# Mstep.hh.SCAD.MSAR: M step of the EM algorithm for fitting homogeneous... In NHMSAR: Non-Homogeneous Markov Switching Autoregressive Models

## Description

M step of the EM algorithm for fitting homogeneous multivariate Markov switching auto-regressive models with penalization of parameters of the VAR(1) models, called in fit.MSAR. Penalized maximum likelihood is used. Penalization may be add to the autoregressive matrices of order 1 and to the precision matrices (inverse of variance of innovation). Ridge, LASSO and SCAD penalization are implmented for the autoregressive matrices and only SCAD for the precision matrices.

## Usage

 `1` ```Mstep.hh.SCAD.MSAR(data, theta, FB, lambda1=.1,lambda2=.1,penalty=,par=NULL) ```

## Arguments

 `data` array of univariate or multivariate series with dimension T x N.samples x d. T: number of time steps of each sample, N.samples: number of realisations of the same stationary process, d: dimension. `theta` model's parameter; object of class MSAR. See also init.theta.MSAR. `FB` Forward-Backward results, obtained by calling Estep.MSAR function `lambda1` penalization constant for the precision matrices. It may be a scalar or a vector of length M (with M the number of regimes). If it is equal to0 no penalization is introduced for the precision matrices. `lambda2` penalization constant for the autoregressive matrices. It may be a scalar or a vector of length M (with M the number of regimes). If it is equal to0 no penalization is introduced for the atoregression matrices. `penalty` choice of the penalty for the autoregressive matrices. Possible values are ridge, lasso or SCAD (default). `par` allows to give an initial value to the precision matrices.

## Details

When LASSO penalty is chosen, the LARS algorithm is used. When SCAD is chosen, a Newton-Raphson algorithm is run with a quadratic approximation of the penalized likelihood. For the precision matrices penalization, the package glasso is used.

Limit of this function: only works for VAR(1) models

## Value

 `A0` intercepts `A` AR coefficients `sigma` variance of innovation `sigma.inv` inverse of variance of innovation `prior` prior probabilities `transmat` transition matrix

## Author(s)

Valerie Monbet, valerie.monbet@univ-rennes1.fr

## References

Efron, B., Hastie, T., Johnstone, I., Tibshirani, R., et al. (2004). Least angle regression. The Annals of statistics, 32(2):407-499.

Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American statistical Association, 96(456):1348-1360.