The Variance Profile

The variance profile is defined as the power mean of the spectral density function of a stationary stochastic process. It is a continuous and non-decreasing function of the power parameter, $p$, which returns the minimum of the spectrum ($prightarrow -infty$), the interpolation error variance (harmonic mean, $p=-1$), the prediction error variance (geometric mean, $p=0$), the unconditional variance (arithmetic mean, $p=1$) and the maximum of the spectrum ($prightarrow infty$). The variance profile provides a useful characterisation of a stochastic process; we focus in particular on the class of fractionally integrated processes. Moreover, it enables a direct and immediate derivation of the Szeg"{o}-Kolmogorov formula and the interpolation error variance formula. The paper proposes a non-parametric estimator of the variance profile based on the power mean of the smoothed sample spectrum, and proves its consistency and its asymptotic normality. From the empirical standpoint, we propose and illustrate the use of the variance profile for estimating the long memory parameter in climatological and financial time series and for assessing structural change.