Reconciling the Gaussian and Whittle likelihood with an application to estimation in the frequency domain

In time series analysis there is an apparent dichotomy between and frequency domain methods. The aim of this paper to draw connections Our focus will be on reconciling the Gaussian likelihood Whittle likelihood. We derive exact, interpretable, bound a second order stationary series. derivation based obtaining transformation which biorthogonal discrete Fourier transform Such yields new decomposition for inverse Toeplitz matrix enables representation within domain. show that difference due omission best linear predictions outside observation in periodogram associated with Based result, we obtain approximation likelihoods terms fitting, finite autoregressive parameters. These approximations are used define two quasi-likelihood criteria. these criteria can yield better spectral divergence criterion, as compared both likelihoods. simulations, proposed estimators have satisfactory sample properties.