params |
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) .
-
- if
cond_dist="Gaussian" or "Student" : \sigma = (vech(\Omega_1),...,vech(\Omega_M))
(Md(d + 1)/2 \times 1) .
- if
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).
-
- if
cond_dist = "Gaussian") : Omit \nu from the parameter vector.
- if
cond_dist="Student" : \nu > 2 is the single degrees of freedom parameter.
- if
cond_dist="ind_Student" : \nu = (\nu_1,...,\nu_d) (d \times 1) , \nu_i > 2 .
- if
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.
- AR_constraints:
Replace \varphi_1,...,\varphi_M with \psi as described in the argument AR_constraints .
- mean_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.
- weight_constraints:
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 .
- B_constraints:
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.
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