in_paramspace {sstvars}R Documentation

Determine whether the parameter vector is in the parameter space

Description

in_paramspace checks whether the parameter vector is in the parameter space.

Usage

in_paramspace(
  p,
  M,
  d,
  params,
  weight_function = c("relative_dens", "logistic", "mlogit", "exponential", "threshold",
    "exogenous"),
  weightfun_pars = NULL,
  cond_dist = c("Gaussian", "Student", "ind_Student"),
  identification = c("reduced_form", "recursive", "heteroskedasticity",
    "non-Gaussianity"),
  B_constraints = NULL,
  other_constraints = NULL,
  all_boldA,
  all_Omegas,
  weightpars,
  distpars,
  transition_weights,
  stab_tol = 0.001,
  posdef_tol = 1e-08,
  distpar_tol = 1e-08,
  weightpar_tol = 1e-08
)

Arguments

p

a positive integer specifying the autoregressive order

M

a positive integer specifying the number of regimes

d

the number of time series in the system, i.e., the dimension

params

a real valued vector specifying the parameter values. Should have the form θ=(ϕ1,0,...,ϕM,0,φ1,...,φM,σ,α,ν)\theta = (\phi_{1,0},...,\phi_{M,0},\varphi_1,...,\varphi_M,\sigma,\alpha,\nu), where (see exceptions below):

  • ϕm,0=\phi_{m,0} = the (d×1)(d \times 1) intercept (or mean) vector of the mmth regime.

  • φm=(vec(Am,1),...,vec(Am,p))\varphi_m = (vec(A_{m,1}),...,vec(A_{m,p})) (pd2×1)(pd^2 \times 1).

  • if cond_dist="Gaussian" or "Student":

    σ=(vech(Ω1),...,vech(ΩM))\sigma = (vech(\Omega_1),...,vech(\Omega_M)) (Md(d+1)/2×1)(Md(d + 1)/2 \times 1).

    if cond_dist="ind_Student":

    σ=(vec(B1),...,vec(BM)\sigma = (vec(B_1),...,vec(B_M) (Md2×1)(Md^2 \times 1).

  • α=\alpha = the (a×1)(a\times 1) vector containing the transition weight parameters (see below).

  • if cond_dist = "Gaussian"):

    Omit ν\nu from the parameter vector.

    if cond_dist="Student":

    ν>2\nu > 2 is the single degrees of freedom parameter.

    if cond_dist="ind_Student":

    ν=(ν1,...,νM)\nu = (\nu_1,...,\nu_M) (M×1)(M \times 1), num>2nu_m > 2.

For models with...

weight_function="relative_dens":

α=(α1,...,αM1)\alpha = (\alpha_1,...,\alpha_{M-1}) (M1×1)(M - 1 \times 1), where αm\alpha_m (1×1),m=1,...,M1(1\times 1), m=1,...,M-1 are the transition weight parameters.

weight_function="logistic":

α=(c,γ)\alpha = (c,\gamma) (2×1)(2 \times 1), where cRc\in\mathbb{R} is the location parameter and γ>0\gamma >0 is the scale parameter.

weight_function="mlogit":

α=(γ1,...,γM)\alpha = (\gamma_1,...,\gamma_M) ((M1)k×1)((M-1)k\times 1), where γm\gamma_m (k×1)(k\times 1), m=1,...,M1m=1,...,M-1 contains the multinomial logit-regression coefficients of the mmth regime. Specifically, for switching variables with indices in I{1,...,d}I\subset\lbrace 1,...,d\rbrace, and with p~{1,...,p}\tilde{p}\in\lbrace 1,...,p\rbrace lags included, γm\gamma_m contains the coefficients for the vector zt1=(1,z~min{I},...,z~max{I})z_{t-1} = (1,\tilde{z}_{\min\lbrace I\rbrace},...,\tilde{z}_{\max\lbrace I\rbrace}), where z~i=(yit1,...,yitp~)\tilde{z}_{i} =(y_{it-1},...,y_{it-\tilde{p}}), iIi\in I. So k=1+Ip~k=1+|I|\tilde{p} where I|I| denotes the number of elements in II.

weight_function="exponential":

α=(c,γ)\alpha = (c,\gamma) (2×1)(2 \times 1), where cRc\in\mathbb{R} is the location parameter and γ>0\gamma >0 is the scale parameter.

weight_function="threshold":

α=(r1,...,rM1)\alpha = (r_1,...,r_{M-1}) (M1×1)(M-1 \times 1), where r1,...,rM1r_1,...,r_{M-1} are the threshold values.

weight_function="exogenous":

Omit α\alpha from the parameter vector.

