get_varying_h adjusts differences for overly large degrees of freedom parameters
for finite difference approximation of the derivatives of the log-likelihood function.
get_varying_h(M, params, model = c("GMVAR", "StMVAR", "G-StMVAR"))Returns a vector with the same length as params. For other parameters than degrees
of freedom parameters larger than 100, the differences will be 6e-6. For the large degrees of
freedom parameters, the difference will be signif(df/1000, digits=2).
a positive integer specifying the number of mixture components.
a size (2x1) integer vector specifying the number of GMVAR type components M1
in the first element and StMVAR type components M2 in the second element. The total number of mixture components
is M=M1+M2.
a real valued vector specifying the parameter values.
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.
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
\(\psi\) \((qx1)\) satisfies (\(\phi\)\(_{1}\)\(,...,\) \(\phi\)\(_{M}) =\) \(C \psi\) where \(C\) is a \((Mpd^2xq)\) constraint matrix.
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, \(\psi\)\( = \) (\(\phi\)\(_{1}\)\(,...,\)\(\phi\)\(_{M})\).
Drop \(\alpha_1,...,\alpha_{M-1}\) from the parameter vector.
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
\(\lambda\)\(_{m}=(\lambda_{m1},...,\lambda_{md})\) 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.
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.
fixed_lambdas:Drop \(\lambda\)\(_{2},...,\) \(\lambda\)\(_{M}\) from the parameter vector.
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.
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first M1 components
are GMVAR type and the rest M2 components are StMVAR type.
This function is used for approximating gradient and Hessian of a StMVAR or G-StMVAR model. Very large degrees of freedom parameters cause significant numerical error if too small differences are used.