**‘Selective linear segmentation for detecting relevant parameter changes’**

Dufays A., Houndetoungan A. and Coën A., Accepted in Journal of Financial Econometrics, 2020.

Change-point processes are one flexible approach to model long time series. We propose a method to uncover which model parameter truly vary when a change-point is detected. Given a set of breakpoints, we use a penalized likelihood approach to select the best set of parameters that changes over time and we prove that the penalty function leads to a consistent selection of the true model. Estimation is carried out via the deterministic annealing expectation-maximization algorithm. Our method accounts for model selection uncertainty and associates a probability to all the possible time-varying parameter specifications. Monte Carlo simulations highlight that the method works well for many time series models including heteroskedastic processes. For a sample of 14 Hedge funds (HF) strategies, using an asset based style pricing model, we shed light on the promising ability of our method to detect the time-varying dynamics of risk exposures as well as to forecast HF returns.

**‘Relevant parameter changes in structural break models’**

Dufays A., Rombouts J., Journal of Econometrics, 2020, 217 (1), 46-78.

Structural break time series models, which are commonly used in macroeconomics and finance, capture unknown structural changes by allowing for abrupt changes to model parameters. However, many specifications suffer from an over-parametrization issue, since typically all parameters have to change when a break occurs. We introduce a sparse change-point model to detect which parameters change over time. We propose a shrinkage prior distribution, which controls model parsimony by limiting the number of parameters that change from one structural break to another. We develop a Bayesian sampler for inference on the sparse change-point model. An extensive simulation study based on AR, ARMA and GARCH processes highlights the excellent performance of the sampler. We provide several empirical applications including an out-of-sample forecasting exercise showing that the Sparse change-point framework compares favourably with other recent time-varying parameter processes.

**‘Modeling macroeconomic series with regime-switching models characterized by a high-dimensional state space’ **

Augustinyak, M. and Dufays A., Economic Letters, 2018, 170, 122-126.

The Markov-switching multifractal process, and recent extensions such as the factorial hidden Markov volatility model, correspond to tightly parametrized hidden Markov models characterized by a high-dimensional state space. Because the central component in these models is a Markov chain restricted to have positive support, the applicability of such models has been so far limited to the modeling of positive processes such as volatilities, inter-trade durations and trading volumes. By adapting the factorial hidden Markov volatility model, we develop a new regime-switching process for capturing time variation in the conditional mean of a time series with support on the whole real line. We show its promising performance to fit 21 widely used macroeconomic data sets.

**‘A new approach to volatility modelling: the factorial hidden Markov volatility model’ **

Augustinyak, M. and Bauwens, L. and Dufays A., Journal of Business and Economic Statistics, 2018, 1-14.

A new process, the factorial hidden Markov volatility (FHMV) model, is proposed to model financial returns or realized variances. Its dynamics are driven by a latent volatility process specified as a product of three components: a Markov chain controlling volatility persistence, an independent discrete process capable of generating jumps in the volatility, and a predictable (data-driven) process capturing the leverage effect. An economic interpretation is attached to each one of these components. Moreover, the Markov chain and jump components allow volatility to switch abruptly between thousands of states, and the transition matrix of the model is structured to generate a high degree of volatility persistence. An empirical study on six financial time series shows that the FHMV process compares favorably to state-of-the-art volatility models in terms of in-sample fit and out-of-sample forecasting performance over time horizons ranging from one to one hundred days.

**‘Sparse Change-point HAR Models for Realized Variance’ **

Dufays A. and Rombouts, J., Econometric Reviews, 2018, 1-24

Change-point time series specifications constitute flexible models that capture unknown structural changes by allowing for switches in the model parameters. Nevertheless most models suffer from an over-parametrization issue since typically only one latent state variable drives the switches in all parameters. This implies that all parameters have to change when a break happens.

To gauge whether and where there are structural breaks in realized variance, we introduce the sparse change-point HAR model. The approach controls for model parsimony by limiting the number of parameters which evolve from one regime to another. Sparsity is achieved thanks to employing a nonstandard shrinkage prior distribution. We derive a Gibbs sampler for inferring the parameters of this process. Simulation studies illustrate the excellent performance of the sampler.

Relying on this new framework, we study the stability of the HAR model using realized variance series of several major international indices between January 2000 and August 2015.

