Market Efficiency in spot and futures Market of precious metal

Paper details

Chose an asset to test market Efficiency in The Derivatives Market for market efficiency in the spot and futures market. use time series data on the spot and futures price of their

asset obtained from Bloomberg, Google Finance or Data Stream. Students will be introduced to the concepts of unit root testing and co-integration analysis, to how these techniques can be easily

applied in Excel. Find data of spot and future price from Bloomberg and use Eviews to do the unit root testing and co-integration test. Key reading: Dougherty, C. (2011). Introduction to

Econometrics (4th Edition). Oxford University Press.Chapter 13.

Arbitrage, Risk Premium, and
Cointegration Tests of the
Efficiency of Futures Markets
Ying-Foon Chow*

1. INTRODUCTION
   The aim of this article is to provide a new perspective on testing
   the efficiency of commodity futures markets. As discussed in
   Fama and French (1987), there are two popular theories on
   commodity futures prices. The risk premium hypothesis splits a
   futures price into a forecast for a future spot price and an
   expected risk premium. The cost-of-carry model, or the theory
   of storage, assigns the difference between the current spot and
   futures prices to interest foregone in storing the commodity,
   warehousing costs, and a convenience yield from holding an
   inventory. Using various aspects of these two models, many
   empirical studies have tested the efficiency of futures markets.
   The issue of efficiency, however, still remains unresolved, due
   in part to the use of different methodologies and sample
   periods.
   The lack of a generally accepted model of the risk premium
   and convenience yield, as well as the need to work with a
   Journal of Business Finance & Accounting, 28(5) & (6), June/July 2001, 0306-686X
   ß Blackwell Publishers Ltd. 2001, 108 Cowley Road, Oxford OX4 1JF, UK
   and 350 Main Street, Malden, MA 02148, USA. 693

• The author is from the Department of Finance, The Chinese University of Hong Kong.
  He is grateful to Gary Koop, Thomas McCurdy, Angelo Melino, Peter Pauly, and especially
  James Pesando for many discussions and comments. He also wishes to thank the Editor,
  two anonymous referees, Eric Chang, Roger Huang, Raymond Kan, Michael McAleer,
  Hua Zhang, Xiaodong Zhu, and seminar participants at the Bank of Canada, the Chinese
  University of Hong Kong, the University of Toronto and the University of Western
  Australia for helpful comments and suggestions. Any remaining errors are the author’s
  own. (Paper received November 1998, revised and accepted August 2000)
  Address for correspondence: Ying-Foon Chow, Department of Finance, The Chinese
  University of Homg Kong, Shatin, New Territories, Hong Kong.
  e-mail: [email protected]
  stationary time series, have prevented the authors of many past
  studies from conducting general tests of the Efficient Market
  Hypothesis (EMH) in futures markets. While applying the
  difference operator to each variable can achieve stationarity in
  the data, this imposes too many unit roots in the system if the
  variables are cointegrated, thereby rendering standard methods
  of statistical inference inappropriate. To properly account for the
  non-stationary time series, researchers have begun to use the
  cointegration framework developed by Engle and Granger
  (1987) to test for market efficiency since the 1990s, e.g.,
  MacDonald and Taylor (1988), Sephton and Cochrane (1991),
  Chowdhury (1991), Krehbiel and Adkins (1993) and Moore and
  Cullen (1995). Based on recent developments in econometric
  techniques for models with non-stationary variables, this article
  exploits the non-stationarity of interest rates, spot and futures
  prices to test the EMH using data on levels in a cointegration
  framework.
  The analysis in this article is of interest for several reasons.
  First, Park and Phillips (1989) have shown that a stationary
  variable can be omitted from a cointegrating regression without
  affecting the consistency of the coefficient estimates nor the
  power of statistical procedures for hypothesis testing. In the
  context of a conventional risk premium hypothesis, if the
  subsequent realized spot price is regressed on the futures price
  and a risk premium, the order of variability of the non-stationary
  futures price should dominate the variability of the risk
  premium, which is usually assumed to be stationary. The tests
  in this article thus allow a significant weakening of the usual
  assumption of a constant risk premium, thereby contrasting with
  more general tests which consider a joint hypothesis of market
  efficiency and the specific identification of a risk premium in the
  futures market. In a similar spirit, the theory of storage can be
  tested directly without specifically identifying the dynamics of the
  carrying cost, which include the convenience yield and are
  typically assumed to be stationary. In particular, this article notes
  that the theory of storage can be viewed as a special case of the
  risk premium hypothesis where the `risk premium’ is identified as
  the interest rate. Finally, the methodology is used to examine the
  issue of cross-market or semi-strong form efficiency in the
  commodity futures markets.
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  In the following, the EMH and its implications for futures
  markets are briefly discussed in Section 2, with Sub-sections 2(i)
  and 2(ii) presenting the two alternative hypotheses in terms of a
  cointegrated system. Section 3 describes the data used in this
  article. As discussed in Engle and Granger (1987), the first stage
  in testing cointegration between variables is to determine the
  order of integration for an individual time series. The results for
  different tests of a unit root are reported in Section 4. As the null
  hypothesis of a unit root cannot be rejected for the series
  considered, the tests of cointegration and parameter restrictions
  are reported in Section 5. Empirical findings on the issue of
  cross-market efficiency are discussed in Section 6, followed by
  concluding remarks.

