Consistent Autoregressive Spectral Estimates

Abstract

We consider an autoregressive linear process {xt}, a one-sided moving average, with summable coefficients, of independent identically distributed variables {et} with zero mean and fourth moment, such that {et} is expressible in terms of past values of {xt}. The spectral density of {xt} is assumed bounded and bounded away from zero. Using data x1, * * *, x,n from the process, we fit an autoregression of order k, where k3/n - 0 as n -* oo. Assuming the order k is asymptotically sufficient to overcome bias, the autoregression yields a consistent estimator of the spectral density of {xt}. Furthermore, assuming k goes to infinity so that the bias from using a finite autoregression vanishes at a sufficient rate, the autoregressive spectral estimates are asymptotically normal, uncorrelated at different fixed fre- quencies. The asymptotic variance is the same as for spectral estimates based on a truncated periodogram.