random_covmat generates random VAR model \((dxd)\) error term covariance matrix \(\Omega\)
from (scaled) Wishart distribution that is fairly close to the given matrix.
smart_covmat(d, M, Omega, W_and_lambdas, accuracy, structural_pars = NULL)Returns a \((d(d+1)/2x1)\) vector containing vech-vectorized covariance matrix \(\Omega\).
Returns a length \(d^2 - n_zeros - d*(M - 1)\) vector of the form
\((Wvec(W),\lambda_2,...,\lambda_M)\) where \(\lambda_m=(\lambda_{m1},...,\lambda_{md})\)
contains the eigenvalue parameters of the \(m\)th regime \((m>1)\) and \(n_zeros\) is the number of zero constraints
in \(W\). If lambdas are C_lambda constrained, replace \(d*(M - 1)\) in the length with \(r\) and
\(\lambda_2,...,\lambda_M)\) with \(\gamma\). If fixed_lambdas are used, the \(\lambda_{mi}\) parameters
are not included. The operator \(Wvec()\) vectorizes a matrix and removes zeros.
the number of time series in the system.
a positive integer specifying the number of mixture components.
a size (2x1) integer vector specifying the number of GMVAR type components M1
in the first element and StMVAR type components M2 in the second element. The total number of mixture components
is M=M1+M2.
a symmetric positive definite \((dxd)\) covariance matrix specifying expected value of the matrix to be generated.
the mean of the normal distribution the new parameters are generated from.
a size \((d^2 - n_zeros + d*(M - 1))\) vector \((Wvec(W),\lambda_{2},...,\lambda{M})\), where \(n_zeros\) is the number of zero constraints in \(W\) and \(\lambda_m=(\lambda_{m1},...,\lambda_{md})\).
a size \((d^2 - n_zeros + r)\) vector \((Wvec(W),\gamma)\), where \(C_{\lambda}\gamma =(\lambda_2,....,\lambda_M)\), \(\gamma\) is of the size \((r x 1)\), and \(C_{\lambda}\) of the size \((d*(M - 1) x r\)).
a positive real number adjusting how close to the given covariance matrix the returned individual should be.
For reduced form models standard deviation of each diagonal element is for reduced form models
\(\omega_{i,i}/\)accuracy when accuracy > d/2
and sqrt(2/d)*\(\omega_{i,i}\) when accuracy <= d/2.
Wishart distribution is used for reduced form models, but for more details read the source code.
For structural models, the parameters are generated from normal distribution with mean given
by the argument W_and_lambdas and the standard deviation is sqrt(abs(W_and_lambdas)/(d + accuracy)).
If NULL a reduced form model is considered. Reduced models can be used directly as recursively
identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing
at least the first one of the following elements:
W - a \((dxd)\) matrix with its entries imposing constraints on \(W\): 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.
C_lambda - a \((d(M-1) x r)\) constraint matrix that satisfies (\(\lambda\)\(_{2}\)\(,...,\)
\(\lambda\)\(_{M}) =\) \(C_{\lambda} \gamma\) where \(\gamma\) is the new \((r x 1)\)
parameter subject to which the model is estimated (similarly to AR parameter constraints). The entries of C_lambda
must be either positive or zero. Ignore (or set to NULL) if the eigenvalues \(\lambda_{mi}\)
should not be constrained.
fixed_lambdas - 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. This constraint is alternative C_lambda.
Ignore (or set to NULL) if the eigenvalues \(\lambda_{mi}\) should not be constrained.
See Virolainen (forthcoming) for the conditions required to identify the shocks and for the B-matrix as well (it is \(W\) times a time-varying diagonal matrix with positive diagonal entries).