1. EFFICIENT MARKET HYPOTHESIS AND COINTEGRATION
   As mentioned above, there are two popular models of commodity
   futures prices. While the cost-of-carry view of futures prices is not
   controversial as stated in Fama and French (1987), the risk
   premium hypothesis has been the subject of a long and
   continuing controversy. However, these models should be viewed
   as complementary rather than competing views of futures
   pricing. In the following, these two models are briefly discussed.
   In particular, their alternative formulations in the cointegration
   framework and parameter restrictions implied by the EMH are
   presented.
   (i) Testing the Risk Premium Hypothesis: A Restatement
   Under the conventional risk premium hypothesis as in Fama and
   French (1987), the test of market efficiency in a futures market
   can be examined based on the empirical model:
   st‡k ˆ 0 ‡ 1ft;t‡k ‡ 2t;t‡k ‡ t‡k ; …1†
   where ft;t‡k is the (logarithmic) price of a k-period futures
   contract at time t, st‡k is the (logarithmic) spot price of an asset at
   time t ‡ k, t;t‡k is the mean zero time-varying component of the
   k-period risk premium, and t‡k is a mean zero shock
   uncorrelated with any information publicly available at time t.
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   Market efficiency implies that the error term t‡k should be
   orthogonal to the time t information set, and a testable
   restriction that 0 ˆ Et…t;t‡k †, 1 ˆ 1 and 2 ˆ 1. Note that
   equation (1) obviously requires a model of the risk premium,
   thus the null hypothesis of interest can be re-stated as follows: (a)
   the futures market is informationally efficient, and (b) the risk
   premium is identified by a given pricing model for the futures
   market.
   The theory of cointegrated processes as in Engle and Granger
   (1987) yields a very simple but robust test of a necessary condition
   for market efficiency which does not require identification of the
   risk premium in the conventional framework. The risk premium
   has been typically considered (covariance) stationary on
   theoretical grounds. On the other hand, levels of spot and
   futures prices have been found to follow processes with very
   persistent shocks, or I…1† in the literature. We can then see that
   cointegration between st‡k and ft;t‡k in the form of:
   st‡k ˆ 0 ‡ 1ft;t‡k ‡ ut‡k ; …2†
   where ut‡k ˆ 2t;t‡k ‡ t‡k , is a necessary condition for market
   efficiency under the conventional risk premium hypothesis:
   Assuming the risk premium is stationary, if st‡k and ft;t‡k are not
   cointegrated, then that implies st‡k and ft;t‡k will tend to deviate
   without bounds. It follows that ft;t‡k has little or no power to
   predict the movement of st‡k , which is clearly inconsistent with
   the EMH in the context of futures markets.
   Furthermore, from equation (1) and under the hypothesis that
   st‡k and ft;t‡k are cointegrated, the excess return on a futures
   contract may be expressed as:
   st‡k ÿ ft;t‡k ˆ 0 ‡ 2t;t‡k ‡ wt‡k ;
   where wt‡k ˆ …1 ÿ 1†ft;t‡k ‡ t‡k . If ft;t‡k is I…1† and 1 6ˆ 1, then
   it follows that wt‡k is also I…1†. This can be interpreted as
   implying that new information is not incorporated into the
   market price. That is, the information which causes the forecast
   (ft;t‡k ) and the realisation (st‡k ) to diverge is not incorporated
   into subsequent forecasts (i.e., ft‡i‡kjt‡i). It follows that market
   efficiency, given that the risk premium is stationary, corresponds
   to st‡k and ft;t‡k being cointegrated with a cointegrating
   coefficient of one in equation (2).
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   It should now be obvious that the cointegration framework
   provides a re-interpretation of past tests of the EMH, particularly
   the commonly tested Unbiased Expectations Hypothesis (UEH)
   which represents a joint test of market efficiency and risk
   neutrality.1 With the presence of stochastic trends in the asset
   prices, together with the assumption of a stationary risk
   premium, the EMH can be examined by testing whether (a)
   the futures and spot prices are cointegrated and (b) 1 ˆ 1 in
   equation (2). Only if one is willing to make the additional
   assumption that the stationary risk premium has a zero mean, can
   the UEH then be tested with the additional restriction of 0 ˆ 0
   in equation (2). Even in this case, however, the existence of a
   serial correlation in the residuals of equation (2) does not
   necessarily constitute evidence against an efficient market
   because the dynamics of the zero mean risk premium are simply
   being transferred to the residuals.
   (ii) Testing the Theory of Storage: An Alternative
   In an effort to reconcile the mixed empirical findings with the
   notion of simple efficiency, Brenner and Kroner (1995) derive
   the conditions for futures market efficiency in the cost-of-carry
   framework. This model uses a no-arbitrage argument by
   factoring in the carrying costs involved in holding an
   underlying asset until maturity. The efficiency of the futures
   market can then be examined based on the following empirical
   model:
   st ÿ ft;t‡k ˆ 0 ‡ 1Rt;t‡k ‡ 2Ct;t‡k ‡ “t; …3†
   where st and ft;t‡k are defined as before, Rt;t‡k ˆ krt;t‡k is the
   continuously compounded rate of return on a k-period risk-free
   bond at time t which pays one dollar at time t ‡ k, and
   Ct;t‡k ˆ kct;t‡k is the continuously compounded k-period net cash
   flow yield accruing to the marginal holder of the spot inventory.
