THRESHOLD SPECTRAL-BASED FOURIER REGRESSION MODEL FOR SEASONAL-CYCLICAL TIME SERIES

Abstract

The Threshold Spectral Fourier Regression (TSFR) model is a hybrid model that integrates Fourier methods with the nonlinear dynamics of threshold models. It was developed to simultaneously model a series that exhibits nonlinear, nonstationary, cyclical, and seasonal behaviours, as well as structural breaks. This study presents the theory and development of the TSFR model. The model leverages spectral estimation and converts the dataset from the time domain to the frequency domain using methods such as the Fourier Transform. A threshold value of  is determined using the quantile method; to split the series into two regimes categorised as Low and High, frequencies exceeding the threshold are retained for inclusion in the model. The TSFR model is developed using the retained frequencies and Fourier terms as regressors, while model parameters are estimated using the Ordinary Least Squares (OLS) method. The fitted model is assessed for goodness-of-fit and model accuracy; the residuals are examined to ensure they are randomly distributed, and are  used to predict future values. Nigerian rainfall data, covering the period 1981- May, 2025, obtained from climate hazards, UC Santa Barbara WEP; made up of 533 data points  are used for empirical analysis. The estimated parameters for the rainfall dataset include five values per category. Low rainfall parameters are 0.2169, 0.0045, 0.0055, 0.0013 and -0.0015 with Coefficient of Determination  = 0.8818, Adjusted Coefficient of Determination = 0.7486, Mean Absolute Percentage Error (MAPE) = 0.3, Akaike Information Criterion (AIC) = 6427.32 and Bayesian Information Criterion (BIC) 0.0002; while the parameters for High rainfall are 0.9998, 0.00005, 0.0002, 0.00003 and 0.00006 with  = 0.9484, Adjusted = 0.90388, MAPE = 0.05, AIC = 8832.82 and BIC = -8808. The findings in this study demonstrate that the TSFR model is suitable for modeling time series with pronounced nonlinear and periodic features. The model is flexible and robust, provides a good fit, exhibits minimal overfitting, and achieves better forecasting accuracy than traditional and non-threshold models.