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Change-Point Analysis of Time Series with Evolutionary Spectra

Alessandro Casini, Pierre Perron

Abstract

This paper develops change-point methods for the spectrum of a locally stationary time series. We focus on series with a bounded spectral density that change smoothly under the null hypothesis but exhibits change-points or becomes less smooth under the alternative. We address two local problems. The first is the detection of discontinuities (or breaks) in the spectrum at unknown dates and frequencies. The second involves abrupt yet continuous changes in the spectrum over a short time period at an unknown frequency without signifying a break. Both problems can be cast into changes in the degree of smoothness of the spectral density over time. We consider estimation and minimax-optimal testing. We determine the optimal rate for the minimax distinguishable boundary, i.e., the minimum break magnitude such that we are able to uniformly control type I and type II errors. We propose a novel procedure for the estimation of the change-points based on a wild sequential top-down algorithm and show its consistency under shrinking shifts and possibly growing number of change-points. Our method can be used across many fields and a companion program is made available in popular software packages.

Change-Point Analysis of Time Series with Evolutionary Spectra

Abstract

This paper develops change-point methods for the spectrum of a locally stationary time series. We focus on series with a bounded spectral density that change smoothly under the null hypothesis but exhibits change-points or becomes less smooth under the alternative. We address two local problems. The first is the detection of discontinuities (or breaks) in the spectrum at unknown dates and frequencies. The second involves abrupt yet continuous changes in the spectrum over a short time period at an unknown frequency without signifying a break. Both problems can be cast into changes in the degree of smoothness of the spectral density over time. We consider estimation and minimax-optimal testing. We determine the optimal rate for the minimax distinguishable boundary, i.e., the minimum break magnitude such that we are able to uniformly control type I and type II errors. We propose a novel procedure for the estimation of the change-points based on a wild sequential top-down algorithm and show its consistency under shrinking shifts and possibly growing number of change-points. Our method can be used across many fields and a companion program is made available in popular software packages.

Paper Structure

This paper contains 44 sections, 23 theorems, 209 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. Statistical Environment and the Testing Problems
  3. Segmented Locally Stationary Processes
  4. The Testing Problems
  5. Tests for Changes in the Spectrum and Their Limiting Distributions
  6. The Test Statistics
  7. The Limiting Distribution Under the Null Hypothesis
  8. Consistency and Minimax Optimal Rate of Convergence
  9. Estimation of the Change-Points
  10. Single Break Alternatives $\mathcal{H}_{1}^{\mathrm{B},1}$
  11. Multiple Breaks Alternatives $\mathcal{H}_{1}^{\mathrm{B},m_{0}}$
  12. Implementation
  13. Small-Sample Evaluations
  14. Empirical Application
  15. Conclusions
  16. ...and 29 more sections

Key Result

Lemma 3.1

Let Assumption Assumption WZ 2011 hold and $X_{t,T}\in\mathscr{L}^{r}$, $r\geq2$. We have for $l\geq0$, $j=1,\ldots,\,r-1$, and any $r$-tuple $a_{1},\ldots,a_{r}$.

Figures (1)

  • Figure 1: Plot of one-day changes in the nominal Treasury yields ($\Delta i_{t}$) in the control sample. The sample size is $T_{C}=762$ which corresponds to all Tuesdays and Wednesdays that are not FOMC meeting days from 1/1/2000 to 12/31/2012. Following nakamura/steinsson:2018 we drop the second half of 2008, the first half of 2009 and a 10 day period after 9/11/2001. The red dashed lines are change-point dates estimated using Algorithm \ref{['Algorithm 1']}.

Theorems & Definitions (24)

  • Definition 2.1
  • Lemma 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Theorem 4.2
  • Proposition 5.1
  • Proposition 5.2
  • Theorem S.A.1
  • ...and 14 more