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. 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.

💡 Research Summary

The paper tackles the problem of estimating spectral parameters for a stationary stochastic process that exhibits seasonal persistence, i.e., a pole in its frequency spectrum located away from the origin. Classical Whittle‑type likelihood approximations are built on the assumption that any singularity lies at frequency zero, which leads to bias and inefficiency when the pole is at a non‑zero seasonal frequency (for example, annual or semi‑annual cycles). To overcome this limitation, the authors introduce a novel Whittle‑type likelihood that explicitly incorporates the pole location.

The key methodological innovation is a two‑step transformation of the original time series (X_t). First, the series is demodulated by multiplying with the complex exponential (\exp(-i\lambda_0 t)), where (\lambda_0) denotes the unknown seasonal frequency. This operation shifts the spectral pole to the origin, producing a transformed series (Y_t = X_t e^{-i\lambda_0 t}) that now satisfies the standard zero‑frequency pole assumption. Second, the transformed series is subjected to an inverse discrete Fourier transform (IDFT) and the periodogram is evaluated on a pre‑selected grid of frequencies ({\omega_k}). The resulting periodogram values (I(\omega_k)) are then used in a Whittle‑type log‑likelihood
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