pick_regime picks the regime parameters
\((\phi_{m,0},vec(A_{m,1}),...,vec(A_{m,p}),vech(\Omega_m))\)
pick_regime(
p,
M,
d,
params,
m,
cond_dist = c("Gaussian", "Student", "ind_Student", "ind_skewed_t"),
identification = c("reduced_form", "recursive", "heteroskedasticity",
"non-Gaussianity")
)Returns the vector...
identification == "non-Gaussianity" or cond_dist %in% c("ind_Student", "ind_skewed_t"):\((\phi_{m,0},vec(A_{m,1}),...,vec(A_{m,p}),vec(B_m))\).
\((\phi_{m,0},vec(A_{m,1}),...,vec(A_{m,p}),vech(\Omega_m))\).
Note that neither weight parameters or distribution parameters are picked.
the autoregressive order of the model
the number of regimes
the number of time series in the system, i.e., the dimension
a real valued vector specifying the parameter values. Should have the form \(\theta = (\phi_{1,0},...,\phi_{M,0},\varphi_1,...,\varphi_M,\sigma,\alpha,\nu)\), where (see exceptions below):
\(\phi_{m,0} = \) 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 threshold values.
weight_function="exogenous":Omit \(\alpha\) 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 \(\Omega_m=W\Lambda_mW'\), \(\Lambda_m=diag(\lambda_{m1},...,\lambda_{md})\), \(\lambda_{mi}>0\), \(m=2,...,M\), \(i=1,...,d\).
Above, \(\phi_{m,0}\) 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 \(\phi_{m,0}\) with regimewise mean
\(\mu_{m}\).
which regime?
Constrained models nor structural models are supported.