A fractionally cointegrated VAR model with deterministic trends and application to commodity futures markets

Abstract

We apply the fractionally cointegrated vector autoregressive (FCVAR) model to analyze the relationship between spot and futures prices in five commodity markets (aluminium, copper, lead, nickel, and zinc). To this end, we first extend the FCVAR model to accommodate deterministic trends in the levels of the processes. The methodological contribution is to provide a representation theory for the FCVAR model with deterministic trends, where we show that the presence of the deterministic trend in the process induces both restricted and unrestricted constant terms in the vector error correction model. The consequences for the cointegration rank test are also briefly discussed. In our empirical application we use the data from Figuerola-Ferretti and Gonzalo (2010), who conduct a similar analysis using the usual (non-fractional) cointegrated VAR model. The main conclusion from the empirical analysis is that, when using the FCVAR model, there is more support for the cointegration vector (1, − 1)' in the long-run equilibrium relationship between spot and futures prices, and hence less evidence of long-run backwardation, compared to the results from the non-fractional model. Specifically, we reject the hypothesis that the cointegration vector is (1, − 1)' using standard likelihood ratio tests only for the lead and nickel markets.