pick_Am: Pick coefficient matrices

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

Pick coefficient matrices

Description

pick_Am picks the coefficient matrices A_{m,i} (i=1,..,p) from the given parameter vector so that they are arranged in a 3D array with the third dimension indicating each lag.

Usage

pick_Am(p, M, d, params, m, 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	number of time series in the system, i.e. the dimension.
params	a real valued vector specifying the parameter values. For unconstrained 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 constrained models: 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. For same_means models: 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}). For models with weight_constraints: Drop \alpha_1,...,\alpha_{M-1} from the parameter vector. For structural models: 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 mth mixture component. If AR parameters are constrained: Replace \phi_{1},..., \phi_{M} with \psi (qx1) that satisfies (\phi_{1},..., \phi_{M}) = C \psi, as above. If same_means: Replace (\phi_{1,0},...,\phi_{M,0}) with (\mu_{1},...,\mu_{g}), as above. If W is constrained: Remove the zeros from vec(W) and make sure the other entries satisfy the sign constraints. If \lambda_{mi} are constrained via 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. If \lambda_{mi} are constrained via fixed_lambdas: Drop \lambda_{2},..., \lambda_{M} from the parameter vector. 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.
m	which component?
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

Does not support constrained parameter vectors.

Value

Returns a 3D array containing the coefficient matrices of the given component. A coefficient matrix A_{m,i} can be obtained by choosing [, , i].

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.