We develop a new multivariate generalized ARCH (GARCH) parameterization suitable for testing the hypothesis that the optimal futures hedge ratio is constant over time, given that the joint distribution of cash and futures prices is characterized by autoregressive conditional heteroskedasticity (ARCH). The advantage of the new parameterization is that it allows for a flexible form of time-varying volatility, even under the null of a constant hedge ratio. The model is estimated using weekly corn prices. Statistical tests reject the null hypothesis of a constant hedge ratio and also reject the null that time variation in optimal hedge ratios can be explained solely by deterministic seasonality and time to maturity effects
Hedging with futures contracts is an important risk management strategy for firms dealing with commodities, the prices of which are notoriously volatile. Hedging reduces risk because cash and futures prices for the same commodity tend to move together, so that changes in the value of a cash position are offset by changes in the value of an opposite futures position. Because cash and futures price movements are typically not perfectly correlated (i.e., there is basis risk), risk management requires determination of the “optimal hedge ratio” (the optimal amount of futures bought or sold expressed as a proportion of the cash position). When basis risk is the only source of uncertainty,1 the optimal hedge ratio often can be reduced to a simple ratio of the conditional covariance between cash and futures prices to the conditional variance of futures prices Benninga et al., 1983, Myers, 1991 and Lence, 1995. To estimate such a ratio, early work simply used the slope of an ordinary least squares regression of cash on futures prices. An improved procedure is possible by computing the relevant moments of the price distribution relative to the proper conditional means Myers and Thompson, 1989 and Moschini and Lapan, 1995.2 More generally, estimation of the optimal hedge ratio recognizes that commodity cash and futures prices often display time-varying volatility and relies on techniques consistent with such a hypothesis, such as Engle's (1982) autoregressive conditional heteroskedasticity (ARCH) framework or Bollerslev's (1986) generalized ARCH (GARCH) approach.
ARCH and GARCH models appear ideally useful for estimating time-varying optimal hedge ratios, and a number of applications have concluded that such ratios seem to display considerable variability over time Cecchetti et al., 1988, Baillie and Myers, 1991, Myers, 1991 and Kroner and Sultan, 1993. Yet, no existing study has provided compelling evidence that such time-varying hedge ratios are statistically different from a constant hedge ratio. A time-varying covariance matrix of cash and futures prices, per se, is not sufficient to establish that the optimal hedge ratio is time varying. Constancy of the hedge ratio restricts the ratio of the covariance between cash and futures prices to the variance of futures prices to be constant, but it need not restrict the moments of the joint distribution of cash and futures prices in any other way. Unfortunately, the particular parametric GARCH models that have been used to date admit a constant hedge ratio only under very restrictive conditions, so that the hypothesis of a constant optimal hedge ratio can be tested only jointly with other hypotheses. The main purpose of this article is to develop a more general GARCH parameterization that yields a constant hedge ratio as a special case, while still allowing for a flexible time-varying distribution of cash and futures prices. The model is illustrated with an application to the problem of storage hedging of corn using futures prices from the Chicago Board of Trade and Iowa cash prices for the period 1976–1997.
In this article, we have provided a new GARCH parameterization that modifies the
Engle and Kroner (1995) BEKK formulation. The new parameterization is particularly
useful for estimating time-varying optimal hedge ratios and testing the null hypothesis
that they are constant over time. Our approach overcomes an important limitation of
previous studies, where the null hypothesis of a constant hedge ratio was only identified
jointly with other restrictive conditions (such as, for example, that the distribution of
cash and futures prices is time-invariant). As shown in this article, such additional
restrictive conditions are not necessary to obtain a constant optimal hedge ratio. In
particular, we have developed modified BEKK parameterizations for the bivariate
GARCH( q,r) model that nest the hypothesis of a constant hedge ratio (or of an
exogenously varying hedge ratio) but retain flexible time-varying variances and
covariances, even under the null hypothesis.
These modified BEKK parameterizations were utilized to estimate bivariate GARCH
models for corn cash and futures prices, based on weekly data spanning the period 1976
to 1997. We find significant GARCH effects in the cash and futures prices, and these
GARCH effects are still present even when accounting separately for seasonality and
time to maturity, which are themselves significant components of the time variation in
the covariance matrix. Furthermore, using Wald QMLE tests we formally reject the null
hypothesis that the ratio of conditional covariance of futures and cash prices to conditional variance of futures prices (the optimal hedge ratio) is constant at essentially
any significance level. We also reject the null hypothesis that optimal hedge ratios vary
only systematically with seasonality and time-to-maturity effects at essentially any
significance level. Thus, our statistical tests support the conclusion that optimal hedge
ratios for weekly storage hedging of corn in the Midwest are indeed time varying in ways
that cannot be explained simply by seasonality and time-to-maturity effects.
