pick_Omegas: Pick covariance matrices

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

Pick covariance matrices

Description

pick_Omegas picks the covariance matrices \Omega_{m} or impact matrices B_m from the given parameter vector so that they are arranged in a 3D array with the third dimension indicating each regime.

Usage

pick_Omegas(
  p,
  M,
  d,
  params,
  cond_dist = c("Gaussian", "Student", "ind_Student", "ind_skewed_t"),
  identification = c("reduced_form", "recursive", "heteroskedasticity",
    "non-Gaussianity")
)

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}.

Details

Constrained parameter vectors are not supported.

Value

Returns a 3D array containing...

If identification == "non-Gaussianity" or cond_dist %in% c("ind_Student", "ind_skewed_t"):

the impact matrices of the regimes, B_m in [, , m].

If otherwise:

the covariance matrices of the given model, \Omega_m in [, , m].

Warning

No argument checks!

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