standard_errors numerically approximates standard errors for the given estimates of GMAR, StMAR, or GStMAR model.
standard_errors(
data,
p,
M,
params,
model = c("GMAR", "StMAR", "G-StMAR"),
restricted = FALSE,
constraints = NULL,
conditional = TRUE,
parametrization = c("intercept", "mean"),
custom_h = NULL,
minval
)Returns approximate standard errors of the parameter values in a numeric vector.
a numeric vector or class 'ts' object containing the data. NA values are not supported.
a positive integer specifying the autoregressive order of the model.
a positive integer specifying the number of mixture components.
a size (2x1) integer vector specifying the number of GMAR type components M1 in the
first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.
a real valued parameter vector specifying the model.
Size \((M(p+3)+M-M1-1x1)\) vector \(\theta\)\(=\)(\(\upsilon_{1}\)\(,...,\)\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1},\)\(\nu\)) where
\(\upsilon_{m}\)\(=(\phi_{m,0},\)\(\phi_{m}\)\(,\)\(\sigma_{m}^2)\)
\(\phi_{m}\)\(=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M\)
\(\nu\)\(=(\nu_{M1+1},...,\nu_{M})\)
\(M1\) is the number of GMAR type regimes.
In the GMAR model, \(M1=M\) and the parameter \(\nu\) dropped. In the StMAR model, \(M1=0\).
If the model imposes linear constraints on the autoregressive parameters:
Replace the vectors \(\phi_{m}\) with the vectors \(\psi_{m}\) that satisfy
\(\phi_{m}\)\(=\)\(C_{m}\psi_{m}\) (see the argument constraints).
Size \((3M+M-M1+p-1x1)\) vector \(\theta\)\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(,\) \(\sigma_{1}^2,...,\sigma_{M}^2,\)\(\alpha_{1},...,\alpha_{M-1},\)\(\nu\)), where \(\phi\)=\((\phi_{1},...,\phi_{p})\) contains the AR coefficients, which are common for all regimes.
If the model imposes linear constraints on the autoregressive parameters:
Replace the vector \(\phi\) with the vector \(\psi\) that satisfies
\(\phi\)\(=\)\(C\psi\) (see the argument constraints).
Symbol \(\phi\) denotes an AR coefficient, \(\sigma^2\) a variance, \(\alpha\) a mixing weight, and \(\nu\) a degrees of
freedom parameter. If parametrization=="mean", just replace each intercept term \(\phi_{m,0}\) with the regimewise mean
\(\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})\). In the G-StMAR model, the first M1 components are GMAR type
and the rest M2 components are StMAR type.
Note that in the case M=1, the mixing weight parameters \(\alpha\) are dropped, and in the case of StMAR or G-StMAR model,
the degrees of freedom parameters \(\nu\) have to be larger than \(2\).
is "GMAR", "StMAR", or "G-StMAR" model considered? In the G-StMAR model, the first M1 components
are GMAR type and the rest M2 components are StMAR type.
a logical argument stating whether the AR coefficients \(\phi_{m,1},...,\phi_{m,p}\) are restricted to be the same for all regimes.
specifies linear constraints imposed to each regime's autoregressive parameters separately.
a list of size \((pxq_{m})\) constraint matrices \(C_{m}\) of full column rank satisfying \(\phi_{m}\)\(=\)\(C_{m}\psi_{m}\) for all \(m=1,...,M\), where \(\phi_{m}\)\(=(\phi_{m,1},...,\phi_{m,p})\) and \(\psi_{m}\)\(=(\psi_{m,1},...,\psi_{m,q_{m}})\).
a size \((pxq)\) constraint matrix \(C\) of full column rank satisfying \(\phi\)\(=\)\(C\psi\), where \(\phi\)\(=(\phi_{1},...,\phi_{p})\) and \(\psi\)\(=\psi_{1},...,\psi_{q}\).
The symbol \(\phi\) denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order
is always p for all regimes.
Ignore or set to NULL if applying linear constraints is not desired.
a logical argument specifying whether the conditional or exact log-likelihood function should be used.
is the model parametrized with the "intercepts" \(\phi_{m,0}\) or "means" \(\mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})\)?
a numeric vector with the same length as params specifying the difference 'h' used in finite difference approximation
for each parameter separately. If NULL (default), then the difference used for differentiating overly large degrees of freedom
parameters is adjusted to avoid numerical problems, and the difference is 6e-6 for the other parameters.
this will be returned when the parameter vector is outside the parameter space and boundaries==TRUE.