pick_weightpars: Pick transition weight parameters

In sstvars: Toolkit for Reduced Form and Structural Smooth Transition Vector Autoregressive Models

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")
)

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},...,\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 mth 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 mth 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} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth regime, \Omega_{m} denotes the positive definite error term covariance matrix of the mth regime, and B_m is the invertible (d\times d) impact matrix of the mth regime. \nu_m is the degrees of freedom parameter of the mth regime. If parametrization=="mean", just replace each \phi_{m} 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 mth 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 ith 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 ith 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 ith 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).

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 mth 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).

Warning

No argument checks!

sstvars documentation built on April 11, 2025, 5:47 p.m.