# M step of the EM algorithm for fitting homogeneous multivariate Markov switching auto-regressive models with penalization of parameters of the VAR(1) 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.

### See Also

Mstep.hh.MSAR, fit.MSAR