Unbalanced Fractional Cointegration and the No-Arbitrage Condition on Commodity Markets

Technical abstract: A necessary condition for two time series to be nontrivially cointegrated is the equality of their respective integration orders. Nonetheless, in some cases, the apparent unbalance of integration orders of the observables can be misleading and the cointegration theory applies all the same. This situation refers to unbalanced cointegration in the sense that balanced long run relationship can be recovered by an appropriate filtering of one of the time series. In this paper, we suggest a local Whittle estimator of bivariate unbalanced fractional cointegration systems. Focusing on a degenerating band around the origin, it estimates jointly the unbalance parameter, the long run coefficient and the integration orders of the regressor and the cointegrating errors. Its consistency is demonstrated for the stationary regions of the parameter space and a finite sample analysis is conducted by means of Monte Carlo experiments. An application to the no-arbitrage condition between crude oil spot and futures prices is proposed to illustrate the empirical relevance of the developed estimator. Non-technical abstract: The no-arbitrage condition between spot and future prices implies an analogous condition on their underlying volatilities. Interestingly, the long memory behavior of the volatility series also involves a long-run relationship that allows to test for the no-arbitrage condition by means of cointegration techniques. Unfortunately, the persistent nature of the volatility can vary with the future maturity, thereby leading to unbalanced integration orders between spot and future volatility series. Nonetheless, if a balanced long-run relationship can be recovered by an appropriate filtering of one of the time series, the cointegration theory applies all the same and unbalanced cointegration operates between the raw series. In this paper, we introduce a new estimator of unbalanced fractional cointegration systems that allows to test for the no-arbitrage condition between the crude oil spot and futures volatilities.