get_Sigmas function

Calculate the dp-dimensional covariance matrices $\Sigma_{m,p}$ in the transition weights with weight_function="relative_dens"

get_Sigmas calculatesthe dp-dimensional covariance matrices $\Sigma_{m,p}$ in the transition weights with weight_function="relative_dens" so that the algorithm proposed by McElroy (2017) employed whenever it reduces the computation time.

get_Sigmas(p, M, d, all_A, all_boldA, all_Omegas)

Arguments

• p: a positive integer specifying the autoregressive order
• M: a positive integer specifying the number of regimes
• d: the number of time series in the system, i.e., the dimension
• all_A: 4D array containing all coefficient matrices $A_{m,i}$, obtained from pick_allA.
• all_boldA: 3D array containing the $((dp)x(dp))$ "bold A" (companion form) matrices of each regime, obtained from form_boldA. Will be computed if not given.
• all_Omegas: a [d, d, M] array containing the covariance matrix Omegas

Returns

Returns a [dp, dp, M] array containing the dp-dimensional covariance matrices for each regime.

Details

Calculates the dp-dimensional covariance matrix using the formula (2.1.39) in Lütkepohl (2005) when d*p < 12 and using the algorithm proposed by McElroy (2017) otherwise.

The code in the implementation of the McElroy's (2017) algorithm (in the function VAR_pcovmat) is adapted from the one provided in the supplementary material of McElroy (2017). Reproduced under GNU General Public License, Copyright (2015) Tucker McElroy.

References

• Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
• McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124 , 92-96.