Non-Regular Likelihood Inference for Seasonally Persistent Processes

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. Asymptotically, the estimator of the pole after suitable standardization follows a Cauchy distribution, and for moderate sample sizes, we can use the finite large sample approximation to the distribution of the estimator of the pole corresponding to the ratio of two Gaussian random variables, with sample size dependent means and variances. 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. Inspired by the developments for full likelihood based estimation procedures, usage of profile likelihood and other likelihood based procedures are also discussed. Semi-parametric estimation methods, such as the Geweke-Porter-Hudak estimator of the long memory parameter, inspired by the developed parametric theory are introduced.