A new spectral method in time series analysis using the complete DFT

Abstract

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