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Estimation of Change Points for Non-linear (auto-)regressive processes using Neural Network Functions

Claudia Kirch, Stefanie Schwaar

TL;DR

This work tackles change-point detection in non-linear (auto-)regressive time series by replacing the unknown regression function with a one-layer neural network, enabling a universal test that detects changes in the best-approximating NN parameters rather than only mean shifts. It proves strong consistency and asymptotic normality for the NN parameter estimator, develops a weighted gradient-based test statistic with Brownian-bridge limits, and derives a change-point estimator that attains the optimal rate $OP(1/n)$ with a non-standard limiting distribution. The approach remains robust under model misspecification and is validated through simulations and a financial-data application (DAX), demonstrating practical utility for detecting structural breaks in complex time series. These results advance change-point methodology for non-linear processes and offer a scalable, flexible tool for real-world applications with exogenous regressors and potential misspecification.

Abstract

In this paper, we propose a new test for the detection of a change in a non-linear (auto-)regressive time series as well as a corresponding estimator for the unknown time point of the change. To this end, we consider an at-most-one-change model and approximate the unknown (auto-)regression function by a neuronal network with one hidden layer. It is shown that the test has asymptotic power one for a wide range of alternatives not restricted to changes in the mean of the time series. Furthermore, we prove that the corresponding estimator converges to the true change point with the optimal rate OP (1/n) and derive the asymptotic distribution. Some simulations illustrate the behavior of the estimator with a special focus on the misspecified case, where the true regression function is not given by a neuronal network. Finally, we apply the estimator to some financial data.

Estimation of Change Points for Non-linear (auto-)regressive processes using Neural Network Functions

TL;DR

This work tackles change-point detection in non-linear (auto-)regressive time series by replacing the unknown regression function with a one-layer neural network, enabling a universal test that detects changes in the best-approximating NN parameters rather than only mean shifts. It proves strong consistency and asymptotic normality for the NN parameter estimator, develops a weighted gradient-based test statistic with Brownian-bridge limits, and derives a change-point estimator that attains the optimal rate with a non-standard limiting distribution. The approach remains robust under model misspecification and is validated through simulations and a financial-data application (DAX), demonstrating practical utility for detecting structural breaks in complex time series. These results advance change-point methodology for non-linear processes and offer a scalable, flexible tool for real-world applications with exogenous regressors and potential misspecification.

Abstract

In this paper, we propose a new test for the detection of a change in a non-linear (auto-)regressive time series as well as a corresponding estimator for the unknown time point of the change. To this end, we consider an at-most-one-change model and approximate the unknown (auto-)regression function by a neuronal network with one hidden layer. It is shown that the test has asymptotic power one for a wide range of alternatives not restricted to changes in the mean of the time series. Furthermore, we prove that the corresponding estimator converges to the true change point with the optimal rate OP (1/n) and derive the asymptotic distribution. Some simulations illustrate the behavior of the estimator with a special focus on the misspecified case, where the true regression function is not given by a neuronal network. Finally, we apply the estimator to some financial data.

Paper Structure

This paper contains 26 sections, 14 theorems, 74 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Modeling with neural networks
  3. Model
  4. Properties of the neural network estimator
  5. Change point procedures
  6. Asymptotics of the test-statistic
  7. change point estimator
  8. Simulations and Application
  9. Simulation
  10. Representative of $\tilde{\theta}$
  11. Correctly specified model
  12. Misspecified model
  13. Application
  14. Fit time series
  15. change point detection and estimation
  16. ...and 11 more sections

Key Result

Proposition 2.1

Assume Ass:MomentAssumptions, Ass:independent_regpart_stationary and Ass:differentiableSigmoidFct, Ass:Existence_ParameterMinimizer. Then

Figures (4)

  • Figure 1: Assumptotic and simulated distribution of the change point estimator for model GAR 1, GAR 2, GAR 3 and GAR 4 with 1 hidden layer neural network as regression function and standard normally distributed errors, N=500
  • Figure 2: DAX-data from 01.2000-01.2008 and the corresponding returns
  • Figure 3: DAX and Fullertransformed returns with the detected and estimated change point (blue line), $\alpha=0.05$
  • Figure 4: estimated change point using significance level $\alpha=0.01$ with constant number of hidden neurons $h=1$

Theorems & Definitions (14)

  • Proposition 2.1
  • Theorem 3.1
  • Corollary 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 6.1
  • Proposition 6.1
  • Proposition 6.2
  • Lemma 6.1
  • Lemma 6.2
  • ...and 4 more