get_alpha_mt computes the mixing weights based on
the logarithm of the multivariate normal densities in the definition of
the mixing weights.
get_alpha_mt(
M,
log_mvnvalues,
alphas,
epsilon,
conditional,
to_return,
also_l_0 = FALSE
)Returns the mixing weights a matrix of the same dimension as log_mvnvalues so
that the t:th row is for the time point t and m:th column is for the regime m.
a positive integer specifying the number of mixture components.
a size (2x1) integer vector specifying the number of GMAR type components M1 in the
first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.
\(T x M\) matrix containing the log multivariate normal densities.
\(M x 1\) vector containing the mixing weight pa
the smallest number such that its exponent is wont classified as numerically zero
(around -698 is used).
a logical argument specifying whether the conditional or exact log-likelihood function should be used.
should the returned object be the log-likelihood value, mixing weights, mixing weights including value for \(alpha_{m,T+1}\), a list containing log-likelihood value and mixing weights, the terms \(l_{t}: t=1,..,T\) in the log-likelihood function (see KMS 2015, eq.(13)), the densities in the terms, regimewise conditional means, regimewise conditional variances, total conditional means, total conditional variances, or quantile residuals?
return also l_0 (the first term in the exact log-likelihood function)?
Note that we index the time series as \(-p+1,...,0,1,...,T\) as in Kalliovirta et al. (2015).
Galbraith, R., Galbraith, J. 1974. On the inverses of some patterned matrices arising in the theory of stationary time series. Journal of Applied Probability 11, 63-71.
Kalliovirta L. (2012) Misspecification tests based on quantile residuals. The Econometrics Journal, 15, 358-393.
Kalliovirta L., Meitz M. and Saikkonen P. 2015. Gaussian Mixture Autoregressive model for univariate time series. Journal of Time Series Analysis, 36(2), 247-266.
Meitz M., Preve D., Saikkonen P. 2023. A mixture autoregressive model based on Student's t-distribution. Communications in Statistics - Theory and Methods, 52(2), 499-515.
Virolainen S. 2022. A mixture autoregressive model based on Gaussian and Student's t-distributions. Studies in Nonlinear Dynamics & Econometrics, 26(4) 559-580.
loglikelihood_int