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Estimation of the long-run variance of nonlinear time series with an application to change point analysis

Vaidotas Characiejus, Piotr Kokoszka, Xiangdong Meng

TL;DR

The work addresses distinguishing long-range dependence from mean changes in nonlinear time series by deriving the asymptotic normality of the smoothed periodogram estimator for the long-run variance when the data follow Bernoulli-shift dynamics. The authors develop a general CLT for weighted sums of periodogram ordinates, establish precise bias and variance behavior, and apply these results to a change-point test based on a local Whittle Hurst estimator. Theoretical guarantees are complemented by a small simulation study using nonlinear models (GARCH$(1,1)$ and stochastic volatility), showing good finite-sample performance and sizes near nominal. The paper thus extends long-run variance estimation techniques beyond linear processes and provides a practical tool for detecting changes in mean versus long memory in nonlinear time series.

Abstract

For a broad class of nonlinear time series known as Bernoulli shifts, we establish the asymptotic normality of the smoothed periodogram estimator of the long-run variance. This estimator uses only a narrow band of Fourier frequencies around the origin and so has been extensively used in local Whittle estimation. Existing asymptotic normality results apply only to linear time series, so our work substantially extends the scope of the applicability of the smoothed periodogram estimator. As an illustration, we apply it to a test of changes in mean against long-range dependence. A simulation study is also conducted to illustrate the performance of the test for nonlinear time series.

Estimation of the long-run variance of nonlinear time series with an application to change point analysis

TL;DR

The work addresses distinguishing long-range dependence from mean changes in nonlinear time series by deriving the asymptotic normality of the smoothed periodogram estimator for the long-run variance when the data follow Bernoulli-shift dynamics. The authors develop a general CLT for weighted sums of periodogram ordinates, establish precise bias and variance behavior, and apply these results to a change-point test based on a local Whittle Hurst estimator. Theoretical guarantees are complemented by a small simulation study using nonlinear models (GARCH and stochastic volatility), showing good finite-sample performance and sizes near nominal. The paper thus extends long-run variance estimation techniques beyond linear processes and provides a practical tool for detecting changes in mean versus long memory in nonlinear time series.

Abstract

For a broad class of nonlinear time series known as Bernoulli shifts, we establish the asymptotic normality of the smoothed periodogram estimator of the long-run variance. This estimator uses only a narrow band of Fourier frequencies around the origin and so has been extensively used in local Whittle estimation. Existing asymptotic normality results apply only to linear time series, so our work substantially extends the scope of the applicability of the smoothed periodogram estimator. As an illustration, we apply it to a test of changes in mean against long-range dependence. A simulation study is also conducted to illustrate the performance of the test for nonlinear time series.
Paper Structure (12 sections, 13 theorems, 112 equations, 2 figures, 2 tables)

This paper contains 12 sections, 13 theorems, 112 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The spectral density and the periodogram
  4. Weak dependence
  5. Estimation of the long-run variance
  6. Application to change point analysis
  7. A small simulation study
  8. Proofs of the results of Sections \ref{['s:est']} and \ref{['s:cpa']}
  9. Auxiliary lemmas
  10. Central limit theorem for weighted sums of the periodogram ordinates
  11. Proofs of the theorems of Section \ref{['s:est']}
  12. Proofs of the results of Section \ref{['s:cpa']}

Key Result

Lemma 1

The inequality holds for $t\ge1$ and $p\ge1$ with $X_t$, $X_{t,\{0\}}$, and $X_t^{(t)}$ given by eq:defX_t, eq:Xt0, and eq:Xtm, respectively.

Figures (2)

  • Figure 1: Empirical density functions of $T^{(R)}$ based on $5{,}000$ replications with the standard normal density overlaid in blue. The $r_t$ follow GARCH$(1,1)$ model with $\alpha_0 = 0.5$, $\alpha_1 = 0.2$ and $\beta_1 = 0.4$.
  • Figure 2: The same as \ref{['f:g11']}, but the $r_t$ follow SV model with $\alpha = 0$, $\phi = 0.5$ and $\sigma_w^2 = 1$.

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof : Proof of \ref{['l:comp']}
  • Remark 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Definition 3
  • ...and 19 more