params |
a real valued vector specifying the parameter values.
- For reduced form models:
-
Should be size ((M(pd^2+d+d(d+1)/2+2)-M1-1)x1) and have the form
\theta = (\upsilon _{1} ,
...,\upsilon _{M} , \alpha_{1},...,\alpha_{M-1}, \nu ) , 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,
-
\nu =(\nu_{M1+1},...,\nu_{M})
-
M1 is the number of GMVAR type regimes.
- For structural model:
-
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}, \nu ) ,
where
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, 2022). 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.
In the GMVAR model, M1=M and \nu is dropped from the parameter vector. In the StMVAR model,
M1=0 . In the G-StMVAR model, the first M1 regimes are GMVAR type and the rest M2 regimes are
StMVAR type. In StMVAR and G-StMVAR models, the degrees of freedom parameters in \nu
# should be strictly larger than two.
The notation is similar to the cited literature.
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structural_pars |
If NULL a reduced form model is considered. Reduced models can be used directly as recursively
identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing
at least the first one of 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.
-
fixed_lambdas - a length d(M-1) numeric vector (\lambda _{2} ,...,
\lambda _{M}) with elements strictly larger than zero specifying the fixed parameter values for the
parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda .
Ignore (or set to NULL ) if the eigenvalues \lambda_{mi} should not be constrained.
See Virolainen (forthcoming) 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).
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