   Since the expected return of a hedged position in any
   commodity market should equal the risk-free return, such a noarbitrage condition implies 1 ˆ ÿ1 and 2 ˆ 1. Observe that,
   just as in equation (1) of the risk premium hypothesis, equation
   (3) also requires a model for the convenience yield before the
   model can be formally tested.
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   The cointegration theory, however, also provides an alternative
   way of examining the cost-of-carry model. As in the case of an
   unobserved risk premium, Brenner and Kroner (1995) argued
   that the (net) convenience yields on holding an asset have
   traditionally been assumed to be stationary. Therefore, assuming
   all variables are I…1† except the Ct;t‡k term, then the results of
   Park and Phillips (1989) imply that the efficiency of the futures
   market can be examined based on the model:
   st ˆ 0 ‡ 1ft;t‡k ‡ 2Rt;t‡k ‡ vt; …4†
   with a testable restriction of 1 ˆ 1 and 2 ˆ ÿ1. It should be
   noted that the constant term (0) reflects the marking-to-market
   effect in a futures market and therefore, as argued in Brenner
   and Kroner (1995), need not be zero. The error term (vt) should
   be serially uncorrelated for simple' efficiency; i.e., profitable arbitrage deviations should not be predictable based on past information. However, the error term in model (4) may not be serially uncorrelated if the convenience yield is serially correlated, but this does not imply persistent risk-free arbitrage profit opportunities. An important result from Engle and Granger (1987) is that if any variables are cointegrated, then they are cointegrated at any lead or lag as well. Therefore, if …st; ft;t‡k ; Rt;t‡k † are cointegrated, then the efficiency of a commodity futures market can be tested in the cost-of-carry framework using the model: st‡k ˆ 0 ‡ 1ft;t‡k ‡ 2Rt;t‡k ‡ vt‡k ; …5† where a necessary condition for futures market efficiency is that …st‡k ; ft;t‡k ; Rt;t‡k † be cointegrated with 1 ˆ 1 and 2 ˆ ÿ1 in equation (5). Interestingly, we can view this result as a special case of the risk premium hypothesis by comparing equation (5) and equation (1). The risk premium in previous studies is typically identified through some general equilibrium asset pricing model and thus is (covariance) stationary by construction. In the cost-of-carry framework, therisk premium’
   is essentially identified as the interest rate, i.e., t;t‡k ˆ ÿRt;t‡k .
   That is, the risk premium will have time series properties
   identical to the interest rate under the no-arbitrage condition.
   Despite an inconclusive investigation as to whether interest rates
   contain unit roots, it is generally agreed that they are strongly
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   persistent in levels. If, however, interest rates are better
   approximated with a near-unit root process, then, from the
   discussions of equation (2), it is indeed possible to find that the
   futures price is not cointegrated with the realised spot price, or
   they are cointegrated but the cointegrating coefficient is not
   equal to one due to the omission of a relevant variable.
2. DATA
   Four precious metals with a lengthy history of futures contract
   delivery periods ± gold, silver, palladium and platinum ± are
   chosen here for investigation. Since the price of precious metal
   futures is generally an increasing function of the time to
   maturity, it can be shown that upon maturity the benefits from
   holding the asset (including convenience yield and net of storage
   costs) are less than the risk-free rate. Therefore, it is assumed that
   contracts mature on the first Wednesday of the delivery month as
   in Pindyck (1993). A contract price may sometimes be
   constrained by exchange-imposed limits on daily price moves.
   In those cases, prices for the preceding Tuesday are used. If those
   prices likewise hit the limit-move, prices for the following
   Thursday are used, or if those are constrained, the preceding
   Monday and then the following Friday are chosen. In all cases,
   the settlement price of the gold and silver contracts traded on the
   Commodity Exchange Inc., as well as the palladium and platinum
   futures contracts traded on the New York Mercantile Exchange,
   are collected from the Commodity System Inc. databank.
   Constant-to-maturity futures contracts of one-month maturity
   are collected for gold and silver, and three-month maturity
   contracts are collected for palladium and platinum. That is, the
   futures prices ft;t‡k are obtained for k ˆ 1 (gold and silver) and 3
   (palladium and platinum). Following the tradition of the
   literature, the spot price st is measured by the price of the
   corresponding maturing contract. Thus both the spot and futures
   prices and the time interval between the two delivery dates are
   known for exactly the same commodity. It should be noted that
   the number of observations of the constant-to-maturity contracts
   is always less than the number of months in the sample period,
   since the futures contracts do not mature each month.
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   The risk-free rate series is defined as the interest rate on a US
   Treasury bill (T-bill) that matures nearest to the trading of the
   relevant futures contract. Therefore, the holding periods for the
   T-bill and the futures contract are matched as closely as possible.