**‘Infinite-State Markov-switching for Dynamic Volatility’ **

(Journal version — Complementary file : here — CORE Discussion paper)

Dufays A., Journal of Financial Econometrics, 2016, 14 (2): 418-460

Generalized auto-regressive conditional heteroskedastic model with changing parameters is specified using the sticky infinite hidden Markov-chain framework. Estimation by Bayesian inference determines the adequate number of regimes as well as the optimal specification (Markov-switching or change-point). The new provided algorithms are studied in terms of mixing properties and computational time. Applications highlight the flexibility of the model and compare it to existing parametric alternatives.

**‘Evolutionary Sequential Monte Carlo for Change-point models’**

(Journal version — Matlab program)

Dufays A., Econometrics, 2016, 4(1), 12;

Sequential Monte Carlo (SMC) methods are widely used for non-linear filtering purposes. However, the SMC scope encompasses wider applications such as estimating static model parameters so much that it is becoming a serious alternative to Markov-Chain Monte-Carlo (MCMC) methods. Not only do SMC algorithms draw posterior distributions of static or dynamic parameters but additionally they provide an estimate of the marginal likelihood. The tempered and time (TNT) algorithm, developed in this paper, combines (off-line) tempered SMC inference with on-line SMC inference for drawing realizations from many sequential posterior distributions without experiencing a particle degeneracy problem. Furthermore, it introduces a new MCMC rejuvenation step that is generic, automated and well-suited for multi-modal distributions. As this update relies on the wide heuristic optimization literature, numerous extensions are readily available. The algorithm is notably appropriate for estimating change-point models. As an example, we compare several change-point GARCH models through their marginal log-likelihoods over time.

**‘Autoregressive Moving Average Infinite Hidden Markov-Switching Models’**

Bauwens, L. and Carpantier, J-F. and Dufays, A., Journal of Business and Economic Statistics, 2015, Forthcoming, (DOI:10.1080/07350015.2015.1123636).

Markov-switching models are usually specified under the assumption that all the parameters change when a regime switch occurs. Relaxing this hypothesis and being able to detect which parameters evolve over time is relevant for interpreting the changes in the dynamics of the series, for specifying models parsimoniously, and may be helpful in forecasting. We propose the class of sticky infinite hidden Markov-switching autoregressive moving average models, in which we disentangle the break dynamics of the mean and the variance parameters. In this class, the number of regimes is possibly infinite and is determined when estimating the model, thus avoiding the need to set this number by a model choice criterion. We develop a new Markov chain Monte Carlo estimation method that solves the path dependence issue due to the moving average component. Empirical results on macroeconomic series illustrate that the proposed class of models dominates the model with fixed parameters in terms of point and density forecasts.

**‘Marginal Likelihood Computation for Markov Switching and Change-point GARCH Models’**

Bauwens L., Dufays A. and Rombouts J., Journal of Econometrics, 2013, 178 (3), 508-522 and CORE Discussion paper 2011/13.

GARCH volatility models with fixed parameters are too restrictive for long time series due to breaks in the volatility process. Flexible alternatives are Markov-switching GARCH and change-point GARCH models. They require estimation by MCMC methods due to the path dependence problem. An unsolved issue is the computation of their marginal likelihood, which is essential for determining the number of regimes or change-points. We solve the problem by using particle MCMC. We examine the performance of this new method on simulated data, and we illustrate its use on several return series.

**‘A Bayesian Method of Change-point Estimation with Recurrent Regimes: Application to GARCH Models’**

Bauwens L., Dufays A. and De Backer B., Journal of Empirical Finance, 2014, DOI: 10.1016/j.jempfin.2014.06.008

We present an algorithm, based on a differential evolution MCMC method, for Bayesian inference in GARCH models, possibly including a leverage effect, subject to an unknown number of structural breaks at unknown dates. Break dates are treated as parameters and we determine the number of breaks by allowing the Markov chains to jump between models that entail different number of breaks. We also show how to compute the marginal likelihood and the posterior marginal likelihood of a model given a certain number of breaks as alternatives to determine the unknown number of breaks. We prove the convergence of the algorithm and allow for both pure Change-Point and recurrent regime specifications. We illustrate the efficiency of the algorithm through simulations and we apply it to eight financial time series of daily returns over the period 1987-2011. We find at least three breaks in all series.