  The estimation of parameters in the frequency spectrum of a seasonally
persistent stationary stochastic process is addressed. For seasonal persistence
associated with a pole in the spectrum located away from frequency zero, a new
Whittle-type likelihood is developed that explicitly acknowledges the location
of the pole. This Whittle likelihood is a large sample approximation to the
distribution of the periodogram over a chosen grid of frequencies, and
constitutes an approximation to the time-domain likelihood of the data, via the
linear transformation of an inverse discrete Fourier transform combined with a
demodulation. The new likelihood is straightforward to compute, and as will be
demonstrated has good, yet non-standard, properties. The asymptotic behaviour
of the proposed likelihood estimators is studied; in particular,
$N$-consistency of the estimator of the spectral pole location is established.
Large finite sample and asymptotic distributions of the score and observed
Fisher information are given, and the corresponding distributions of the
maximum likelihood estimators are deduced. A study of the small sample
properties of the likelihood approximation is provided, and its superior
performance to previously suggested methods is shown, as well as agreement with
the developed distributional approximations.
