get_test_Omega computes the covariance matrix Omega used in the
quantile residuals tests described by Kalliovirta and Saikkonen 2010.
get_test_Omega(
data,
p,
M,
params,
model,
conditional,
parametrization,
constraints,
same_means,
weight_constraints,
structural_pars = NULL,
g,
dim_g,
ncores = 1,
stat_tol = 0.001,
posdef_tol = 1e-08,
df_tol = 1e-08
)Returns the covariance matrix Omega described by Kalliovirta and Saikkonen 2010.
a matrix or class 'ts' object with d>1 columns. Each column is taken to represent
a univariate 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 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 real valued vector specifying the parameter values.
Should be size \(((M(pd^2+d+d(d+1)/2+2)-M1-1)x1)\) and have the form \(\theta\)\( = \)(\(\upsilon\)\(_{1}\), ...,\(\upsilon\)\(_{M}\), \(\alpha_{1},...,\alpha_{M-1},\)\(\nu\)\()\), 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,
\(\nu\)\(=(\nu_{M1+1},...,\nu_{M})\)
\(M1\) is the number of GMVAR type regimes.
Should be size \(((M(d+d(d+1)/2+2)+q-M1-1)x1)\) and have the form \(\theta\)\( = (\phi_{1,0},...,\phi_{M,0},\)\(\psi\), \(\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\)\(\nu\)), where
\(\psi\) \((qx1)\) satisfies (\(\phi\)\(_{1}\)\(,...,\) \(\phi\)\(_{M}) =\) \(C \psi\) where \(C\) is a \((Mpd^2xq)\) constraint matrix.
Should have the form \(\theta\)\( = (\)\(\mu\),\(\psi\), \(\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\)\(\nu\)\()\), where
\(\mu\)\(= (\mu_{1},...,\mu_{g})\) where \(\mu_{i}\) is the mean parameter for group \(i\) and \(g\) is the number of groups.
If AR constraints are employed, \(\psi\) is as for constrained models, and if AR constraints are not employed, \(\psi\)\( = \) (\(\phi\)\(_{1}\)\(,...,\)\(\phi\)\(_{M})\).
Drop \(\alpha_1,...,\alpha_{M-1}\) from the parameter vector.
Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector should have the form \(\theta\)\( = (\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(_{1},...,\)\(\phi\)\(_{M}, vec(W),\)\(\lambda\)\(_{2},...,\)\(\lambda\)\(_{M},\alpha_{1},...,\alpha_{M-1},\)\(\nu\)\()\), where
\(\lambda\)\(_{m}=(\lambda_{m1},...,\lambda_{md})\) contains the eigenvalues of the \(m\)th mixture component.
Replace \(\phi\)\(_{1}\)\(,...,\) \(\phi\)\(_{M}\) with \(\psi\) \((qx1)\) that satisfies (\(\phi\)\(_{1}\)\(,...,\) \(\phi\)\(_{M}) =\) \(C \psi\), as above.
Replace \((\phi_{1,0},...,\phi_{M,0})\) with \((\mu_{1},...,\mu_{g})\), as above.
Remove the zeros from \(vec(W)\) and make sure the other entries satisfy the sign constraints.
C_lambda:Replace \(\lambda\)\(_{2},...,\) \(\lambda\)\(_{M}\) with \(\gamma\) \((rx1)\) that satisfies (\(\lambda\)\(_{2}\) \(,...,\) \(\lambda\)\(_{M}) =\) \(C_{\lambda} \gamma\) where \(C_{\lambda}\) is a \((d(M-1) x r)\) constraint matrix.
fixed_lambdas:Drop \(\lambda\)\(_{2},...,\) \(\lambda\)\(_{M}\) from the parameter vector.
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. The \(W\) and \(\lambda_{mi}\) are structural parameters replacing the
error term covariance matrices (see Virolainen, 2022). If \(M=1\), \(\alpha_{m}\) and \(\lambda_{mi}\) are dropped.
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.
In the GMVAR model, \(M1=M\) and \(\nu\) is dropped from the parameter vector. In the StMVAR model,
\(M1=0\). In the G-StMVAR model, the first M1 regimes are GMVAR type and the rest M2 regimes are
StMVAR type. In StMVAR and G-StMVAR models, the degrees of freedom parameters in \(\nu\) should
be strictly larger than two.
The notation is similar to the cited literature.
is "GMVAR", "StMVAR", or "G-StMVAR" model considered? In the G-StMVAR model, the first M1 components
are GMVAR type and the rest M2 components are StMVAR type.
a logical argument specifying whether the conditional or exact log-likelihood function should be used.
"intercept" or "mean" determining whether the model is parametrized with intercept
parameters \(\phi_{m,0}\) or regime means \(\mu_{m}\), m=1,...,M.
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 related 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.
Restrict the mean parameters of some regimes to be the same? 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
the same 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".
a numeric vector of length \(M-1\) specifying fixed parameter values for the mixing weight parameters \(\alpha_m, \ m=1,...,M-1\). Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.
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).
function g specifying the transformation.
output dimension of the transformation g.
the number of CPU cores to be used in numerical differentiation. Multiple cores are not supported on Windows, though.
numerical tolerance for stationarity of the AR parameters: if the "bold A" matrix of any regime
has eigenvalues larger that 1 - stat_tol the model is classified as non-stationary. Note that if the
tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error.
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 model is classified as not satisfying positive definiteness assumption. Note that if the tolerance is too small, numerical evaluation of the log-likelihood might fail and cause error.
the parameter vector is considered to be outside the parameter space if all degrees of
freedom parameters are not larger than 2 + df_tol.
Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Kalliovirta L. and Saikkonen P. 2010. Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247.
Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.
Virolainen S. 2025. A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics, 43, 1, 44-54.