Box–Cox transforms for realized volatility

Abstract

The log transformation of realized volatility is often preferred to the raw version of realized volatility because of its superior finite sample properties. One of the possible explanations for this finding is the fact the skewness of the log transformed statistic is smaller than that of the raw statistic. Simulation evidence presented here shows that this is the case. It also shows that the log transform does not completely eliminate skewness in finite samples. This suggests that there may exist other nonlinear transformations that are more effective at reducing the finite sample skewness. The main goal of this paper is to study the accuracy of a new class of transformations for realized volatility based on the Box–Cox transformation. This transformation is indexed by a parameter β and contains as special cases the log (when β = 0 ) and the raw (when β = 1 ) versions of realized volatility. Based on the theory of Edgeworth expansions, we study the accuracy of the Box–Cox transforms across different values of β . We derive an optimal value of β that approximately eliminates skewness. We then show that the corresponding Box–Cox transformed statistic outperforms other choices of β , including β = 0 (the log transformation). We provide extensive Monte Carlo simulation results to compare the finite sample properties of different Box–Cox transforms. Across the models considered in this paper, one of our conclusions is that β = − 1 (i.e. relying on the inverse of realized volatility also known as realized precision) is the best choice if we want to control the coverage probability of 95% level confidence intervals for integrated volatility.