get_varying_h: Get differences 'h' which are adjusted for overly large...

In saviviro/gmvarkit: Estimate Gaussian and Student's t Mixture Vector Autoregressive Models

Get differences 'h' which are adjusted for overly large degrees of freedom parameters

Description

get_varying_h adjusts differences for overly large degrees of freedom parameters for finite difference approximation of the derivatives of the log-likelihood function.

Usage

get_varying_h(M, params, model = c("GMVAR", "StMVAR", "G-StMVAR"))

Arguments

M	For GMVAR and StMVAR models: a positive integer specifying the number of mixture components. For G-StMVAR models: 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.
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 \psi (qx1) satisfies (\phi_{1},..., \phi_{M}) = C \psi where C is a (Mpd^2xq) constraint matrix. 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 \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth mixture component. 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.
model	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.

Details

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.

Value

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).

saviviro/gmvarkit documentation built on Jan. 4, 2025, 12:52 p.m.