params |
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) .
-
- 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_{1,0},...,\phi_{M,0} 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,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} .
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.
|
weight_function |
What type of transition weights \alpha_{m,t} should be used?
"relative_dens" :\alpha_{m,t}=
\frac{\alpha_mf_{m,dp}(y_{t-1},...,y_{t-p+1})}{\sum_{n=1}^M\alpha_nf_{n,dp}(y_{t-1},...,y_{t-p+1})} , where
\alpha_m\in (0,1) are weight parameters that satisfy \sum_{m=1}^M\alpha_m=1 and
f_{m,dp}(\cdot) is the dp -dimensional stationary density of the m th regime corresponding to p
consecutive observations. Available for Gaussian conditional distribution only.
"logistic" :M=2 , \alpha_{1,t}=1-\alpha_{2,t} ,
and \alpha_{2,t}=[1+\exp\lbrace -\gamma(y_{it-j}-c) \rbrace]^{-1} , where y_{it-j} is the lag j
observation of the i th variable, c is a location parameter, and \gamma > 0 is a scale parameter.
"mlogit" :\alpha_{m,t}=\frac{\exp\lbrace \gamma_m'z_{t-1} \rbrace}
{\sum_{n=1}^M\exp\lbrace \gamma_n'z_{t-1} \rbrace} , where \gamma_m are coefficient vectors, \gamma_M=0 ,
and z_{t-1} (k\times 1) is the vector containing a constant and the (lagged) switching variables.
"exponential" :M=2 , \alpha_{1,t}=1-\alpha_{2,t} ,
and \alpha_{2,t}=1-\exp\lbrace -\gamma(y_{it-j}-c) \rbrace , where y_{it-j} is the lag j
observation of the i th variable, c is a location parameter, and \gamma > 0 is a scale parameter.
"threshold" :\alpha_{m,t} = 1 if r_{m-1}<y_{it-j}\leq r_{m} and 0 otherwise, where
-\infty\equiv r_0<r_1<\cdots <r_{M-1}<r_M\equiv\infty are thresholds y_{it-j} is the lag j
observation of the i th variable.
"exogenous" :Exogenous nonrandom transition weights, specify the weight series in weightfun_pars .
See the vignette for more details about the weight functions.
|
AR_constraints |
a size (Mpd^2 \times q) constraint matrix C specifying linear constraints
to the autoregressive parameters. The constraints are of the form
(\varphi_{1},...,\varphi_{M}) = C\psi , where \varphi_{m} = (vec(A_{m,1}),...,vec(A_{m,p})) \ (pd^2 \times 1),\ m=1,...,M ,
contains the coefficient matrices and \psi (q \times 1) contains the related parameters.
For example, to restrict the AR-parameters to be the identical across the regimes, set C =
[I:...:I ]' (Mpd^2 \times pd^2) where I = diag(p*d^2) .
|