On the behavior of the DFA and DCCA in trend-stationary processes

TL;DR

This paper develops the asymptotic theory for DFA and DCCA in trend-stationary processes, providing new insights into their statistical properties without distributional assumptions, supported by simulations and real data.

Contribution

It generalizes and improves existing results on DFA and DCCA by deriving their asymptotic properties under broad conditions for trend-stationary processes.

Findings

Proves stationarity of DFA and DCCA as stochastic processes

Provides closed-form expressions for moments and covariance structures

Validates theoretical results with simulations and empirical data

Abstract

In this work, we develop the asymptotic theory of the Detrended Fluctuation Analysis (DFA) and Detrended Cross-Correlation Analysis (DCCA) for trend-stationary stochastic processes without any assumption on the specific form of the underlying distribution. All results are presented and derived under the general framework of potentially overlapping boxes for the polynomial fit. We prove the stationarity of the DFA and DCCA, viewed as stochastic processes, obtain closed forms for moments up to second order, including the covariance structure for DFA and DCCA and a miscellany of law of large number related results. Our results generalize and improve several results presented in the literature. To verify the behavior of our theoretical results in small samples, we present a Monte Carlo simulation study and an empirical application to econometric time series.