pick_weightpars: Pick transition weight parameters

Description

pick_weightpars picks the transition weight parameters from the given parameter vector.

Usage

pick_weightpars(
  p,
  M,
  d,
  params,
  weight_function = c("relative_dens", "logistic", "mlogit", "exponential", "threshold",
    "exogenous"),
  weightfun_pars = NULL,
  cond_dist = c("Gaussian", "Student", "ind_Student", "ind_skewed_t")
)

Value

If weight_function = "relative_dens":: Returns a length $M$ vector containing the transition weight parameters $\alpha_{m}, m=1,...,M$, including the non-parametrized $\alpha_{M}$.

weight_function="logistic":

Returns a length two vector $(c,\gamma)$, where $c\in\mathbb{R}$ is the location parameter and $\gamma >0$ is the scale parameter.

If weight_function = "mlogit":

Returns a length $(M-1)k$ vector $(\gamma_1,...,\gamma_M)$, where $\gamma_m$ $(k\times 1)$, $m=1,...,M-1$ ($\gamma_M=0$) contains the mlogit-regression coefficients of the $m$th regime. Specifically, for switching variables with indices in $J\subset\lbrace 1,...,d\rbrace$, 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 $\tilde{z}_{i} =(y_{j,t-1},...,y_{j,t-\tilde{p}})$, $i\in I$. So $k=1+|I|\tilde{p}$ where $|I|$ denotes the number of elements in $I$.

weight_function="exponential":

Returns a length two vector $(c,\gamma)$, where $c\in\mathbb{R}$ is the location parameter and $\gamma >0$ is the scale parameter.

weight_function="threshold":

Returns a length $M-1$ vector $(r_1,...,r_{M-1})$, where $r_1,...,r_{M-1}$ are the threshold values.

weight_function="exogenous":

Returns numeric(0).

Arguments

p

the autoregressive order of the model

M

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 $\theta = (\phi_{1,0},...,\phi_{M,0},\varphi_1,...,\varphi_M,\sigma,\alpha,\nu)$, 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 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 $\mu_{m}$.

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.

weightfun_pars

If weight_function == "relative_dens":: Not used.

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

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

The second element $lags:

an integer in $\lbrace 1,...,p \rbrace$ 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 $t$ and regime $m$. 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", "ind_Student", or "ind_skewed_t", where "ind_Student" the Student's $t$ distribution with independent components, and "ind_skewed_t" is the skewed $t$ distribution with independent components (see Hansen, 1994).

Warning

No argument checks!