AR_constraints:

Replace φ1,...,φM\varphi_1,...,\varphi_M with ψ\psi as described in the argument AR_constraints.

mean_constraints:

Replace ϕ1,0,...,ϕM,0\phi_{1,0},...,\phi_{M,0} with (μ1,...,μg)(\mu_{1},...,\mu_{g}) where μi, (d×1)\mu_i, \ (d\times 1) is the mean parameter for group ii and gg is the number of groups.

weight_constraints:

If linear constraints are imposed, replace α\alpha with ξ\xi as described in the argument weigh_constraints. If weight functions parameters are imposed to be fixed values, simply drop α\alpha from the parameter vector.

identification="heteroskedasticity":

σ=(vec(W),λ2,...,λM)\sigma = (vec(W),\lambda_2,...,\lambda_M), where WW (d×d)(d\times d) and λm\lambda_m (d×1)(d\times 1), m=2,...,Mm=2,...,M, satisfy Ω1=WW\Omega_1=WW' and Ωm=WΛmW\Omega_m=W\Lambda_mW', Λm=diag(λm1,...,λmd)\Lambda_m=diag(\lambda_{m1},...,\lambda_{md}), λmi>0\lambda_{mi}>0, m=2,...,Mm=2,...,M, i=1,...,di=1,...,d.

B_constraints (only for structural models identified by heteroskedasticity):

Replace vec(W)vec(W) with vec~(W)\tilde{vec}(W) that stacks the columns of the matrix WW in to vector so that the elements that are constrained to zero are not included.

Above, ϕm,0\phi_{m,0} is the intercept parameter, Am,iA_{m,i} denotes the iith coefficient matrix of the mmth regime, Ωm\Omega_{m} denotes the positive definite error term covariance matrix of the mmth regime, and BmB_m is the invertible (d×d)(d\times d) impact matrix of the mmth regime. νm\nu_m is the degrees of freedom parameter of the mmth regime. If parametrization=="mean", just replace each ϕm,0\phi_{m,0} with regimewise mean μm\mu_{m}. vec()vec() is vectorization operator that stacks columns of a given matrix into a vector. vech()vech() stacks columns of a given matrix from the principal diagonal downwards (including elements on the diagonal) into a vector. Bvec()Bvec() is a vectorization operator that stacks the columns of a given impact matrix BmB_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 αm,t\alpha_{m,t} should be used?

"relative_dens":

αm,t=αmfm,dp(yt1,...,ytp+1)n=1Mαnfn,dp(yt1,...,ytp+1)\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 αm(0,1)\alpha_m\in (0,1) are weight parameters that satisfy m=1Mαm=1\sum_{m=1}^M\alpha_m=1 and fm,dp()f_{m,dp}(\cdot) is the dpdp-dimensional stationary density of the mmth regime corresponding to pp consecutive observations. Available for Gaussian conditional distribution only.

"logistic":

M=2M=2, α1,t=1α2,t\alpha_{1,t}=1-\alpha_{2,t}, and α2,t=[1+exp{γ(yitjc)}]1\alpha_{2,t}=[1+\exp\lbrace -\gamma(y_{it-j}-c) \rbrace]^{-1}, where yitjy_{it-j} is the lag jj observation of the iith variable, cc is a location parameter, and γ>0\gamma > 0 is a scale parameter.

"mlogit":

αm,t=exp{γmzt1}n=1Mexp{γnzt1}\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 γm\gamma_m are coefficient vectors, γM=0\gamma_M=0, and zt1z_{t-1} (k×1)(k\times 1) is the vector containing a constant and the (lagged) switching variables.

"exponential":

M=2M=2, α1,t=1α2,t\alpha_{1,t}=1-\alpha_{2,t}, and α2,t=1exp{γ(yitjc)}\alpha_{2,t}=1-\exp\lbrace -\gamma(y_{it-j}-c) \rbrace, where yitjy_{it-j} is the lag jj observation of the iith variable, cc is a location parameter, and γ>0\gamma > 0 is a scale parameter.

"threshold":

αm,t=1\alpha_{m,t} = 1 if rm1<yitjrmr_{m-1}<y_{it-j}\leq r_{m} and 00 otherwise, where r0<r1<<rM1<rM-\infty\equiv r_0<r_1<\cdots <r_{M-1}<r_M\equiv\infty are thresholds yitjy_{it-j} is the lag jj observation of the iith variable.

