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Asymptotic results for Fourier‐PARMA time series

Periodically stationary times series are useful to model physical systems whose mean behavior and covariance structure varies with the season. The Periodic Auto‐Regressive Moving Average (PARMA) process provides a powerful tool for modelling periodically stationary series. Since the process is non‐stationary, the innovations algorithm is useful to obtain parameter estimates. Fitting a PARMA model to high‐resolution data, such as weekly or daily time series, is problematic because of the large number of parameters. To obtain a more parsimonious model, the discrete Fourier transform (DFT) can be used to represent the model parameters. This article proves asymptotic results for the DFT coefficients, which allow identification of the statistically significant frequencies to be included in the PARMA model.