pick_lambdas picks the structural parameters eigenvalue 'lambdas' from the parameter vector
of a structural model identified by heteroskedasticity.
pick_lambdas(
p,
M,
d,
params,
identification = c("reduced_form", "recursive", "heteroskedasticity",
"non-Gaussianity")
)Returns the length (d*(M - 1)) vector \((\lambda_{2},...,\lambda_{M})\)
(see the argument params) for structural models identified by heteroskedasticity,
numeric(0) if \(M=1\), and NULL for other models.
a positive integer specifying the autoregressive order
a positive integer specifying the number of regimes
a real valued vector specifying the parameter values. Should have the form \(\theta = (\phi_{1},...,\phi_{M},\varphi_1,...,\varphi_M,\sigma,\alpha,\nu)\), where (see exceptions below):
\(\phi_{m} = \) the \((d \times 1)\) intercept (or mean) vector of the \(m\)th regime.
\(\varphi_m = (vec(A_{m,1}),...,vec(A_{m,p}))\) \((pd^2 \times 1)\).
cond_dist="Gaussian" or "Student":\(\sigma = (vech(\Omega_1),...,vech(\Omega_M))\) \((Md(d + 1)/2 \times 1)\).
cond_dist="ind_Student" or "ind_skewed_t":\(\sigma = (vec(B_1),...,vec(B_M)\) \((Md^2 \times 1)\).
\(\alpha = \) the \((a\times 1)\) vector containing the transition weight parameters (see below).
cond_dist = "Gaussian"):Omit \(\nu\) from the parameter vector.
cond_dist="Student":\(\nu > 2\) is the single degrees of freedom parameter.
cond_dist="ind_Student":\(\nu = (\nu_1,...,\nu_d)\) \((d \times 1)\), \(\nu_i > 2\).
cond_dist="ind_skewed_t":\(\nu = (\nu_1,...,\nu_d,\lambda_1,...,\lambda_d)\) \((2d \times 1)\), \(\nu_i > 2\) and \(\lambda_i \in (0, 1)\).
For models with...
weight_function="relative_dens":\(\alpha = (\alpha_1,...,\alpha_{M-1})\) \((M - 1 \times 1)\), where \(\alpha_m\) \((1\times 1), m=1,...,M-1\) are the transition weight parameters.
weight_function="logistic":\(\alpha = (c,\gamma)\) \((2 \times 1)\), where \(c\in\mathbb{R}\) is the location parameter and \(\gamma >0\) is the scale parameter.
weight_function="mlogit":\(\alpha = (\gamma_1,...,\gamma_M)\) \(((M-1)k\times 1)\), where \(\gamma_m\) \((k\times 1)\), \(m=1,...,M-1\) contains the multinomial logit-regression coefficients of the \(m\)th regime. Specifically, for switching variables with indices in \(I\subset\lbrace 1,...,d\rbrace\), and with \(\tilde{p}\in\lbrace 1,...,p\rbrace\) lags included, \(\gamma_m\) contains the coefficients for the vector \(z_{t-1} = (1,\tilde{z}_{\min\lbrace I\rbrace},...,\tilde{z}_{\max\lbrace I\rbrace})\), where \(\tilde{z}_{i} =(y_{it-1},...,y_{it-\tilde{p}})\), \(i\in I\). So \(k=1+|I|\tilde{p}\) where \(|I|\) denotes the number of elements in \(I\).
weight_function="exponential":\(\alpha = (c,\gamma)\) \((2 \times 1)\), where \(c\in\mathbb{R}\) is the location parameter and \(\gamma >0\) is the scale parameter.
weight_function="threshold":\(\alpha = (r_1,...,r_{M-1})\) \((M-1 \times 1)\), where \(r_1,...,r_{M-1}\) are the thresholds.
weight_function="exogenous":Omit \(\alpha\) from the parameter vector.
Replace \(\varphi_1,...,\varphi_M\) with \(\psi\) as described in the argument AR_constraints.
Replace \(\phi_{1},...,\phi_{M}\) with \((\mu_{1},...,\mu_{g})\) where \(\mu_i, \ (d\times 1)\) is the mean parameter for group \(i\) and \(g\) is the number of groups.
If linear constraints are imposed, replace \(\alpha\) with \(\xi\) as described in the
argument weigh_constraints. If weight functions parameters are imposed to be fixed values, simply drop \(\alpha\)
from the parameter vector.
identification="heteroskedasticity":\(\sigma = (vec(W),\lambda_2,...,\lambda_M)\), where \(W\) \((d\times d)\) and \(\lambda_m\) \((d\times 1)\), \(m=2,...,M\), satisfy \(\Omega_1=WW'\) and \(\Omega_m=W\Lambda_mW'\), \(\Lambda_m=diag(\lambda_{m1},...,\lambda_{md})\), \(\lambda_{mi}>0\), \(m=2,...,M\), \(i=1,...,d\).
For models identified by heteroskedasticity, replace \(vec(W)\) with \(\tilde{vec}(W)\) that stacks the columns of the matrix \(W\) in to vector so that the elements that are constrained to zero are not included. For models identified by non-Gaussianity, replace \(vec(B_1),...,vec(B_M)\) with similarly with vectorized versions \(B_m\) so that the elements that are constrained to zero are not included.
Above, \(\phi_{m}\) is the intercept parameter, \(A_{m,i}\) denotes the \(i\)th coefficient matrix of the \(m\)th
regime, \(\Omega_{m}\) denotes the positive definite error term covariance matrix of the \(m\)th regime, and \(B_m\)
is the invertible \((d\times d)\) impact matrix of the \(m\)th regime. \(\nu_m\) is the degrees of freedom parameter
of the \(m\)th regime.
If parametrization=="mean", just replace each \(\phi_{m}\) 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. \(Bvec()\)
is a vectorization operator that stacks the columns of a given impact matrix \(B_m\) into a vector so that the elements
that are constrained to zero by the argument B_constraints are excluded.
is it reduced form model or an identified structural model; if the latter, how is it identified (see the vignette or the references for details)?
"reduced_form":Reduced form model.
"recursive":The usual lower-triangular recursive identification of the shocks via their impact responses.
"heteroskedasticity":Identification by conditional heteroskedasticity, which imposes constant relative impact responses for each shock.
"non-Gaussianity":Identification by non-Gaussianity; requires mutually independent non-Gaussian shocks, thus,
currently available only with the conditional distribution "ind_Student".
Constrained parameter vectors are not supported. Not even constraints in \(W\)!
Lütkepohl H., Netšunajev A. 2017. Structural vector autoregressions with smooth transition in variances. Journal of Economic Dynamics & Control, 84, 43-57.