params |
a real valued vector specifying the parameter values.
Should have the form θ=(ϕ1,0,...,ϕM,0,φ1,...,φM,σ,α,ν) ,
where (see exceptions below):
ϕm,0= the (d×1) intercept (or mean) vector of the m th regime.
φm=(vec(Am,1),...,vec(Am,p)) (pd2×1) .
-
- if
cond_dist="Gaussian" or "Student" : σ=(vech(Ω1),...,vech(ΩM))
(Md(d+1)/2×1) .
- if
cond_dist="ind_Student" : σ=(vec(B1),...,vec(BM) (Md2×1) .
α= the (a×1) vector containing the transition weight parameters (see below).
-
- if
cond_dist = "Gaussian") : Omit ν from the parameter vector.
- if
cond_dist="Student" : ν>2 is the single degrees of freedom parameter.
- if
cond_dist="ind_Student" : ν=(ν1,...,νM) (M×1) , num>2 .
For models with...
weight_function="relative_dens" :α=(α1,...,αM−1)
(M−1×1) , where αm (1×1),m=1,...,M−1 are the transition weight parameters.
weight_function="logistic" :α=(c,γ)
(2×1) , where c∈R is the location parameter and γ>0 is the scale parameter.
weight_function="mlogit" :α=(γ1,...,γM) ((M−1)k×1) ,
where γm (k×1) , m=1,...,M−1 contains the multinomial logit-regression coefficients
of the m th regime. Specifically, for switching variables with indices in I⊂{1,...,d} , and with
p~∈{1,...,p} lags included, γm contains the coefficients for the vector
zt−1=(1,z~min{I},...,z~max{I}) , where
z~i=(yit−1,...,yit−p~) , i∈I . So k=1+∣I∣p~
where ∣I∣ denotes the number of elements in I .
weight_function="exponential" :α=(c,γ)
(2×1) , where c∈R is the location parameter and γ>0 is the scale parameter.
weight_function="threshold" :α=(r1,...,rM−1)
(M−1×1) , where r1,...,rM−1 are the threshold values.
weight_function="exogenous" :Omit α from the parameter vector.
- AR_constraints:
Replace φ1,...,φM with ψ as described in the argument AR_constraints .
- mean_constraints:
Replace ϕ1,0,...,ϕM,0 with (μ1,...,μg) where
μi, (d×1) is the mean parameter for group i and g is the number of groups.
- weight_constraints:
If linear constraints are imposed, replace α with ξ as described in the
argument weigh_constraints . If weight functions parameters are imposed to be fixed values, simply drop α
from the parameter vector.
identification="heteroskedasticity" :σ=(vec(W),λ2,...,λM) , where
W (d×d) and λm (d×1) , m=2,...,M , satisfy
Ω1=WW′ and Ωm=WΛmW′ , Λm=diag(λm1,...,λmd) ,
λmi>0 , m=2,...,M , i=1,...,d .
- B_constraints (only for structural models identified by heteroskedasticity):
Replace vec(W) with 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.
Above, ϕm,0 is the intercept parameter, Am,i denotes the i th coefficient matrix of the m th
regime, Ωm denotes the positive definite error term covariance matrix of the m th regime, and Bm
is the invertible (d×d) impact matrix of the m th regime. νm is the degrees of freedom parameter
of the m th regime.
If parametrization=="mean" , just replace each ϕm,0 with regimewise mean μm .
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 Bm into a vector so that the elements
that are constrained to zero by the argument B_constraints are excluded.
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weight_function |
What type of transition weights αm,t should be used?
"relative_dens" :αm,t=∑n=1Mαnfn,dp(yt−1,...,yt−p+1)αmfm,dp(yt−1,...,yt−p+1) , where
αm∈(0,1) are weight parameters that satisfy ∑m=1Mαm=1 and
fm,dp(⋅) 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 , α1,t=1−α2,t ,
and α2,t=[1+exp{−γ(yit−j−c)}]−1 , where yit−j is the lag j
observation of the i th variable, c is a location parameter, and γ>0 is a scale parameter.
"mlogit" :αm,t=∑n=1Mexp{γn′zt−1}exp{γm′zt−1} , where γm are coefficient vectors, γM=0 ,
and zt−1 (k×1) is the vector containing a constant and the (lagged) switching variables.
"exponential" :M=2 , α1,t=1−α2,t ,
and α2,t=1−exp{−γ(yit−j−c)} , where yit−j is the lag j
observation of the i th variable, c is a location parameter, and γ>0 is a scale parameter.
"threshold" :αm,t=1 if rm−1<yit−j≤rm and 0 otherwise, where
−∞≡r0<r1<⋯<rM−1<rM≡∞ are thresholds yit−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.
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