random_covmat generates random VAR model \((dxd)\) error term covariance matrix \(\Omega\)
from (scaled) Wishart distribution for reduced form models and the parameters \(W\),\(\lambda_1,...,\lambda_M\)
for structural models (from normal distributions).
random_covmat(d, M, omega_scale, W_scale, lambda_scale, structural_pars = NULL)the number of time series in the system.
a positive integer specifying the number of mixture components.
a size \((dx1)\) strictly positive vector specifying the scale and variability of the
random covariance matrices in random mutations. The covariance matrices are drawn from (scaled) Wishart
distribution. Expected values of the random covariance matrices are diag(omega_scale). Standard
deviations of the diagonal elements are sqrt(2/d)*omega_scale[i]
and for non-diagonal elements they are sqrt(1/d*omega_scale[i]*omega_scale[j]).
Note that for d>4 this scale may need to be chosen carefully. Default in GAfit is
var(stats::ar(data[,i], order.max=10)$resid, na.rm=TRUE), i=1,...,d. This argument is ignored if
structural model is considered.
a size \((dx1)\) strictly positive vector partly specifying the scale and variability of the
random covariance matrices in random mutations. The elements of the matrix \(W\) are drawn independently
from such normal distributions that the expectation of the main diagonal elements of the first
regime's error term covariance matrix \(\Omega_1 = WW'\) is W_scale. The distribution of \(\Omega_1\)
will be in some sense like a Wishart distribution but with the columns (elements) of \(W\) obeying the given
constraints. The constraints are accounted for by setting the element to be always zero if it is subject to a zero
constraint and for sign constraints the absolute value or negative the absolute value are taken, and then the
variances of the elements of \(W\) are adjusted accordingly. This argument is ignored if reduced form model
is considered.
a length \(M - 1\) vector specifying the standard deviation of the mean zero normal
distribution from which the eigenvalue \(\lambda_{mi}\) parameters are drawn from in random mutations.
As the eigenvalues should always be positive, the absolute value is taken. The elements of lambda_scale
should be strictly positive real numbers with the \(m-1\)th element giving the degrees of freedom for the \(m\)th
regime. The expected value of the main diagonal elements \(ij\) of the \(m\)th \((m>1)\) error term covariance
matrix will be W_scale[i]*(d - n_i)^(-1)*sum(lambdas*ind_fun) where the \((d x 1)\) vector lambdas is
drawn from the absolute value of the t-distribution, n_i is the number of zero constraints in the \(i\)th
row of \(W\) and ind_fun is an indicator function that takes the value one iff the \(ij\)th element of
\(W\) is not constrained to zero. Basically, larger lambdas (or smaller degrees of freedom) imply larger variance.
If the lambda parameters are constrained with the \((d(M - 1) x r)\) constraint matrix \(C_lambda\), then provide a length \(r\) vector specifying the standard deviation of the (absolute value of the) mean zero normal distribution each of the \(\gamma\) parameters are drawn from (the \(\gamma\) is a \((r x 1)\) vector). The expected value of the main diagonal elements of the covariance matrices then depend on the constraints.
This argument is ignored if \(M==1\) or a reduced form model is considered. Default is rep(3, times=M-1)
if lambdas are not constrained and rep(3, times=r) if lambdas are constrained.
As with omega_scale and W_scale, this argument should be adjusted carefully if specified by hand. NOTE that if lambdas are constrained in some other way than restricting some of them to be identical, this parameter should be adjusted accordingly in order to the estimation succeed!
If NULL a reduced form model is considered. For structural model, should be a list containing
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.
See Virolainen (2020) 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).
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 constrained, replacce \(d*(M - 1)\) in the length with \(r\) and \(\lambda_2,...,\lambda_M)\) with \(\gamma\). The operator \(Wvec()\) vectorizes a matrix and removes zeros.