n = 11) (0.31) (0.04) (0.34) (1.21) (0.13) (0.23) (0.86) (0.10) (0.43) (1.01) (0.12) (0.90) (1.78) (0.22) (0.96) (1.81) (0.21) (0.68) (2.85) (0.38) (0.32) (2.08) (0.23) (0.43) (2.45) (0.30) (0.35) (2.79) (0.30) (0.17) (2.94) (0.35) (0.08) (2.36) (0.27) (0.25) March Contract (n = 13) September Contract (n = 13) (1.19) (0.13) (0.02)* (0.70) (0.09) (0.18) (0.93) (0.11) (0.40) (1.03) (0.13) (0.13) (1.27) (0.15) (0.76) (0.64) (0.08) (0.01)** (1.90) (0.23) (0.99) (0.93) (0.11) (0.02)* (2.15) (0.25) (0.93) (2.05) (0.26) (0.27) (2.80) (0.35) (0.48) (1.72) (0.21) (0.12) May Contract (n = 13) November Contract (n = 13) d (1.06) (0.13) (0.52) (1.45) (0.18) (0.87) (1.38) (0.16) (0.14) (2.12) (0.27) (0.18) (1.60) (0.17) (0.04)* (2.42) (0.30) (0.29) (1.65) (0.19) (0.14) (2.31) (0.29) (0.11) (1.68) (0.19) (0.23) (2.20) (0.27) (0.13) (2.44) (0.27) (0.33) (2.37) (0.29) (0.09) a Standard errors of a and b parameter estimates are provided in parentheses. b F is the joint F test of rejecting the null hypothesis of a = 0 and b = 1(p value of F test is given below F statistic in parentheses). c Note that the July contract has fewer observations because it did not start trading until d The one- and two-month horizons for the November contract contain 14 observations (2 observations from 1986). indicates significant at the 5% level, ** indicates significant at the 1% level.

13 486 Review of Agricultural Economics forecast horizons, the intercept terms are usually not significant, while the slope coefficients are significant and close to one in most regressions. The F test rejects the joint null hypotheses (a = 0 and b = 1) in only four equations. The March contract with a one-month lag, the May contract with a three-month lag, and the September contract with a four-month lag are all rejected at the 5% significance level. The September contract with a three-month lag is rejected at the 1% level. Interestingly, two of the equations in which we reject unbiasedness relate to futures prices observed during February. This price forecasting bias may be attributed to the futures market s inability to adequately account for production shocks, resulting from March time harvests in the Southern Hemisphere rice markets. Residual diagnostics reveal no evidence of serial correlation or heteroskedasticity. The results suggest that rice futures are, in general, weak-form efficient with respect to the long-run unbiasedness hypothesis across both different contract months and forecast horizons. However, these results should be interpreted with some caution due to the small sample size used and uncertainty with respect to the stationary nature of the price series. With these two cautionary notes in mind, we further explored the issue of futures unbiasedness and market efficiency in the following section by using the pooled data set previously described. This alternative data set provided a larger sample size, which allowed us to determine the order of integration underlying the true data-generating process and enabled us to conduct more appropriate tests of the long-run unbiasedness hypothesis. Turning to our pooled data, Dickey Fuller unit root tests were used to determine if the series contained a unit root. The estimated model included a constant term and a trend term. Ljung-Box and Lagrange Multiplier statistics indicated that no lags are necessary to remove residual autocorrelation. The results indicate that both futures and cash prices from this pooled data series are nonstationary (table 2). The results are not sensitive to the removal of the trend term from the model. One problem associated with traditional unit root tests such as the Dickey Fuller test is that the null hypothesis is based on the assumption of a unit root. As a result, nonrejection of the null hypothesis does not imply acceptance of a unit root. Table 2. Unit root tests Price Series Test Statistic Critical Value at 5% Level Dickey Fuller t tests a S t F t S t F t Johansen unit root tests b S t F t a All tests include both a constant term and a time trend. H 0 : series contains a unit root. b All tests include a constant term in the cointegration space. VAR lag length = 1. H 0 : series is stationary.

