Akaike's information criterion correction for the least-squares autoregressive spectral estimator

In this article we propose a new correction for the penalty term of the Akaike's information criterion (AIC), when it is used in the context of order selection for an autoregressive fit of the spectral density of a stationary time series. The classical AIC penalty term may be viewed as an approximation of an appropriate target quantity. Simulations show that the quality of this approximation strongly depends on the type of autoregressive estimator used, as well as on the discrepancy used. Therefore here we consider the least squares autoregressive estimator and the Whittle discrepancy only. In this context we propose a closed formula correction of the AIC penalty term. We also develop asymptotic theory which justifies this proposal: an asymptotically valid second-order expansion of a stochastic approximation of the target quantity. This expansion assumes a non-parametric framework: it does not assume gaussianity of the process and only requires its spectral density to be smooth enough. Simulations show that, as compared to previously introduced corrections, this new correction performs similarly to finite sample information criterion, while they both outperform AIC corrected and AIC.