loglikelihood_int computes log-likelihood of a GMVAR model.
loglikelihood_int(
data,
p,
M,
params,
conditional = TRUE,
parametrization = c("intercept", "mean"),
constraints = NULL,
structural_pars = NULL,
to_return = c("loglik", "mw", "mw_tplus1", "loglik_and_mw", "terms", "regime_cmeans",
"total_cmeans", "total_ccovs"),
check_params = TRUE,
minval = NULL
)a matrix or class 'ts' object with d>1 columns. Each column is taken to represent
a single time series. NA values are not supported.
a positive integer specifying the autoregressive order of the model.
a positive integer specifying the number of mixture components.
a real valued vector specifying the parameter values.
Should be size \(((M(pd^2+d+d(d+1)/2+1)-1)x1)\) and have form \(\theta\)\( = \)(\(\upsilon\)\(_{1}\), ...,\(\upsilon\)\(_{M}\), \(\alpha_{1},...,\alpha_{M-1}\)), where
\(\upsilon\)\(_{m}\) \( = (\phi_{m,0},\)\(\phi\)\(_{m}\)\(,\sigma_{m})\)
\(\phi\)\(_{m}\)\( = (vec(A_{m,1}),...,vec(A_{m,p})\)
and \(\sigma_{m} = vech(\Omega_{m})\), m=1,...,M.
Should be size \(((M(d+d(d+1)/2+1)+q-1)x1)\) and have form \(\theta\)\( = (\phi_{1,0},...,\phi_{M,0},\)\(\psi\) \(,\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1})\), where
\(\psi\) \((qx1)\) satisfies (\(\phi\)\(_{1}\)\(,...,\) \(\phi\)\(_{M}) =\) \(C \psi\) where \(C\) is \((Mpd^2xq)\) constraint matrix.
Should have the form \(\theta\)\( = (\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(_{1},...,\)\(\phi\)\(_{M}, vec(W),\)\(\lambda\)\(_{2},...,\)\(\lambda\)\(_{M},\alpha_{1},...,\alpha_{M-1})\), where
\(\lambda\)\(_{m}=(\lambda_{m1},...,\lambda_{md})\) contains the eigenvalues of the \(m\)th mixture component.
Replace \(\phi\)\(_{1}\)\(,...,\) \(\phi\)\(_{M}\) with \(\psi\) \((qx1)\) that satisfies (\(\phi\)\(_{1}\)\(,...,\) \(\phi\)\(_{M}) =\) \(C \psi\), as above.
Remove the zeros from \(vec(W)\) and make sure the other entries satisfy the sign constraints.
Replace \(\lambda\)\(_{2},...,\)\(\lambda\)\(_{M}\) with \(\gamma\) \((rx1)\) that satisfies (\(\lambda\)\(_{2}\)\(,...,\) \(\lambda\)\(_{M}) =\) \(C_{\lambda} \gamma\) where \(C_{\lambda}\) is a \((d(M-1) x r)\) constraint matrix.
Above, \(\phi_{m,0}\) is the intercept parameter, \(A_{m,i}\) denotes the \(i\)th coefficient matrix of the \(m\)th
mixture component, \(\Omega_{m}\) denotes the error term covariance matrix of the \(m\):th mixture component, and
\(\alpha_{m}\) is the mixing weight parameter. The \(W\) and \(\lambda_{mi}\) are structural parameters replacing the
error term covariance matrices (see Virolainen, 2020). If \(M=1\), \(\alpha_{m}\) and \(\lambda_{mi}\) are dropped.
If parametrization=="mean", just replace each \(\phi_{m,0}\) with regimewise mean \(\mu_{m}\).
\(vec()\) is vectorization operator that stacks columns of a given matrix into a vector. \(vech()\) stacks columns
of a given matrix from the principal diagonal downwards (including elements on the diagonal) into a vector.
The notation is in line with the cited article by Kalliovirta, Meitz and Saikkonen (2016) introducing the GMVAR model.
a logical argument specifying whether the conditional or exact log-likelihood function should be used.
"mean" or "intercept" determining whether the model is parametrized with regime means \(\mu_{m}\) or
intercept parameters \(\phi_{m,0}\), m=1,...,M.
a size \((Mpd^2 x q)\) constraint matrix \(C\) specifying general linear constraints
to the autoregressive parameters. We consider constraints of form
(\(\phi\)\(_{1}\)\(,...,\)\(\phi\)\(_{M}) = \)\(C \psi\),
where \(\phi\)\(_{m}\)\( = (vec(A_{m,1}),...,vec(A_{m,p}) (pd^2 x 1), m=1,...,M\),
contains the coefficient matrices and \(\psi\) \((q x 1)\) contains the related parameters.
For example, to restrict the AR-parameters to be the same for all regimes, set \(C\)=
[I:...:I]' \((Mpd^2 x pd^2)\) where I = diag(p*d^2).
Ignore (or set to NULL) if linear constraints should not be employed.
If NULL a reduced form model is considered. For structural model, should be a list containing
the following elements:
W - a \((dxd)\) matrix with its entries imposing constraints on \(W\): NA indicating that the element is
unconstrained, a positive value indicating strict positive sign constraint, a negative value indicating strict
negative sign constraint, and zero indicating that the element is constrained to zero.
C_lambda - a \((d(M-1) x r)\) constraint matrix that satisfies (\(\lambda\)\(_{2}\)\(,...,\)
\(\lambda\)\(_{M}) =\) \(C_{\lambda} \gamma\) where \(\gamma\) is the new \((r x 1)\)
parameter subject to which the model is estimated (similarly to AR parameter constraints). The entries of C_lambda
must be either positive or zero. Ignore (or set to NULL) if the eigenvalues \(\lambda_{mi}\)
should not be constrained.
See Virolainen (2020) for the conditions required to identify the shocks and for the B-matrix as well (it is \(W\) times a time-varying diagonal matrix with positive diagonal entries).
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, or
the terms \(l_{t}: t=1,..,T\) in the log-likelihood function (see KMS 2016, eq.(9))? Or should
the regimewise conditional means, total conditional means, or total conditional covariance matrices
be returned? Default is the log-likelihood value ("loglik").
should it be checked that the parameter vector satisfies the model assumptions? Can be skipped to save computation time if it does for sure.
the value that will be returned if the parameter vector does not lie in the parameter space (excluding the identification condition).
log-likelihood value of the specified GMVAR model,
to_return=="mw":a size ((n_obs-p)xM) matrix containing the mixing weights: for m:th component in m:th column.
to_return=="mw_tplus1":a size ((n_obs-p+1)xM) matrix containing the mixing weights: for m:th component in m:th column. The last row is for \(\alpha_{m,T+1}\)
to_return=="terms":a size ((n_obs-p)x1) numeric vector containing the terms \(l_{t}\).
to_return=="loglik_and_mw":a list of two elements. The first element contains the log-likelihood value and the second element contains the mixing weights.
to_return=="regime_cmeans":an [T-p, d, M] array containing the regimewise conditional means
(the first p values are used as the initial values).
to_return=="total_cmeans":a [T-p, d] matrix containing the conditional means of the process
(the first p values are used as the initial values).
to_return=="total_ccov":an [d, d, T-p] array containing the conditional covariance matrices of the process
(the first p values are used as the initial values).
loglikelihood_int takes use of the function dmvn from the package mvnfast
to cut down computation time. Values extremely close to zero are handled with the package Brobdingnag.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
L<U+00FC>tkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.
Virolainen S. 2020. Structural Gaussian mixture vector autoregressive model. Unpublished working paper, available as arXiv:2007.04713.