   The data are extracted from the Center for Research on Security
   Prices (CRSP) bond data tape. The sample period for all series is
   February 1975 to February 2000 for gold, February 1970 to
   February 2000 for silver, June 1977 to March 2000 for palladium
   and April 1970 to April 2000 for platinum. Descriptive statistics of
   the data are presented in Table 1. Finally, with the exception of
   the interest rates, the price series used in this article are the
   natural logarithmic values of the actual prices.
   Table 1
   Descriptive Statistics of the Data
   Market Variable N Mean Std. Dev. Min. Max.
   Gold Spot Price 301 344.88 108.11 104.80 713.00
   Futures Price 346.40 109.89 105.00 695.50
   Interest Rate (%) 0.55 0.23 0.22 1.35
   Silver Spot Price 361 602.02 427.38 130.00 3885.00
   Futures Price 602.30 411.35 130.40 3637.00
   Interest Rate (%) 0.53 0.22 0.22 1.35
   Palladium Spot Price 92 143.25 90.01 42.40 690.90
   Futures Price 136.67 68.81 43.00 394.50
   Interest Rate (%) 1.77 0.73 0.75 4.01
   Platinum Spot Price 121 348.92 146.97 101.20 806.50
   Futures Price 349.89 147.17 102.50 722.60
   Interest Rate (%) 1.69 0.68 0.68 3.85
   Notes:
   The gold and platinum contracts are quoted in dollars per troy ounce, the silver contracts
   are quoted in cents per troy ounce, and the palladium contracts are quoted in dollars per
   ounce. The standard delivery months are February, April, June, August, October and
   December for gold; January, March, May, July, September and December for silver;
   March, June, September and December for palladium; and January, April, July and
   October for platinum. In addition, there are monthly supplemental contracts in gold and
   silver to fill in the months between the standard contracts.
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3. TESTS OF UNIT ROOT AND STATIONARITY
   Before estimating cointegrating regressions, it is conventional to
   test the order of integration for the variables used. The
   augmented Dicky±Fuller (ADF) tests are often used in the
   literature to test for the possible existence of unit roots.
   Alternatively, as in this article, one can use the non-parametric
   Zt and Za tests of Phillips and Perron (1988), which allow for
   heterogeneously distributed, and possibly dependent, innovation
   sequences in the series considered. To examine the possible
   existence of deterministic trends in the data, the testing strategy
   suggested by Dolado, Jenkinson and Sosvilla-Rivero (1990) is
   used in choosing the appropriate regression model. The longrun' variance of the regression residuals is estimated using the optimal bandwidth selection procedure of Andrews and Monahan (1992). Furthermore, the first-order autoregression prewhitened QS kernel estimator is used for Zt and Za tests, since Maekawa (1994) shows that the prewhitened kernel estimator significantly improves the performance of these unit root tests. On the other hand, it is at least conceptually preferable to test for the stationarity of an observed time series since many other tests have failed to reject the null hypothesis of integration. In particular, the frequency of incorrect conclusions is decreased when a test of stationarity is conducted in conjunction with a standard unit root test, relative to the application of unit root tests alone. The G…0; q† test proposed in Park (1990) and the  test proposed in Kwiatkowski, Phillips, Schmidt and Shin (1992) for the null hypothesis of level stationary series are conducted here tocross validate’ the unit root tests, after the procedure of
   Dolado, Jenkinson and Sosvilla-Rivero (1990) revealed no
   evidence of a trend in the series considered. As suggested in
   Ogaki and Park (1997), q is chosen to be 3 for the G…0; q† test.
   The results of the Zt, Za, G…0; 3† and  tests are reported in
   Table 2. The results from the ADF tests are similar to the Zt and
   Za tests, and are therefore not reported here. These results are
   consistent with the hypothesis that the data considered have only
   one unit root. While not reported here, the first difference in all
   series considered significantly rejects the null hypothesis of a unit
   root.
