Consistent Autoregressive Spectral Estimates

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, , xₙ from the process, we fit an autoregression of order k, where k³/n 0 as n. 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 frequencies. The asymptotic variance is the same as for spectral estimates based on a truncated periodogram.