Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Frequency-Domain NonStationarity Test for dependent data

Mohamedou Ould Haye, Anne Philippe

TL;DR

This work tackles the problem of distinguishing long-memory dependence from nonstationarity in dependent data, a challenging task because both produce slowly decaying autocovariances. It introduces a parameter-free frequency-domain test based on epoch-wise periodograms, yielding a tractable limiting distribution in the form of a finite mixture of chi-square variables, thereby avoiding estimation of the memory parameter $d$. The main theoretical result, Theorem 1, shows that $Q_{n,m}(s,d)$ converges to $Q(s,d)=\sum_{i=1}^{2s} \zeta_i(d) Q_i$ with $Q_i\sim\chi^2(1)$ and weights given by eigenvalues of $\Sigma(d)D^{-1}$, unifying stationary ($I(0)$) and integrated ($I(1)$) regimes across $d\in(-1/2,3/2)$. Empirical studies indicate good size control and power, with $s=2$ offering a practical balance, and the method comparing favorably to existing approaches like V/S, providing explicit critical values and robust performance near the stationarity boundary.

Abstract

Distinguishing long-memory behaviour from nonstationarity is challenging, as both produce slowly decaying sample autocovariances. Existing stationarity tests either fail to account for long-memory processes or exhibit poor empirical size, particularly near the boundary between stationarity and nonstationarity. We propose a new, parameter-free testing procedure based on the evaluation of periodograms across multiple epochs. The limiting distributions derived here are obtained under stationarity and nonstationarity assumptions and analytically tractable, expressed as finite sums of weighted independent $χ^2$ random variables. Simulation studies indicate that the proposed method performs favorably compared to existing approaches.

A Frequency-Domain NonStationarity Test for dependent data

TL;DR

This work tackles the problem of distinguishing long-memory dependence from nonstationarity in dependent data, a challenging task because both produce slowly decaying autocovariances. It introduces a parameter-free frequency-domain test based on epoch-wise periodograms, yielding a tractable limiting distribution in the form of a finite mixture of chi-square variables, thereby avoiding estimation of the memory parameter . The main theoretical result, Theorem 1, shows that converges to with and weights given by eigenvalues of , unifying stationary () and integrated () regimes across . Empirical studies indicate good size control and power, with offering a practical balance, and the method comparing favorably to existing approaches like V/S, providing explicit critical values and robust performance near the stationarity boundary.

Abstract

Distinguishing long-memory behaviour from nonstationarity is challenging, as both produce slowly decaying sample autocovariances. Existing stationarity tests either fail to account for long-memory processes or exhibit poor empirical size, particularly near the boundary between stationarity and nonstationarity. We propose a new, parameter-free testing procedure based on the evaluation of periodograms across multiple epochs. The limiting distributions derived here are obtained under stationarity and nonstationarity assumptions and analytically tractable, expressed as finite sums of weighted independent random variables. Simulation studies indicate that the proposed method performs favorably compared to existing approaches.

Paper Structure

This paper contains 6 sections, 5 theorems, 99 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Definition of the critical region
  4. Superimposed Cumulative Distribution Functions (CDF)
  5. Empirical Size and Power
  6. Proofs of Lemmas

Key Result

Theorem 1

Let $(X_t)$ be an $I(0)$ or $I(1)$ stochastic process with memory parameter $d \in (-1/2,\,3/2)$. Then, where $\{Q_i\}$ are i.i.d. $\chi^2(1)$ random variables, and $\{\zeta_i(d)\}$ are the eigenvalues of the matrix $\Sigma(d)D^{-1}$, where $\Sigma(d)$ is the block matrix and $D$ is the diagonal matrix with diagonal entries and for $i,j=1,\ldots,s$, where and similarly,

Figures (2)

  • Figure 1: Superimposed CDFs of $Q_{n,m}(2,d)$: finite-sample (red) vs. limiting distribution (blue) for $d=0.45$, $0.5$, and $1.0$ ($n=2000$, $\ell=10$, $m=200$).
  • Figure 2: Empirical power functions of the statistic $Q_{n,m}(s)$ for two DGPs: (Left) FARIMA$(0,d,0)$ with $d \leq \tfrac{1}{2}$ and (Right) AR$(1)$ with $\phi \in [0,1]$. Simulations used $n=2000$, $\ell=10$, $m=200$, $s=2$, and $3000$ Monte Carlo replications.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • proof
  • proof
  • proof