Section: New Results

Markov–switching vector autoregressive models

Participant : Valérie Monbet.

This is a collaboration with Pierre Ailliot (université de Bretagne Occidentale), Julie Bessac (Argonne National Laboratory, Chicago) and Julien Cattiaux (Météo–France, Toulouse).

Multivariate time series are of interest in many fields including economics and environment. The most popular tools for studying multivariate time series are the vector autoregressive (VAR) models because of their simple specification and the existence of efficient methods to fit these models. However, the VAR models do not allow to describe time series mixing different dynamics. For instance, when meteorological variables are observed, the resulting time series exhibit an alternance of different temporal dynamics corresponding to weather regimes. The regime is often not observed directly and is thus introduced as a latent process in time series models in the spirit of hidden Markov models. Markov switching vector autoregressive (MSVAR) models have been introduced as a generalization of autoregressive models and hidden Markov models. They lead to flexible and interpretable models. In this mutivariate context, several questions occur.

• The discrete hidden variable also called regime has to be correctly defined. Indeed the regime can be local (e.g. link to a subset of the variables) or global (e.g. the same for all the variables). It can also be observed and inferred a priori or hidden. In the second case, it has to be estimated at the same time as the model parameters.

  The question of the definition of the regime is investigated in [26] for the specific problem of multi site wind modeling.

• Markov Switching VAR models (MSVAR) suffer of the same dimensionality problem as VAR models. For large (and even moderate) dimensions, the number of autoregressive coefficients in each regime can be prohibitively large which results in noisy estimates. When the variables are correlated, which is the standard situation in multivariate time series, over–learning is frequent. The estimated parameters contains spurious non–zero coefficients and are then difficult to interpret. The predictions associated to the model are usually unstable. Collinearity causes also ill–conditioning of the innovation covariance. In [29] , we propose a likelihood penalization method with hard thresholding for MSVAR models leading to sparse MSVAR. Both autoregressive matrices and precision matrices are penalized using smoothly clipped absolute deviation (SCAD) penalties.