Local Whittle Analysis of Stationary Fractional Cointegration and the Implied–Realized Volatility Relation

Abstract

I consider local Whittle analysis of a stationary fractionally cointegrated model. The local Whittle quasi maximum likelihood estimator is proposed to jointly estimate the integration orders of the regressors, the integration order of the errors, and the cointegration vector. The proposed estimator is semiparametric in the sense that it employs local assumptions on the joint spectral density matrix of the regressors and the errors near the zero frequency. I show that the estimator is consistent under weak regularity conditions, and, under an additional local orthogonality condition between the regressors and the cointegration errors, I show asymptotic normality. Indeed, the estimator is asymptotically normal for the entire stationary region of the integration orders, and, thus, for a wider range of integration orders than the narrow-band frequency domain least squares estimator of the cointegration vector, and it is superior to the latter estimator with respect to asymptotic variance. Monte Carlo evidence documenting the finite-sample feasibility of the new methodology is presented. In an application to financial volatility series, I examine the unbiasedness hypothesis in the implied–realized volatility relation.

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