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Estimating weak Markov-switching AR(1) models

Yacouba Boubacar Mainassara, Landy Rabehasaina, Armel Bra

TL;DR

The paper develops a moment-based estimation framework for weak Markov-switching AR(1) models with errors that are uncorrelated but potentially dependent. By formulating a moment map from autocovariances and solving a Newton-type system, the authors prove consistency and asymptotic normality of the estimator under strong mixing and finite-moment assumptions, and they derive an explicit sandwich form for the asymptotic covariance. They also propose a spectral-based, weakly consistent estimator for the covariance matrix, enabling valid inference, and they illustrate performance through Monte Carlo simulations and an application to hourly meteorological wind-speed data. The work broadens the applicability of Markov regime-switching autoregressions by accommodating conditional heteroscedasticity and dependent innovations, and it provides practical tools for variance estimation and regime determination in real-world time series. Overall, the results advance methodology for inference in nonlinear time series with regime changes under weaker independence assumptions, with demonstrated robustness in finite samples and real data.

Abstract

In this paper, we present the asymptotic properties of the moment estimator for autoregressive (AR for short) models subject to Markovian changes in regime under the assumption that the errors are uncorrelated but not necessarily independent. We relax the standard independence assumption on the innovation process to extend considerably the range of application of the Markov-switching AR models. We provide necessary conditions to prove the consistency and asymptotic normality of the moment estimator in a specific case. Particular attention is paid to the estimation of the asymptotic covariance matrix. Finally, some simulation studies and an application to the hourly meteorological data are presented to corroborate theoretical work.

Estimating weak Markov-switching AR(1) models

TL;DR

The paper develops a moment-based estimation framework for weak Markov-switching AR(1) models with errors that are uncorrelated but potentially dependent. By formulating a moment map from autocovariances and solving a Newton-type system, the authors prove consistency and asymptotic normality of the estimator under strong mixing and finite-moment assumptions, and they derive an explicit sandwich form for the asymptotic covariance. They also propose a spectral-based, weakly consistent estimator for the covariance matrix, enabling valid inference, and they illustrate performance through Monte Carlo simulations and an application to hourly meteorological wind-speed data. The work broadens the applicability of Markov regime-switching autoregressions by accommodating conditional heteroscedasticity and dependent innovations, and it provides practical tools for variance estimation and regime determination in real-world time series. Overall, the results advance methodology for inference in nonlinear time series with regime changes under weaker independence assumptions, with demonstrated robustness in finite samples and real data.

Abstract

In this paper, we present the asymptotic properties of the moment estimator for autoregressive (AR for short) models subject to Markovian changes in regime under the assumption that the errors are uncorrelated but not necessarily independent. We relax the standard independence assumption on the innovation process to extend considerably the range of application of the Markov-switching AR models. We provide necessary conditions to prove the consistency and asymptotic normality of the moment estimator in a specific case. Particular attention is paid to the estimation of the asymptotic covariance matrix. Finally, some simulation studies and an application to the hourly meteorological data are presented to corroborate theoretical work.

Paper Structure

This paper contains 35 sections, 19 theorems, 245 equations, 7 figures, 12 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model and assumptions
  3. Estimation of the ARHMC$(1)$ model parameters
  4. Asymptotic properties
  5. Consistency and asymptotic normality of the moments estimator
  6. An example where $\mathbf{r}(\theta_0) < K^2 + K$
  7. Expression of the matrix $I$ when $(\eta_t)_{t\in\mathbb{Z}}$ is assumed i.i.d.
  8. $\diamond$ Case 1: Expression for $\mathbb{E}\left(\prod_{i=1}^{s} c^{\iota_i}_{\kappa_i}f_{-\iota_i-1}\right)$.
  9. $\diamond$ Case 2: Expression for $\mathbb{E}\left(\prod_{i=1}^{2} c^{\iota_i}_{\kappa_i}f_{-\iota_i-1}\prod_{i=3}^{4}c^{\iota_i}_{\kappa_i}f_{-\iota_i-1}\right)$.
  10. Estimation of the asymptotic covariance matrix
  11. Numerical illustrations
  12. Application to real data
  13. Conclusion
  14. Proofs
  15. Proof of Theorem \ref{['thm:1']}
  16. ...and 20 more sections

