From nonstationarity to stationarity via $1/f$ noise: discrete Fourier transforms and sample mean asymptotics for testing

Technical Summary: From Nonstationarity to Stationarity via 1/f Noise

Problem Statement
The paper addresses the fundamental challenge in time series analysis of distinguishing between genuine nonstationarity and long-memory stationarity. Both phenomena exhibit slowly decaying sample autocovariances and similar low-frequency behavior, making them difficult to differentiate. While extensive literature exists for stationary processes with a memory parameter $d \in (-1/2, 1/2)$, and for nonstationary processes with $d \in [1/2, 3/2)$, the boundary case $d = 1/2$ (known as 1/f noise or the "just nonstationary" case) presents specific theoretical difficulties. Existing tests, such as the time-domain V/S test proposed by Giraitis et al. (2006), suffer from inflated empirical sizes as $d$ approaches $1/2$, failing to distinguish subtle differences between regimes. Furthermore, fully parametric approaches like those of Robinson (1994, 2003) rely on specific linear filter assumptions and require prewhitening steps that may degrade performance when short-memory components are present.

Methodology
The authors propose a nonparametric testing procedure that does not require the estimation of the memory parameter $d$. The methodology is built upon the following theoretical and statistical components:

1. Unified Asymptotic Framework: The paper derives the joint limiting distribution of the Discrete Fourier Transforms (DFT) of a time series at a fixed number of Fourier frequencies for the entire range $d \in (-1/2, 3/2)$. This covers both stationary ($d < 1/2$) and nonstationary ($d \ge 1/2$) regimes, including the boundary $d=1/2$.

2. Logarithmic Corrections: A key theoretical insight is that at the boundary $d=1/2$, standard normalization is insufficient for the sample mean and sample variance to admit nondegenerate limits. The authors demonstrate that a normalization factor of $\log(n)$ is required for these statistics to converge to explicit asymptotic distributions.

3. Test Statistic Construction: The proposed test statistic, $Q_n(s)$, combines three elements:

   • The sample mean (specifically a difference of partial sums, $D_n$).
   • The sample variance.
   • A finite number of low-frequency periodogram ordinates ($I_n(\lambda_j)$).

   The statistic is defined as:
   $$Q_n(s) := \frac{D_n^2}{\frac{1}{n}\sum_{k=1}^n (X_k - \bar{X}_n)^2} + \sum_{j=1}^s \frac{I_n(\lambda_j)}{\frac{1}{n}\sum_{k=1}^n (X_k - \bar{X}_n)^2} \log(k+1)$$
   where $D_n$ represents the difference between the mean of the first $m$ observations and the last $m$ observations (with $m \approx \sqrt{n}$).

Key Contributions and Results

• Theorem 2.1 (Joint DFT Limit): Establishes the asymptotic joint normality of the real and imaginary parts of the DFT for processes with $d \in (-1/2, 3/2)$. The limiting distribution is Gaussian with an explicit covariance structure $\Sigma(d)$, which varies depending on whether $d$ is in the stationary, boundary, or nonstationary range.
• Theorem 2.2 (Sample Mean and Variance at $d=1/2$): Proves that for $d=1/2$, the sample mean $\bar{X}_n$ normalized by $\sqrt{\log(n)}$ converges to a normal distribution. Furthermore, it establishes that the sample variance, when normalized by $n \log(n)$, converges to a constant, and the ratio of the squared mean to the variance converges to a $\chi^2(1)$ distribution.
• Proposition 3.1 (Limiting Distribution of the Test Statistic):
  • Under the null hypothesis ($H_0$: nonstationarity with $d \ge 1/2$ or deterministic trends), the statistic $Q_n(s)$ diverges to infinity if $d > 1/2$. If $d = 1/2$, it converges to a finite weighted sum of independent $\chi^2(1)$ variables. If a deterministic trend exists, the statistic also diverges.
  • Under the alternative hypothesis ($H_1$: stationarity with $d < 1/2$), the statistic converges in probability to 0.
  • This allows for the construction of a consistent test with a rejection region $R = \{Q_n(s) < q(\alpha, s)\}$, where $q(\alpha, s)$ is the quantile of the limiting distribution under $d=1/2$.

Simulation Results
The paper validates the proposed method through extensive Monte Carlo simulations comparing $Q_n(s)$ against Robinson's (1994) efficient nonstationarity test ($\tilde{r}$) and its prewhitened version.

• Robustness to Short Memory: The proposed statistic $Q_n(s)$ maintains good power and size control even when the data-generating process includes short-memory AR components. In contrast, Robinson's test $\tilde{r}$ performs well only for pure FARIMA models but deteriorates significantly in the presence of AR components, even with prewhitening.
• Aggregated Processes: The test remains robust for aggregated random-coefficient AR(1) processes, which mimic long memory but are not fractional.
• Structural Breaks: The test successfully detects nonstationarity arising from deterministic structural breaks in the mean, whereas Robinson's test often fails to reject the null of nonstationarity in these scenarios (incorrectly accepting the null).
• Boundary Performance: At the critical boundary $d=1/2$, the empirical size of $Q_n(s)$ remains close to the nominal level (e.g., 0.05), whereas other methods often exhibit high empirical sizes.

Significance and Claims
The authors claim that the primary contribution of this work is the derivation of a unified asymptotic framework for DFTs across stationary and nonstationary regimes, specifically resolving the behavior at the $d=1/2$ boundary. The resulting test statistic offers a computationally efficient, parameter-free method for distinguishing stationary behavior from fractional or deterministic nonstationarity. Unlike previous approaches, this method does not rely on parametric assumptions regarding the linear filter or require prewhitening steps, making it more robust to model misspecification and short-memory dynamics. The paper positions this as a solution to the "subtle" distinction between stationary and nonstationary behavior near the $1/f$ noise boundary, providing a tractable limit distribution (a weighted sum of $\chi^2$ variables) that contrasts with the stochastic integral limits typical of classical time-domain tests.