14 Unbiasedness and the U.S. Rice Futures Market 487 Bearing this in mind, further unit root testing was conducted using the Johansen cointegration procedure. This is a multivariate approach based on deriving maximum likelihood estimates of the cointegrating regression. In this case, the null hypothesis of stationarity is assumed for each of the variables included in the system. Hansen and Juselius designed the test to indicate if the individual series are stationary. Test results, provided in the lower half of table 2, indicate that the null hypothesis of stationarity can be rejected for both the futures and cash price series. The VAR specification was estimated using one to four lags. The optimal number of lags was chosen by using likelihood ratio tests to determine the validity of restrictions imposed by successive reductions of the model s order in conjunction with multivariate Ljung-Box and Lagrange Multiplier tests for autocorrelation. The most appropriate specification was deemed to be a VAR(1). Further Dickey Fuller test results indicate that first differencing is adequate to render the series stationary (table 2). Given that the cash and futures prices are I (1), cointegration techniques can be used to determine if a long-run relationship exists between the cash and futures prices. The Johansen procedure was used to test for cointegration between cash and futures prices. The maximal eigenvalues and trace statistics indicate that the null hypothesis of no cointegration is rejected at 10% significance (table 3). The terms a and b shown in table 3 are the normalized intercept and futures price coefficients in the cointegrating regression. These coefficients appear to be close to the (0, 1) restriction of unbiasedness. Formal testing of the long-run unbiasedness hypothesis was conducted using Johansen likelihood ratio tests on the implied (0, 1) restrictions of a and b. The joint null hypothesis that a = 0 and b = 1 is strongly rejected at the 1% significance level, suggesting there is some evidence that rice futures prices may have been biased predictors of two-month-ahead Arkansas long-grain cash prices. The two individual restrictions of a = 0 and b = 1 were also tested for separately. In this case, the restriction that b = 1 could not be rejected at the 5% significance level. The restriction that a = 0 was again rejected at the 5% level, but not at the 1% level. The results indicate the possibility of either some degree of inefficiency and/or the existence of a small risk premium. Hakkio and Rush point out that cointegration theory is based on long-term relationships between variables, but as mentioned previously, the rice futures contract has only existed since As such, only 78 observations were used in our estimation, which may not provide Table 3. Cointegration test results trace max a b k= 0 k = 0 a = 0, b = 1 a = 0 b = (17.79) (10.29) (0.00) (0.03) (0.11) The trace and max critical values at the 10% level are shown in parentheses below the test statistics. The cointegrating vector is normalized with respect to S t. P values for the various hypotheses are shown in parentheses below the test statistics. VAR lag length = 1.

15 488 Review of Agricultural Economics Table 4. Error-correction model results: S t = λ + ρe t 1 + β F t 1 + v t λ ρ β Wald Test H 0 : β = 1 H 0 : ρ = (0.08) (0.34) (0.34) (0.17) (0.80) (0.82) Residual diagnostics R 2 W B-P D-W LM(4) (0.01) (0.01) (0.20) Standard errors are shown in parentheses below the coefficient estimates for λ, ρ, and β. T statistics and p values (in parentheses) are shown for individual hypotheses ρ = 1and β = 1. P values for joint hypothesis: ρ = 1, β = 1, are shown in parentheses below the Wald-statistic distributed χ2 2 (chi-square with 2 degrees of freedom). B-P stands for Breusch Pagan heteroskedasticity test statistic. W stands for White heteroskedasticity test statistic. D-W stands for Durbin Watson statistic. LM(4) stands for Lagrange Multiplier test statistic for fourth-order serial correlation. enough information to make any strong conclusions about long-run unbiasedness. We return to the issue of long-run unbiasedness when discussing the results of our dynamic error-correction model specifications below. Two different approaches were followed to test for short-run unbiasedness. We first imposed the long-run unbiasedness (0, 1) restrictions in our ECM specification. The ECM was initially estimated with zero to six lags of S t 1 and F t 1. Following Engle and Granger, only significant lags were retained. No lags of S t 1 or F t 1 were found to be significant, suggesting all relevant information from past prices is incorporated into current futures prices. Final results for the model with no dynamic lag structure of past cash and futures prices are presented in table 4. Residual diagnostics reported in the lower half of table 4 reveal no evidence of serial correlation. Breusch Pagan and White tests indicated possible heteroskedasticity, so t and Wald statistics were calculated using White s heteroskedastic-consistent standard errors. The joint null hypothesis that ρ = 1, β = 1 cannot be rejected with a χ2 2 statistic of Similarly, we fail to reject either of the hypotheses restrictions when testing for them separately. These results strongly support the hypothesis that the rice futures market is unbiased, and hence efficient in the short-run. The parameter estimates for the coefficient on the error-correction term and the first differenced futures price term are numerically close to negative one and unity, respectively. The magnitude of the error-correction term coefficient indicates the speed of adjustment of any disequilibrium towards the long-run equilibrium state of unbiasedness and market efficiency. The parameter estimate of the error-correction term coefficient suggests that the futures prices adjust instantaneously to any temporary perturbation to the long-run equilibrium relationship. The coefficient on the error-correction term, as well as being close to negative one, is also highly significant. This is consistent with the result that the futures and cash markets are

