data |
a matrix or class 'ts' object with d>1 columns. Each column is taken to represent
a univariate time series. Missing values are not supported.
|
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.
|
parametrization |
"intercept" or "mean" determining whether the model is parametrized with intercept
parameters \phi_{m,0} or regime means \mu_{m}, m=1,...,M.
|
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".
|
AR_constraints |
a size (Mpd^2 x q) constraint matrix C specifying linear constraints
to the autoregressive parameters. The constraints are of the form
(\varphi_{1},...,\varphi_{M}) = C\psi, where \varphi_{m} = (vec(A_{m,1}),...,vec(A_{m,p})) \ (pd^2 x 1),\ m=1,...,M,
contains the coefficient matrices and \psi (q x 1) contains the related parameters.
For example, to restrict the AR-parameters to be the identical across the regimes, set C =
[I:...:I]' (Mpd^2 x pd^2) where I = diag(p*d^2).
|
mean_constraints |
Restrict the mean parameters of some regimes to be identical? Provide a list of numeric vectors
such that each numeric vector contains the regimes that should share the common mean parameters. For instance, if
M=3, the argument list(1, 2:3) restricts the mean parameters of the second and third regime to be
identical but the first regime has freely estimated (unconditional) mean. Ignore or set to NULL if mean parameters
should not be restricted to be the same among any regimes. This constraint is available only for mean parametrized models;
that is, when parametrization="mean".
|
weight_constraints |
a list of two elements, R in the first element and r in the second element,
specifying linear constraints on the transition weight parameters \alpha.
The constraints are of the form \alpha = R\xi + r, where R is a known (a\times l)
constraint matrix of full column rank (a is the dimension of \alpha), r is a known (a\times 1) constant,
and \xi is an unknown (l\times 1) parameter. Alternatively, set R=0 in order to constrain the
the weight parameter to the constant r (in this case, \alpha is dropped from the constrained parameter vector).
|
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.
|
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.
|
Throws an informative error if there is something wrong with the parameter vector.