Semiparametric Analysis of Stationary Fractional Cointegration and the Implied-Realized Volatility Relation in High-Frequency Options Data

We consider semiparametric frequency domain analysis of cointegration between long memory processes, i.e. fractional cointegration. This concept allows derivation of useful long-run relations even among stationary long memory processes. The approach uses a degenerating part of the periodogram near the origin to form a frequency domain least squares (FDLS) estimator of the cointegrating relation. The resulting estimator is consistent for arbitrary short-run dynamics, whereas the latter would have to be specified correctly in any parametric approach. We derive the asymptotic distribution theory for the FDLS estimator of the cointegration vector in the stationary long memory case. The new theory requires a general theorem on the asymptotic order of the covariance between the cross-periodograms of stationary long memory processes, which we provide. The motivating example is the relation between the volatility realized in the stock market and the associated implicit volatility derived from option prices. An application to high-frequency U.S. stock index and option data is offered.