"exogenous":

Exogenous nonrandom transition weights, specify the weight series in weightfun_pars.

See the vignette for more details about the weight functions.

weightfun_pars
If weight_function == "relative_dens":

Not used.

If weight_function %in% c("logistic", "exponential", "threshold"):

a numeric vector with the switching variable i{1,...,d}i\in\lbrace 1,...,d \rbrace in the first and the lag j{1,...,p}j\in\lbrace 1,...,p \rbrace in the second element.

If weight_function == "mlogit":

a list of two elements:

The first element $vars:

a numeric vector containing the variables that should used as switching variables in the weight function in an increasing order, i.e., a vector with unique elements in {1,...,d}\lbrace 1,...,d \rbrace.

The second element $lags:

an integer in {1,...,p}\lbrace 1,...,p \rbrace specifying the number of lags to be used in the weight function.

If weight_function == "exogenous":

a size (nrow(data) - p x M) matrix containing the exogenous transition weights as [t, m] for time tt and regime mm. Each row needs to sum to one and only weakly positive values are allowed.

cond_dist

specifies the conditional distribution of the model as "Gaussian", "Student", or "ind_Student", where the latest is the Student's tt distribution with independent components.

identification

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".

B_constraints

a (d×d)(d \times d) matrix with its entries imposing constraints on the impact matrix BtB_t: NA indicating that the element is unconstrained, a positive value indicating strict positive sign constraint, a negative value indicating strict negative sign constraint, and zero indicating that the element is constrained to zero. Currently only available for models with identification="heteroskedasticity" or "non-Gaussianity" due to the (in)availability of appropriate parametrizations that allow such constraints to be imposed.

other_constraints

A list containing internally used additional type of constraints (see the options below).

$fixed_lambdas (only if identification="heteroskedasticity"):

a length d(M1)d(M-1) numeric vector (λ\lambda2_{2},...,,..., λ\lambdaM)_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters λmi\lambda_{mi} should be constrained to.

$B1_constraints (only if identification="non-Gaussianity"):

set to the string "fixed_sign_and_order" to impose the constraints that the elements of the first impact matrix B1B_1 are strictly positive and that they are in a decreasing order.

all_boldA

3D array containing the ((dp)x(dp))((dp)x(dp)) "bold A" (companion form) matrices of each regime, obtained from form_boldA. Will be computed if not given.

all_Omegas

A 3D array containing the covariance matrix parameters obtain from pick_Omegas...

If cond_dist %in% c("Gaussian", "Student"):

all covariance matrices Ωm\Omega_{m} in [, , m].

If cond_dist=="ind_Student":

all impact matrices BmB_m of the regimes in [, , m].

weightpars

numerical vector containing the transition weight parameters, obtained from pick_weightpars.

distpars

A numeric vector containing the distribution parameters...

If cond_dist=="Gaussian":

Not used, i.e., a numeric vector of length zero.

If cond_dist=="Student":

The degrees of freedom parameter, i.e., a numeric vector of length one.

transition_weights

(optional; only for models with cond_dist="ind_Student" or identification="non-Gaussianity") A T×MT \times M matrix containing the transition weights. If cond_dist="ind_Student" checks that the impact matrix m=1Mαm,t1/2Bm\sum_{m=1}^M\alpha_{m,t}^{1/2}B_m is invertible for all t=1,...,Tt=1,...,T.

stab_tol

numerical tolerance for stability of condition of the regimes: if the "bold A" matrix of any regime has eigenvalues larger that 1 - stat_tol the parameter is considered to be outside the parameter space. Note that if tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error.

posdef_tol

numerical tolerance for positive definiteness of the error term covariance matrices: if the error term covariance matrix of any regime has eigenvalues smaller than this, the parameter is considered to be outside the parameter space. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error.

distpar_tol

the parameter vector is considered to be outside the parameter space if the degrees of freedom parameters is not larger than 2 + distpar_tol (applies only if cond_dist="Student").

weightpar_tol

numerical tolerance for weight parameters being in the parameter space. Values closer to to the border of the parameter space than this are considered to be "outside" the parameter space.

Details

The parameter vector in the argument params should be unconstrained and reduced form.

Value

Returns TRUE if the given parameter values are in the parameter space and FALSE otherwise. This function does NOT consider identification conditions!

References

@keywords internal


[Package sstvars version 1.0.1 Index]