Averaged periodogram spectral estimation with long-memory conditional heteroscedasticity (Q2744934)

A covariance stationary process is considered described by the model NEWLINE\[NEWLINEx_t=E(X_t)+\sum_{j=0}^\infty \alpha_j\varepsilon_{t-j},\qquad \alpha_0=1,\quad \sum_{j=0}^\infty\alpha_j^2<\infty,NEWLINE\]NEWLINE \(E(\varepsilon_t |{\mathcal F}_{t-1})=0\), \(E(\varepsilon^2_t |{\mathcal F}_{t-1})=\sigma^2_t\) a.s., where \(\sigma_t^2=\beta+\sum_{j=1}^\infty \psi_j\widetilde\varepsilon^2_{t-j}\), and \(\widetilde\varepsilon_j\) are i.i.d. For such processes an averaged periodogram \(\widehat f(0)=m^{-1}\sum_{j=1}^m I_x(2\pi j/n)\) is considered (\(I_x(\lambda)\) being the periodogram of \(x_t\), \(t=1,\dots,n\)) as an estimator of the spectral density \(f(0)\) of \(x_t\). The long-memory case \(f(\lambda)\sim L(1/\lambda)\lambda^{-2d}\) is investigated, where \(L\) is a slowly varying function, \(0<d<1/2\). Results on consistency of \(\widehat f(0)\) and asymptotic expansions are derived in terms of coefficients \(\psi_j\), index \(d\) and moments of \(\widetilde\varepsilon_j\). The Robinson estimator \(\widehat d_x\) for \(d\) is considered and its consistency is demonstrated. Results of simulations are presented.