params |
a real valued vector specifying the parameter values.
- For unconstrained 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 constrained models:
-
Should be size ((M(d+d(d+1)/2+2)+q-M1-1)x1) and have the form
\theta = (\phi_{1,0},...,\phi_{M,0},\psi,
\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where
- For same_means models:
-
Should have the form
\theta = (\mu,\psi,
\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where
-
\mu= (\mu_{1},...,\mu_{g}) where
\mu_{i} is the mean parameter for group i and
g is the number of groups.
If AR constraints are employed, \psi is as for constrained
models, and if AR constraints are not employed, \psi =
(\phi_{1},...,\phi_{M}).
- For models with weight_constraints:
Drop \alpha_1,...,\alpha_{M-1} from
the parameter vector.
- For structural models:
-
Reduced form models can be directly used as recursively identified structural models. If the structural model is
identified by conditional heteroskedasticity, the parameter vector 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
- If AR parameters are constrained:
Replace \phi_{1},...,
\phi_{M} with \psi (qx1) that satisfies (\phi_{1},...,
\phi_{M}) = C \psi, as above.
- If same_means:
Replace (\phi_{1,0},...,\phi_{M,0}) with (\mu_{1},...,\mu_{g}),
as above.
- If
W is constrained: Remove the zeros from vec(W) and make sure the other entries satisfy
the sign constraints.
- If
\lambda_{mi} are constrained via C_lambda: 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.
- If
\lambda_{mi} are constrained via fixed_lambdas: Drop \lambda_{2},...,
\lambda_{M} from the parameter vector.
Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth
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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