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.
翻译:在这项工作中,我们开发了分解结构分析(DFA)和分解交叉校正分析(DCCA)的无症状理论,用于趋势静止的切除过程,而没有假定基本分布的具体形式;所有结果均在多面体可能重叠的方框总体框架之下提出和得出;我们证明DFA和DCCA的固定性,被视为随机过程,在第二顺序之前获得封闭形式,包括DFA和DCCA的共变结构和大量相关结果的错误法律。我们的结果概括并改进了文献中介绍的若干结果。为了核实小样本中我们理论结果的行为,我们提出蒙特卡洛模拟研究和计量时间序列的经验应用。