Key Result

Theorem 1

Under Assumptions $(\mathbf{A_1})$, $(\mathbf{A_2})$, $(\mathbf{A_3})$ and $(\mathbf{A_6})$, the joint moments of the process $(X_t)_{t\in\mathbb{Z}}$ defined in (eq:1) satisfy for all $k\in\mathbb{N}$: where $I_K$ denotes the identity matrix of size $K$, $\bm{1}' := (1, \dots, 1)$ is a row vector of dimension $K$ and $\bm{\pi}_{f^2} := \{f^2(1)\pi(1), \dots, f^2(K)\pi(K)\}'$ is a column vector o

Figures (7)

  • Figure 1: An illustration example of the sets $\overline{\mathcal{P}_s}$, $\mathcal{L}(\mathcal{P}_s)$, and $\mathcal{I}(\mathcal{P}_s)$ for $s=4$. This shows how each element of $\overline{\mathcal{P}_s}$ is positioned on the number line $\mathcal{L}(\mathcal{P}_s)$ and the intervals between consecutive points represent the elements of $\mathcal{I}(\mathcal{P}_s)$, demonstrating their sequential relationships and distribution.
  • Figure 2: Simulation of length 400 of model (\ref{['eq:1']}) with $\theta_0: = \left(\alpha_{11},\alpha_{22},\beta_{11},\beta_{21},\gamma_{11},\gamma_{22}\right)' = \left(-0.4,0.3,0.3,0.2,1.0,0.5\right)'$ and $(\omega_0, a_0, \beta_0) = (0.2,0.1,0.5)$.
  • Figure 3: Boxplots and distribution of errors $\hat{\theta}_n(i) - \theta_0(i)$ for $i = 1, \ldots, 6$, where the noise $\eta_t$ is defined as $u_t$, $u_t u_{t-1}$, $u_t^2 u_{t-1}$, and $(\omega_0 + a_0 \eta_{t-1}^2 + \beta_0 h_{t-1} \, u_t)^{1/2}$, respectively. The kernel density estimate is displayed in full line and the centered Gaussian density with the same variance is plotted in dotted line.
  • Figure 4: QQ-plots of errors $\hat{\theta}_n(i) - \theta_0(i)$ for $i = 1, \ldots, 6$, where the noise $\eta_t$ is defined as $u_t$, $u_t u_{t-1}$, $u_t^2 u_{t-1}$, and $(\omega_0 + a_0 \eta_{t-1}^2 + \beta_0 h_{t-1} \, u_t)^{1/2}$, respectively.
  • Figure 5: Comparison of standard and modified estimates of the asymptotic covariance matrix $\Omega$ of the moment estimator on the simulated model presented in Fig. \ref{['fig: density']} and Fig. \ref{['fig: qqplot']} . Strong ARHMC corresponds to the model (\ref{['eq:1']}) with the noise Strong, Weak 1 ARHMC corresponds to the model (\ref{['eq:1']}) with the noise Weak 1, and Weak 2 ARHMC corresponds to the model (\ref{['eq:1']}) with the noise Weak 2. The red diamond symbols represent the mean over $R=1,000$ replications of standardized squared errors: $n\{\hat{\alpha}_{11}+0.4\}^2$ for (a), $n\{\hat{\alpha}_{22}-0.3\}^2$ for (b), $n\{\hat{\beta}_{11}-0.3\}^2$ for (c), $n\{\hat{\beta}_{22}-0.3\}^2$ for (d), $n\{\hat{\gamma}_{11}-1.0\}^2$ for (e), and $n\{\hat{\gamma}_{22}-0.5\}^2$ for (f).
  • ...and 2 more figures

Theorems & Definitions (23)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Theorem 4
  • Lemma 1: francq2004estimation
  • Lemma 2: davydov1968convergence
  • ...and 13 more