sort_components: Sort components in parameter vector according to mixing...

In saviviro/gmvarkit: Estimate Gaussian and Student's t Mixture Vector Autoregressive Models

Sort components in parameter vector according to mixing weights into a decreasing order

Description

sort_components sorts mixture components in the parameter vector according to mixing weights into a decreasing order.

Usage

sort_components(
  p,
  M,
  d,
  params,
  model = c("GMVAR", "StMVAR", "G-StMVAR"),
  structural_pars = NULL
)

Arguments

p	a positive integer specifying the autoregressive order of the model.
M	For GMVAR and StMVAR models: a positive integer specifying the number of mixture components. For G-StMVAR models: 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.
d	the number of time series in the system.
params	a real valued vector specifying the parameter values. For reduced form models: 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. For structural model: 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 mth mixture component. Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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.
structural_pars	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).

Details

Constrained parameter vectors are not supported (expect for constraints in W but including constraining some mean parameters to be the same among different regimes)! For structural models, sorting the regimes in a decreasing order requires re-parametrizing the decomposition of the covariance matrices if the first regime changes. As a result, the sorted parameter vector will differ from the given one not only by the ordering of the elements but also by some of the parameter values.

Value

Returns sorted parameter vector of the form described for the argument params, with the mixture components sorted so that \alpha_{1}>...>\alpha_{M} for GMVAR and StMVAR models, and \alpha_{1}>...>\alpha_{M1} and \alpha_{M1+1}>...>\alpha_{M} for G-StMVAR models.

Warning

No argument checks!

References

• Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.

• Virolainen S. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.

• 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.

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saviviro/gmvarkit documentation built on March 8, 2024, 4:15 a.m.