standard_errors numerically approximates standard errors for the given estimates of GMVAR model using square
roots of the diagonal of inverse of observed information matrix.
standard_errors(data, p, M, params, conditional = TRUE,
parametrization = c("intercept", "mean"), constraints = NULL, minval)a matrix or class 'ts' object with d>1 columns. Each column is taken to represent
a single time series. 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 real valued vector specifying the parameter values.
Should be size \(((M(pd^2+d+d(d+1)/2+1)-1)x1)\) and have form \(\theta\)\( = \)(\(\upsilon\)\(_{1}\), ...,\(\upsilon\)\(_{M}\), \(\alpha_{1},...,\alpha_{M-1}\)), where:
\(\upsilon\)\(_{m}\) \( = (\phi_{m,0},\)\(\phi\)\(_{m}\)\(,\sigma_{m})\)
\(\phi\)\(_{m}\)\( = (vec(A_{m,1}),...,vec(A_{m,p})\)
and \(\sigma_{m} = vech(\Omega_{m})\), m=1,...,M.
Should be size \(((M(d+d(d+1)/2+1)+q-1)x1)\) and have form \(\theta\)\( = (\phi_{1,0},...,\phi_{M,0},\)\(\psi\) \(,\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1})\), where:
\(\psi\) \((qx1)\) satisfies (\(\phi\)\(_{1}\)\(,...,\) \(\phi\)\(_{M}) =\) \(C \psi\). Here \(C\) is \((Mpd^2xq)\) constraint matrix.
Above \(\phi_{m,0}\) is the intercept parameter, \(A_{m,i}\) denotes the \(i\):th coefficient matrix of the \(m\):th
mixture component, \(\Omega_{m}\) denotes the error term covariance matrix of the \(m\):th mixture component and
\(\alpha_{m}\) is the mixing weight parameter.
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.
The notations are in line with the cited article by Kalliovirta, Meitz and Saikkonen (2016).
a logical argument specifying whether the conditional or exact log-likelihood function
should be used. Default is TRUE.
"mean" or "intercept" determining whether the model is parametrized with regime means \(\mu_{m}\) or
intercept parameters \(\phi_{m,0}\), m=1,...,M. Default is "intercept".
a size \((Mpd^2 x q)\) constraint matrix \(C\) specifying general linear constraints
to the autoregressive parameters. We consider constraints of form
(\(\phi\)\(_{1}\)\(,...,\)\(\phi\)\(_{M}) = \)\(C \psi\),
where \(\phi\)\(_{m}\)\( = (vec(A_{m,1}),...,vec(A_{m,p}) (pd^2 x 1), m=1,...,M\)
contains the coefficient matrices and \(\psi\) \((q x 1)\) contains the constrained parameters.
For example, to restrict the AR-parameters to be the same for all regimes, set \(C\)=
[I:...:I]' \((Mpd^2 x pd^2)\) where I = diag(p*d^2).
Ignore (or set to NULL) if linear constraints should not be employed.
value that will be returned if the parameter vector does not lie in the parameter space (excluding the identification condition).
a vector containing the approximate standard errors of the estimates
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.