  In time series analysis there is an apparent dichotomy between time and
frequency domain methods. The aim of this paper is to draw connections between
frequency and time domain methods. Our 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
time series. 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-likelihoods 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.
