Long-memory functional time series modeling for electricity spot price forecasting using fractionally integrated functional autoregression

Electricity spot prices are found to be highly long-range dependent, with hyperbolically decaying instead of geometrically decaying autocorrelation functions that challenge traditional functional time series models of forecasting. In existing hybrid forecasting frameworks, the stochastic intraday part is usually modeled by first-order Functional Autoregressive (FAR(1)) models, which have a short memory structure which fails to capture the temporal dependence adequately. To overcome this limitation, we introduce a new forecasting model, the Fractionally Integrated Functional Autoregressive model, FFAR( d ), which introduces a long-memory dynamics into the functional autoregression. The proposed methodology is to estimate the fractional differencing parameter for the functional principal component scores retained after the reduction, to model the fractional differenced series by short memory autoregression and to obtain forecasts of the original series using fractional integration. The framework is easily incorporated into a generalized additive model (GAM)-based deseasonalization pipeline for exogenous variables, creating the extended FFARX( d ) model. In order to make better-informed estimates more reliable, we detect and correct for the small-sample bias of the Geweke–Porter–Hudak estimator, develop asymptotic inference for memory parameters and use residual whiteness tests to evaluate model appropriateness. In extensive simulation studies using synthesized functional electricity prices with known long-memory properties, the local Whittle estimator proves to have a good ability to recover the fractional memory parameters, and FFAR( d ), FFARX( d ), consistently outperforms the conventional FAR(1), FARX(1), seasonal benchmark models as well as the functional neural-network approach over various forecasting horizons. Improvements are substantiated by evaluations based on the rolling-origin and the corresponding RMSE, MAE, MASE, sMAPE, prediction interval coverage and CRPS; statistically significant gains are corroborated by paired hypothesis tests. Robustness and efficiency of the proposed framework is further validated by additional sensitivity analyses, ablation experiments and computational complexity assessments. While the empirical validation is still hampered by the lack of a public long-term hourly electricity price database, the proposed FFAR( d ) methodology gives a rigorous and practically feasible approach for modelling persistent dependence in functional time series and is a promising starting point for long-memory forecasting applications in the energy markets and other complex functional data environments.