get_Sigmas: Calculate the dp-dimensional covariance matrices Sigma_{m,p}...

In gmvarkit: Estimate Gaussian and Student's t Mixture Vector Autoregressive Models

Calculate the dp-dimensional covariance matrices \Sigma_{m,p} in the mixing weights of the GMVAR, StMVAR, or G-StMVAR model.

Description

get_Sigmas calculates the dp-dimensional covariance matrices \Sigma_{m,p} in the mixing weights of the GMVAR, StMVAR, or G-StMVAR model so that the algorithm proposed by McElroy (2017) employed whenever it reduces the computation time.

Usage

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

Arguments

p	a positive integer specifying the autoregressive order of the model.
M	For GMVAR and StMVAR models: a positive integer specifying the number of mixture components. For G-StMVAR models: a size (2\times 1) integer vector specifying the number of GMVAR type components M1 in the first element and StMVAR type components M2 in the second element. The total number of mixture components is M=M1+M2.
d	the number of time series in the system.
all_A	4D array containing all coefficient matrices A_{m,i}, obtained from pick_allA.
all_boldA	3D array containing the ((dp)\times (dp)) "bold A" matrices related to each mixture component VAR-process, obtained from form_boldA. Will be computed if not given.
all_Omega	a [d, d, M] array containing the covariance matrix Omegas

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.

Value

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

References

• Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.

• 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.

• Virolainen S. 2025. A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics. 43:1, 44-54.

• Virolainen S. in press. A Gaussian and Student’s mixture vector autoregressive model with an application to monetary policy shocks. Econometrics and Statistics.

gmvarkit documentation built on Nov. 5, 2025, 6:15 p.m.