Model selection for broadband semiparametric estimation of long memory in time series (Q2784955)

A Gaussian time series with spectral density \(f(\omega)=|1-e^{-i\omega}|^{-2d}f^*(\omega)\) is considered, where \(f^*\) is positive, even, continuous, bounded above and bounded away from zero, and \(d\) is the long-memory parameter. The problem is to estimate \(d\). The author uses a semiparametric approach based on the expansion of \(\log f^*=\sum_{k=0}^\infty\gamma_k\cos(k\omega)\). This series is truncated at \(k=h\) and an ordinary least squares estimator \(\hat\beta=(\hat\gamma_0,\dots,\hat\gamma_h, -2\hat d)\) is used for the parameter \(\beta=(\gamma_0,\dots,\gamma_h, -2d)\). The truncation parameter \(h\) is selected to minimize the \(MSE=E(\hat d- d)^2\). The author derives asymptotic approximations of the MSE and constructs an algorithm for selection of \(h\). Some Monte Carlo results are presented.