Paper: arXiv 2509.21326

Authors: Yuelong Li

Abstract

This paper develops a rigorous functional-analytic framework for the MACD (Moving Average Convergence Divergence) indicator, a classical tool in technical analysis. We show that MACD, commonly defined as the difference between two moving averages, can be precisely interpreted as a phase-corrected, smoothed derivative operator. By analyzing nested and recursive moving averages, we establish that MACD is structurally equivalent to a band-pass filter and derive exact formulas expressing it as a finite difference of delayed and doubly averaged signals. We prove new operator identities demonstrating that MACD corresponds to the derivative of a phase-centered, double-smoothed average, appropriately delayed to correct for asymmetries introduced by causal averaging. This characterization unifies MACD with concepts from harmonic analysis and operator theory, providing a principled basis for understanding its role in signal detection, filtering, and trend analysis. The framework naturally generalizes to recursive decompositions, culminating in an expansion that expresses MACD as a weighted sum of delayed, smoothed derivatives, thereby revealing the true analytical structure underlying this widely used yet traditionally heuristic indicator.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous functional-analytic reinterpretation of the classic MACD indicator as a phase-corrected, smoothed derivative operator with complete proofs. However, it is entirely theoretical, lacking any empirical backtesting, financial data validation, or empirical rigor.

Research Flowchart

  flowchart TD
    A["Research Goal: Establish functional-analytic framework for MACD"] --> B["Input: Raw Time Series & Causal Moving Averages"]
    B --> C["Methodology: Operator Theory & Harmonic Analysis"]
    C --> D["Computational Process: Phase Correction & Derivative Generalization"]
    D --> E["Finding 1: MACD structurally equivalent to Band-Pass Filter"]
    D --> F["Finding 2: MACD represents delayed derivative of double-smoothed average"]
    E & F --> G["Outcome: Recursive expansion unifying MACD with signal detection theory"]