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   Table 2
   Tests of Unit Roots
   Regression model: yt ˆ 0 ‡ 1ytÿ1 ‡ ut
   H0: Unit Root H0: Level Stationary
   Zt Za G(0,3) 
   Gold Spot ÿ2.017 ÿ6.369 11.175 0.412z
   (0.280) (0.316) (0.011)
   Futures ÿ1.965 ÿ6.100 11.067 0.419z
   (0.302) (0.336) (0.011)
   Interest rate ÿ2.050 ÿ9.674 10.119 0.603y
   (0.266) (0.143) (0.018)
   Silver Spot ÿ2.347 ÿ8.899 13.870 0.374y
   (0.158) (0.173) (0.003)
   Futures ÿ2.329 ÿ8.586 13.806 0.382y
   (0.164) (0.187) (0.003)
   Interest rate ÿ2.383 ÿ9.934 8.164 0.382y
   (0.147) (0.135) (0.043)
   Palladium Spot ÿ0.633 ÿ2.433 15.093 0.771*
   (0.857) (0.722) (0.002)
   Futures ÿ1.337 ÿ5.050 13.806 0.704y
   (0.609) (0.421) (0.003)
   Interest rate ÿ2.025 ÿ7.587 12.371 1.032*
   (0.276) (0.229) (0.006)
   Platinum Spot ÿ1.470 ÿ4.033 7.086 0.532y
   (0.546) (0.529) (0.069)
   Futures ÿ1.618 ÿ4.333 6.922 0.523y
   (0.470) (0.496) (0.074)
   Interest rate ÿ2.328 ÿ10.014 8.678 0.400z
   (0.165) (0.127) (0.034)
   Notes:
   The p-values are reported in parentheses. The p-values for the Zt and Za tests are
   computed using the results of MacKinnon (1996); while G(0,3) has an asymptotic 2…3†
   distribution under the null hypothesis. The 1%, 5% and 10% critical values of the
   Kwiatkowski, Phillips, Schmidt and Shin (1992)  test are 0.739, 0.463 and 0.347, so *, y
   and z indicate the rejection of the null hypothesis at the 1%, 5% and 10% levels. The
   regression model is chosen by the testing strategy suggested in Dolado, Jenkinson and
   Sosvilla-Rivero (1990). The `long-run’ variance of ut is estimated using the QS kernel with
   the optimal bandwidth selection procedure of Andrews and Monahan (1992), and a firstorder autoregression prewhitening method for the Zt and Za , tests as suggested in
   Maekawa (1994).
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4. TESTS OF COINTEGRATION & PARAMETER RESTRICTIONS
   (i) Tests of Cointegration
   The residual-based Zt and Za tests of Phillips and Ouliaris (1990)
   are first conducted to assess the adequacy of the cointegrating
   models (2) and (5) over the sample period. Again, the long-run' variance of the regression residuals is estimated using the prewhitened QS kernel estimator with the optimal bandwidth selection procedure of Andrews and Monahan (1992). Note that these tests address the question of testing the null hypothesis of no cointegration rather than cointegration between the variables considered. Since the hypothesis of cointegration is of primary interest, it is often argued that cointegration would be a more natural choice of the null hypothesis. Therefore, the variableaddition H…p; q† test of Park (1990) and the residual-based C test of Shin (1994) are used to test cointegration as the null hypothesis for the models (2) and (5). As discussed in Ogaki and Park (1997), the H…0; q† statistic tests the deterministic cointegration restriction, while the H…1; q† statistic tests for stochastic cointegration between variables, and the choice of q should be small. Therefore, following Ogaki and Park (1997), the H…0; 1† and H…1; 3† tests are performed in this article. The results of the above tests are presented in Table 3. The results of the Zt and Za tests seem to support the alternative of cointegration between futures and spot prices in all markets with or without the interest rates. The direct tests of cointegration also imply similar results: Under the assumption that the variables are stochastically cointegrated, the deterministic cointegration restriction cannot be rejected in all markets for both models in terms of the H…0; 1† tests. The results of the H…1; 3† tests for stochastic cointegration indicate that the cointegration hypothesis can be rejected for the palladium and the platinum markets in both the stationary risk premium framework and the cost-of-carry framework. In addition, the C tests cannot reject cointegration in model (2) for all markets, while model (5) is rejected marginally for the silver market. Although the results may seem to indicate a certainfragility’ in
   the cointegration hypothesis, there is evidence to support the
   idea that both model (2) and model (5) are cointegrated systems.
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   Table 3
   Tests of Cointegration
   Model (2): st ˆ 0 ‡ 1ftÿk;t ‡ ut
   H0: No Cointegration H0: Cointegration
   Zt Za H(0,1) H(1,3) C
   Gold ÿ14.720 ÿ257.040 0.569 0.244 0.085
   (<0.001) (<0.001) (0.451) (0.885)
   Silver ÿ16.025 ÿ305.723 0.175 3.114 0.064
   (<0.001) (<0.001) (0.676) (0.211)
   Palladium ÿ8.840 ÿ86.676 2.210 7.796 0.184
   (<0.001) (<0.001) (0.137) (0.020)
   Platinum ÿ10.682 ÿ118.926 1.690 6.770 0.185
   (<0.001) (<0.001) (0.194) (0.034)
   Model (5): st ˆ 0 ‡ 1ftÿk;t ‡ 2Rtÿk;t ‡ vt
   H0: No Cointegration H0: Cointegration
   Zt Za H(0,1) H(1,3) C
