An Improved Formula for the Asymptotic Variance of Spectrum Estimates

Abstract

When proving results on the asymptotic behavior of estimates of the spectrum of a stationary time-series, it is invariably assumed that as the sample size $T$ tends to infinity, so does the truncation point $M_T$, but at a slower rate, so that $M_TT^{-1}$ tends to zero. This is a convenient assumption mathematically in that, in particular, it ensures consistency of the estimates, but it is unrealistic when such results are used as approximations to the finite case where the value of $M_TT^{-1}$ cannot be zero. We derive a formula for the asymptotic variance on the assumption that $M_TT^{-1}$ tends to a constant $\gamma$; a more accurate approximation to the variance in the finite case is then obtained by using this formula with $\gamma$ equal to the actual value of $M_TT^{-1}$. Numerical comparisons are made in the white noise case.