16 Unbiasedness and the U.S. Rice Futures Market 489 cointegrated and prices adjust quickly to any temporary disequilibrium. This failure to reject short-run unbiasedness is inconsistent with our previous result of long-run biasedness. To readdress the conflicting evidence with respect to our long-run unbiasedness results, we also used a second approach developed by Beck. We reestimated the ECM, instead imposing the long-run restriction, a = 0.52 and b = 1.05, derived from our cointegration results. Following this approach, we allow for the possible existence of a risk premium term that may have led to the rejection of long-run unbiasedness in our cointegration tests. In this case, the ECM was specified without the constant term, λ, as we now have a nonzero constant term, a, embedded in the error-correction term: (8) m S t = ρ e t 1 + β F t 1 + β i F t i + i=2 k ψ j S t j + v t. j=1 As previously, short-run unbiasedness requires the restrictions ρ = 1, β = 1, and β i = ψ i = 0 from equation (8) to hold. The results presented at the top of table 5, with standard errors adjusted for heteroskedasticity, indicate that the joint restrictions are strongly rejected by the Wald test, χ3 2 = In addition, we find a significant coefficient on cash price changes lagged one period, ψ 1 = 0.45, which would suggest relevant information contained in past cash prices could be used to predict the current period s change in the cash price. This would suggest that the rice futures market is, in fact, biased, assuming this particular model specification. However, estimation of this alternative model specification failed to yield a significant error-correction term, ρ = 0.14, which is inconsistent with our previous result of cointegration between the futures and cash prices series. This inconsistency implies the long-run restrictions, a = 0.52 and b = 1.05, are misspecified. This result provides us with additional support that the true longrun cointegrating relationship conforms to the long-run unbiasedness restrictions, a = 0 and b = 1. Therefore, we tentatively conclude that although the hypothesis of long-run unbiasedness was formally rejected by the Johansen cointegration tests, this rejection may be attributable to our small sample size. For comparative purposes, we also show unbiasedness test results in table 5 for the first differences regression (equation (5)) and for the basis regression (equation (6)). The unbiasedness hypothesis is strongly rejected with respect to parameter estimates derived from the first differences equation. However, this model specification performs comparatively poorly in terms of R 2 and residual correlation criteria suggesting it misspecifies the dynamics through omission of an error-correction term. In contrast, we fail to reject unbiasedness using the basis equation. Standard errors were again adjusted for heteroskedasticity using White s correction. Thus, it can be seen that the basis model specification gives us equivalent results as our ECM specification in equation (3). This is consistent with the fact that our ECM specification nests the basis equation. We now turn our attention to the forecasting test results of short-run, weak-form market efficiency (table 6). The futures market forecasts are found to be superior to the ARIMA forecasts for all forecasting horizons using both the RMSE and Theil s inequality coefficient criteria. In terms of RMSE, the forecasting accuracy of both

17 490 Review of Agricultural Economics Table 5. A comparison of unbiasedness tests using various specifications S t = ρe t 1 + β F t 1 + ψ 1 S t 1 + v t ρ β ψ 1 Wald Test a H 0 : β = 1 H 0 : ρ = (0.13) (0.24) (0.22) (0.00) (0.71) (0.00) Residual diagnostics R 2 W B-P DW LM(4) (0.01) (0.07) (0.49) Equation (5): S t = A + B F t 1 + u t A B Wald Test b H 0 : A = 0 H 0 : B = (0.13) (0.12) (0.00) (0.96) (0.00) Residual diagnostics R 2 W B-P DW LM(4) (0.52) (0.40) (0.07) Equation (6): S t = A + B( F t 1 S t 1 ) + u t A B Wald Test b H 0 : A = 0 H 0 : B = (0.08) (0.34) (0.15) (0.23) (0.95) Residual diagnostics R 2 W B-P DW LM(4) (0.00) (0.00) (0.15) a P values for the joint hypothesis, ρ = 1, β = 1, ψ 1 = 0, are shown in parentheses below the Wald statistic. b P values for the joint hypothesis, A = 0 and B = 1, are shown in parentheses below the Wald statistic. All Wald statistics are distributed: χ2 2 (chi-square with 2 degrees of freedom). the futures prices and the ARIMA model decline over time. Based on Theil s U coefficients, the futures price is superior to a no-change random walk forecast for all forecasting horizons. On the other hand, the ARIMA model failed to beat the random walk model over all forecasting horizons. This result is not entirely inconsistent with earlier studies employing ARIMA forecasting models. For example, Rauser and Carter obtained Theil inequality coefficients in excess of unity using