   Gold ÿ14.764 ÿ257.859 0.444 2.448 0.128
   (<0.001) (<0.001) (0.505) (0.294)
   Silver ÿ16.226 ÿ309.625 0.885 3.072 0.209z
   (<0.001) (<0.001) (0.347) (0.215)
   Palladium ÿ9.035 ÿ87.456 0.533 9.860 0.144
   (<0.001) (<0.001) (0.465) (0.007)
   Platinum ÿ11.060 ÿ123.699 0.979 9.508 0.110
   (<0.001) (<0.001) (0.322) (0.009)
   Notes:
   The p-values are reported in parentheses. The p-values for the Zt and Za tests are
   computed using the results of MacKinnon (1996); while H…p; q† has an asymptotic
   2…q ÿ p† distribution under the null hypothesis. The 1%, 5% and 10% critical values of
   Shin (1994) C test are 0.533, 0.314 and 0.231 for model (2), and 0.380, 0.221 and 0.163
   for model (5), so z indicates the rejection of the null hypothesis at the 10% level. The
   long-run' covariance parameters of ut and vt are estimated using the QS kernel with the optimal bandwidth selection procedure of Andrews and Monahan (1992), and a firstorder autoregression prewhitening method for the Zt and Za tests, as suggested in Maekawa (1994). 704 CHOW ß Blackwell Publishers Ltd 2001 This indicates that the futures price can indeed help predict about the future movements of the spot price in terms of the conventional risk premium hypothesis represented by model (2). While in the cost-of-carry framework of model (5), the variables comprising the no-arbitrage condition do seem to form a stationary equilibrium in the long run. (ii) Tests of Parameter Restriction Despite their popularity, however, a limitation of the Engle± Granger type of residual-based tests, such as the Zt and Za tests above, is that no strong statistical inference can be drawn with respect to the parameters, e.g., 1 and 2 in model (5), which are the main interest here. Although the ordinary least squares (OLS) estimator can be shown to be super-consistent under cointegration, Stock (1987) has argued that the estimated standard errors may be misleading for hypothesis testing. There have been several procedures developed recently to estimate long-run relationships, which allow one to formally conduct asymptotic chi-square tests on the cointegrating parameters. The dynamic OLS (DOLS) methodology of Stock and Watson (1993) is used here, since Stock and Watson (1993) and Montalvbo (1995) suggest that the DOLS estimator performs systematically better than the other estimators based on Monte Carlo simulation experiments. The parameter estimates and the hypothesis test results of model (2) are reported in the upper panel of Table 4. It can be seen that the estimates for 1 are very close to the hypothesized value of one, ranging from 0.997 (gold) to 1.036 (palladium). Observe that the estimates of 0 are fairly close to zero, ranging from ÿ0:181 (palladium) to 0.010 (gold). Thus, it may be of interest to see if there is any support for theunbiased
   expectations’ hypothesis where 1 ˆ 1 and 0 ˆ 0 in model (2).
   It can be seen that this hypothesis is significantly rejected in all
   markets. As often argued in the literature, such rejections do not
   mean that the markets are inefficient. Rather, it is more likely
   that the risk premium, if stationary, has a non-zero mean over
   time. However, the hypothesis that 1 ˆ 1 is also rejected for all
   markets except the silver market. Therefore, it seems that the
   futures markets for these precious metals do not conform to the
   EFFICIENCY OF FUTURES MARKETS 705
   ß Blackwell Publishers Ltd 2001
   necessary condition of the EMH under the risk premium model
   (2).
   The lower panel of Table 4 reports the parameter estimates
   and test results of the cost-of-carry model (5). The estimates of 1
   are again very close to the hypothesized value of one, ranging
   from 0.999 (gold) to 1.026 (palladium). On the other hand,
   there is a considerable variation in the estimates of 2, ranging
   Table 4
   Tests of Cointegrating Parameters
   Model (2): st ˆ 0 ‡ 1ftÿk;t ‡ ut
   H0 : 0 ˆ 0
   b0 b1 H0 : 1 ˆ 1 and 1 ˆ 1
   Gold 0.010 0.997 ÿ2.248 164.989
   [0.007] 0.001 (<0.001)
   Silver ÿ0.008 1.000 0.206 36.628
   [0.011] 0.002 (<0.001)
   Palladium ÿ0.181 1.036 3.020 17.059
   [0.057] 0.012 (<0.001)
   Platinum ÿ0.082 1.012 2.711 63.653
   [0.025] 0.004 (<0.001)
   Model (5): st ˆ 0 ‡ 1ftÿk;t ‡ 2Rtÿk;t ‡ vt
   H0 : 1 ˆ 1
   b0 b1 b2 H0 : 1 ˆ 1 H0 : 2 ˆ ÿ1 and 2 ˆ ÿ1
   Gold 0.003 0.999 ÿ1.070 ÿ0.429 ÿ0.480 0.616
   [0.005] [0.001] 0.145 (0.631) (0.735)
   Silver ÿ0.033 1.006 ÿ2.048 2.992 ÿ1.949 8.996
   [0.011] [0.002] 0.538 (0.052) (0.011)
   Palladium ÿ0.111 1.026 ÿ1.098 2.306 ÿ0.245 6.576
   [0.057] [0.011] 0.401 (0.808) (0.037)
   Platinum ÿ0.083 1.016 ÿ1.365 4.609 ÿ1.406 21.368
   [0.020] [0.003] 0.260 (0.163) (<0.001)
   Notes:
   The standard errors and p-values are, respectively, reported in brackets and parentheses.
   The standard errors are calculated using the method outlined in Stock and Watson
   (1993).
