A computationally efficient approximation for fractional differencing: First-order operators

This paper introduces the First-Order Fractional Differencing (FOFD) operator that substantially reduces the computational burden of fractional differencing for large-scale applications. While the standard Grünwald–Letnikov (GL) operator requires O(T2) operations for a series of length T, and recent FFT-based methods achieve O(TlogT), our FOFD operator requires only O(T) operations through a simple two-point recursion. We develop an optimal weight calibration framework that ensures this computational efficiency does not compromise statistical accuracy, deriving a general formula wopt=d⋅(1−0.9ρ)β(p) that adapts to the persistence structure of autoregressive processes. Empirical applications demonstrate substantial improvements: for the Chicago Fed National Financial Conditions Index with extreme persistence (ρ=0.992), optimal weight calibration reduces approximation error by 93% while preserving the autocorrelation structure of the GL operator. For a series of 10,000 observations, our method requires 20,000 operations compared to 530,000 for FFT-based methods and 50 million for standard implementations—enabling fractional differencing in real-time and high-frequency contexts previously infeasible due to computational constraints. The method’s simplicity, requiring no specialized libraries and providing direct implementation through our calibration formula, makes it immediately accessible to practitioners while maintaining the long-memory properties essential for financial time series modeling.