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 \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 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": \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_M) (M \times 1), nu_m > 2.
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.
- AR_constraints:
Replace \varphi_1,...,\varphi_M with \psi as described in the argument AR_constraints.
- mean_constraints:
Replace \phi_{1,0},...,\phi_{M,0} with (\mu_{1},...,\mu_{g}) where
\mu_i, \ (d\times 1) is the mean parameter for group i and g is the number of groups.
- weight_constraints:
If linear constraints are imposed, replace \alpha with \xi as described in the
argument weigh_constraints. If weight functions parameters are imposed to be fixed values, simply drop \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.
- B_constraints (only for structural models identified by heteroskedasticity):
Replace vec(W) with \tilde{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, \phi_{m,0} 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,0} with regimewise mean \mu_{m}.
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 B_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 \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", or "ind_Student",
where the latest is the Student's t 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 \times d) matrix with its entries imposing constraints on the impact matrix B_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(M-1) numeric vector
(\lambda_{2},..., \lambda_{M}) with elements strictly larger
than zero specifying the fixed parameter values for the parameters \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 B_1 are strictly positive and that they
are in a decreasing order.
|
all_boldA |
3D array containing the ((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 \Omega_{m} in [, , m].
- If
cond_dist=="ind_Student": all impact matrices B_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 \times M matrix containing the transition weights. If cond_dist="ind_Student" checks that the impact matrix
\sum_{m=1}^M\alpha_{m,t}^{1/2}B_m is invertible for all t=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.
|