   706 CHOW
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   from ÿ1:098 (palladium) to ÿ2:048 (silver). The hypothesis that
   1 ˆ 1 and 2 ˆ ÿ1 jointly is then tested under the assumption
   of a stationary convenience yield. While this hypothesis is not
   rejected for the gold market, it is rejected for the other markets
   under conventional significance levels. The hypotheses that
   1 ˆ 1 and 2 ˆ ÿ1 are also tested individually. It can be seen
   from Table 4 that 1 ˆ 1 can be rejected in all markets except the
   gold market. This may indicate that the futures price is not an
   unbiased estimate of the future spot price for these markets. On
   the other hand, despite the relatively high standard errors of the
   estimates, the hypothesis of 2 ˆ ÿ1 can only be rejected
   marginally for the silver market and cannot be rejected for the
   other metals. Based on these results, one might conclude that the
   no-arbitrage condition is not satisfied in at least one market
   considered here.
   In summary, when comparing the results in Table 4, it can be
   seen that while the gold futures market may not be consistent
   with the EMH under the risk premium framework, it is consistent
   with the no-arbitrage condition as implied by the cost-of-carry
   model. On the other hand, even though the no-arbitrage
   condition is rejected in the silver market, such results do not
   indicate market inefficiencies. Rather, it is more likely that the
   risk premium model is more appropriate than the cost-of-carry
   model for the silver futures market. However, the results in Table
   4 do indicate that neither the risk premium hypothesis nor the
   cost-of-carry framework could model the platinum futures market
   satisfactorily.
5. TESTS OF CROSS-MARKET EFFICIENCY
   Using data from the London Metal Exchange, Hsieh and
   Kulatilaka (1982) and Canarella and Pollard (1986) provided
   multi-market tests of market efficiency, arguing that the metals
   under study are traded in the same location and hence a wider
   view of efficiency is required. The issue examined in these studies
   was whether past forecast errors across markets are useful in
   predicting errors in other markets, since traders should be aware
   of activities across the exchange. That is, the multi-market test
   may be considered a semi-strong form test. Since the tests in this
   EFFICIENCY OF FUTURES MARKETS 707
   ß Blackwell Publishers Ltd 2001
   article allow for risk premiums to be incorporated in the error
   term and the intercept, the models used here will provide less
   ambiguous tests of market efficiency.
   As the maturity cycle for palladium contracts does not
   correspond to that of platinum, we shall concentrate on crossmarket efficiency in the gold and silver markets instead. It is
   commonly believed that there may be a stable or semi-stable longrun relationship between these prices. If they are then
   cointegrated, however, the extent to which either can be forecast
   is expected to be limited due to the standard EMH. Thus, if gold
   and silver prices are indeed generated by an efficient market,
   they cannot move closely together in the long-run, or one can be
   used to help forecast the other as argued in Escribano and
   Granger (1998).
   We first test if gold and silver prices are cointegrated in either
   the spot or the futures market with the following cointegrating
   regressions:
   s
   i
   t ˆ 0 ‡ 1s
   j
   t ‡ us
   t ; …6†
   f i
   t;t‡1 ˆ 0 ‡ 1f
   j
   t;t‡1 ‡ uf
   t ; …7†
   for i; j ˆ gold and silver. The results presented in Table 5 show
   strong support of the EMH in that there is no evidence of
   cointegration between the gold and silver prices in either market.
   These results are consistent with that of Escribano and Granger
   (1998), who found that cointegration between these prices could
   have occurred during some periods between 1971 to 1990, but
   the two markets are becoming separated thereafter.
   Now we turn to the issue of multi-market efficiency in the
   futures markets. To this end, Table 6 presents the results of the
   following regressions:
   s
   i
   t‡1 ˆ 0 ‡ 1f i
   t;t‡1 ‡ 2f
   j
   t;t‡1 ‡ ut‡1; …8†
   for i; j ˆ gold and silver. The results of the no-cointegration tests
   in the top panel of Table 6 seem to support the alternative of
   cointegration in equation (8). The direct tests of cointegration,
   however, provide somewhat contradictory evidence: Although
   the H…0; 1† tests indicate that a deterministic cointegration
   restriction cannot be rejected in either market when the variables
   are assumed to be stochastically cointegrated, the H…1; 3† tests
   708 CHOW
   ß Blackwell Publishers Ltd 2001
   indicate that the hypothesis of stochastic cointegration among
   these variables can be rejected at conventional levels. On the
   other hand, the C tests indicate that the null hypothesis of
   cointegration cannot be rejected.
