reform_constrained_pars reforms constrained parameter vector
into the form that corresponds to unconstrained parameter vectors.
reform_constrained_pars(
p,
M,
d,
params,
constraints = NULL,
same_means = NULL,
structural_pars = NULL,
change_na = FALSE
)a positive integer specifying the autoregressive order of the model.
a positive integer specifying the number of mixture components.
the number of time series in the system.
a real valued vector specifying the parameter values.
Should be size \(((M(pd^2+d+d(d+1)/2+1)-1)x1)\) and have the 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 the 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 \(\theta\)\( = (\)\(\mu\),\(\psi\), \(\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1})\), 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, \(\psi\)\( = \) (\(\phi\)\(_{1}\)\(,...,\)\(\phi\)\(_{M})\).
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.
Replace \((\phi_{1,0},...,\phi_{M,0})\) with \((\mu_{1},...,\mu_{g})\), 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 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.
Restrict the mean parameters of some regimes to be the same? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
M=3, the argument list(1, 2:3) restricts the mean parameters of the second and third regime to be
the same but the first regime has freely estimated (unconditional) mean. Ignore or set to NULL if mean parameters
should not be restricted to be the same among any regimes. This constraint is available only for mean parametrized models;
that is, when parametrization="mean".
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).
change NA parameter values of constrained models to -9.999?
Returns "regular model" parameter vector corresponding to the constraints.
No argument checks!
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Virolainen S. 2020. Structural Gaussian mixture vector autoregressive model. Unpublished working paper, available as arXiv:2007.04713.