An efficient taper for potentially overdifferenced long-memory time series (Q2703252)

The authors consider the problem of long-memory parameter \(d\) estimation for processes with \(d\in(-0.5,1.5)\), i.e. these processes may be nonstationary, and additive deterministic linear trend is possible. The first order differences of the data process \(y_t\) will have the memory parameter \(d^*=d-1\in(-1.5,0.5)\). The estimator for \(d^*\) is defined as \( \hat d^*=arg min R(d)\), NEWLINE\[NEWLINE R(d)=\log \hat G(d)-2dm^{-1}\sum_{j=1}^m\log\lambda_{\tilde j},\quad \hat G(d)=m^{-1}\sum_{j=1}^m \lambda^{2d}_{\tilde j}I^T(\lambda_j),NEWLINE\]NEWLINE where \(\tilde j=[(j-1)/2]\), \(\lambda_j=2\pi j/n\), \(I^T\) is a periodogram of \(y(j)\), \(j=1,\dots,n\), tapered by the data taper \( h_t=0.5 [1-\exp(i2\pi(t-1/2)/n)] \), and \(m\) is a fixed algorithm parameter. Consistency and asymptotic normality of \(\hat d^*\) are demonstrated. Simulation results are presented. The estimator is applied to some climatological and economic time series. A problem of estimator invariance to polynomial trends is considered.