Fractional Cointegration and Level Shifts in the Realized and Implied Volatility Relation

The market efficiency hypothesis predicts that implied volatility is an unbiased forecast of the future realized volatility of the spot price.
The relationship realized-implied volatility can be analyzed in terms of fractional cointegration, supposing that the dynamics of the two
series are driven by the same fractional stochastic trend, providing support for a long-run unbiasedness of implied volatility as a predictor of realized volatility. In this paper, we test for the presence of fractional cointegration according both to a regression setup, as in Christensen and Nielsen (2006), and to a full parametric specification, say a fractional VECM, evaluating the respective performances in small samples with a Monte Carlo exercise. Moreover, we consider the possibility that the persistence of implied and realized volatility is induced by a common level shift process. We provide a new inference technique to deal with the estimation of the probability of occurrence of the shift, and we show that usual fractional cointegration and common level shifts are observationally equivalent.