   The lower panel of Table 6 provides a possible explanation of
   such findings. The joint hypothesis that 1 ˆ 1 and 2 ˆ 0 cannot
   be rejected in either market under conventional significance
   levels, and the same conclusions can also be made when the
   hypotheses that 1 ˆ 1 and 2 ˆ 0 are tested individually. Note
   that with the addition of the silver futures price, as in the case of
   model (5), the gold futures price is now an unbiased estimator of
   Table 5
   Tests of Cointegration Across Spot Markets and Futures Markets
   Equation (6): si
   t ˆ 0 ‡ 1s
   j
   t ‡ us
   t
   Dependent H0: No Cointegration H0: Cointegration
   Variable
   Zt Za H(0,1) H(1,3) C
   Gold ÿ1.510 ÿ3.374 23.659 101.606 0.265z
   (0.759) (0.853) (<0.001) (<0.001)
   Silver ÿ2.243 ÿ9.944 6.909 111.831 0.257z
   (0.403) (0.362) (0.009) (<0.001)
   Equation (7): f i
   t;t‡1 ˆ 0 ‡ 1f
   j
   t;t‡1 ‡ uf
   t
   Dependent H0: No Cointegration H0: Cointegration
   Variable
   Zt Za H(0,1) H(1,3) C
   Gold ÿ1.560 ÿ3.761 26.934 109.868 0.265z
   (0.740) (0.826) (<0.001) (<0.001)
   Silver ÿ2.181 ÿ9.433 6.726 135.356 0.247z
   (0.435) (0.393) (0.009) (<0.001)
   Notes:
   The p-values are reported in parentheses. The p-values for the Zt and Za tests are
   computed using the results of MacKinnon (1996); while H…p; q† has an asymptotic
   2…q ÿ p† distribution under the null hypothesis. The 1%, 5% and 10% critical values of
   Shin (1994) C test are 0.533, 0.314 and 0.231, so z indicates the rejection of the null
   hypothesis at the 10% level. The `long-run’ covariance parameters of us
   t and uf
   t are
   estimated using the QS kernel with the optimal bandwidth selection procedure of
   Andrews and Monahan (1992), and a first-order autoregression prewhitening method for
   the Zt and Za tests, as suggested in Maekawa (1994).
   EFFICIENCY OF FUTURES MARKETS 709
   ß Blackwell Publishers Ltd 2001
   the future spot price. Nevertheless, these results imply that while
   the futures price from the other market may seem to co-move
   with the own spot price, such co-movements are statistically and
   economically insignificant and may even be spurious. Therefore,
   we still cannot reject the restrictions associated with market
   efficiency for the gold and silver markets.
6. CONCLUDING REMARKS
   To account for the non-stationarity in the data, this article has
   applied recently developed econometric techniques to a test of
   the Efficient Market Hypothesis (EMH) in the conventional risk
   Table 6
   Cointegration Tests of Multi-market Efficiency
   Equation (8): si
   t‡1 ˆ 0 ‡ 1f i
   t;t‡1 ‡ 2f
   j
   t;t‡1 ‡ ut
   Dependent H0: No Cointegration H0 : Cointegration
   Variable
   Zt Za H(0,1) H(1,3) C
   Gold ÿ14.705 ÿ256.677 0.002 4.938 0.090
   (<0.001) (<0.001) (0.963) (0.084)
   Silver ÿ14.545 ÿ254.089 0.001 6.144 0.066
   (<0.001) (<0.001) (0.986) (0.046)
   Dependent H0 : 1 ˆ 1
   Variable b0 b1 b2 H0 : 1 ˆ 1 H0 : 2 ˆ 0 and 2 ˆ 0
   Gold 0.030 1.001 ÿ0.006 0.519 ÿ1.213 1.636
   [0.006] [0.001] 0.005 (0.226) (0.441)
   Silver 0.024 0.996 ÿ0.001 ÿ1.261 ÿ0.065 1.625
   [0.018] [0.003] 0.015 (0.948) (0.444)
   Notes:
   The standard errors and p-values are, respectively, reported in brackets and parentheses.
   The p-values for the Zt and Za tests are computed using the results of MacKinnon (1996);
   while H…p; q† has an asymptotic 2…q ÿ p† distribution under the null hypothesis. The 1%,
   5% and 10% critical values of Shin (1994) C test are 0.380, 0.221 and 0.163. The longrun' covariance parameters of ut are estimated using the QS kernel with the optimal bandwidth selection procedure of Andrews and Monahan (1992), and a first-order autoregression prewhitening method for the Zt and Za tests, as suggested in Maekawa (1994). The standard errors are calculated using the method outlined in Stock and Watson (1993). 710 CHOW ß Blackwell Publishers Ltd 2001 premium framework, and to a test of the complementary hypothesis of the no-arbitrage condition in the cost-of-carry framework, for four precious metal futures markets. The cointegration techniques allows one to test market efficiency under both hypotheses without relying on any specific assumed form of the risk premium or the convenience yield, other than the stationary assumption. In addition, it is argued that the costof-carry model (5) can be viewed as a special case of the risk premium model (2) where therisk premium’ is identified as the
   interest rate. As such, the possibility of a non-stationary risk
   premium in the futures market cannot be ruled out since the
   interest rate may be better approximated with a unit root
   distribution. In this case, model (2) is mis-specified and the
   estimates are therefore inconsistent.
   The empirical results show supporting evidence of cointegration among the series considered in all markets. This is
   consistent with a necessary condition for the EMH that the
   futures market provides useful information on future movements
   in the spot market. However, it is found that the parameter
   restrictions are not satisfied for the cointegrating parameters in
   some markets under either the conventional risk premium
   hypothesis (2), or the cost-of-carry model (5), or both. The lack
   of supportive evidence for both the risk premium hypothesis and
   the cost-of-carry model in these futures markets is puzzling. Such
   evidence implies that either the traders of these futures contracts
   ignore the availability of a positive risk-free return ± thus
   rejecting rationality on the part of the participants in these
   markets ± or that there exists an important omitted variable from
   the risk premium or no-arbitrage relationships.
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   EFFICIENCY OF FUTURES MARKETS 713
   ß Blackwell Publishers Ltd 2001
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