# get_regime_autocovs_int: Calculate regimewise autocovariance matrices In saviviro/gmvarkit: Estimate Gaussian or Student's t Mixture Vector Autoregressive Model

## Description

`get_regime_autocovs_int` calculates the regimewise autocovariance matrices Γ_{m}(j) j=0,1,...,p for the given GSMVAR model.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```get_regime_autocovs_int( p, M, d, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), constraints = NULL, same_means = NULL, 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 unconstrained models: Should be size ((M(pd^2+d+d(d+1)/2+2)-M1-1)x1) and have the form θ = (υ_{1}, ...,υ_{M}, α_{1},...,α_{M-1},ν), where υ_{m} = (φ_{m,0},φ_{m},σ_{m}) φ_{m} = (vec(A_{m,1}),...,vec(A_{m,p}) and σ_{m} = vech(Ω_{m}), m=1,...,M, ν=(ν_{M1+1},...,ν_{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 θ = (φ_{1,0},...,φ_{M,0},ψ, σ_{1},...,σ_{M},α_{1},...,α_{M-1},ν), where ψ (qx1) satisfies (φ_{1},..., φ_{M}) = C ψ where C is a (Mpd^2xq) constraint matrix. For same_means models: Should have the form θ = (μ,ψ, σ_{1},...,σ_{M},α_{1},...,α_{M-1},ν), where μ= (μ_{1},...,μ_{g}) where μ_{i} is the mean parameter for group i and g is the number of groups. If AR constraints are employed, ψ is as for constrained models, and if AR constraints are not employed, ψ = (φ_{1},...,φ_{M}). For structural models: Should have the form θ = (φ_{1,0},...,φ_{M,0},φ_{1},...,φ_{M}, vec(W),λ_{2},...,λ_{M},α_{1},...,α_{M-1},ν), where λ_{m}=(λ_{m1},...,λ_{md}) contains the eigenvalues of the mth mixture component. If AR parameters are constrained: Replace φ_{1},..., φ_{M} with ψ (qx1) that satisfies (φ_{1},..., φ_{M}) = C ψ, as above. If same_means: Replace (φ_{1,0},...,φ_{M,0}) with (μ_{1},...,μ_{g}), as above. If W is constrained:Remove the zeros from vec(W) and make sure the other entries satisfy the sign constraints. If λ_{mi} are constrained:Replace λ_{2},...,λ_{M} with γ (rx1) that satisfies (λ_{2},..., λ_{M}) = C_{λ} γ where C_{λ} is a (d(M-1) x r) constraint matrix. Above, φ_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth mixture component, Ω_{m} denotes the error term covariance matrix of the m:th mixture component, and α_{m} is the mixing weight parameter. The W and λ_{mi} are structural parameters replacing the error term covariance matrices (see Virolainen, 2020). If M=1, α_{m} and λ_{mi} are dropped. If `parametrization=="mean"`, just replace each φ_{m,0} with regimewise mean μ_{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 ν 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 ν should be strictly larger than two. The notation is similar to the cited literature. `model` 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. `constraints` a size (Mpd^2 x q) constraint matrix C specifying general linear constraints to the autoregressive parameters. We consider constraints of form (φ_{1},...,φ_{M}) = C ψ, where φ_{m} = (vec(A_{m,1}),...,vec(A_{m,p}) (pd^2 x 1), m=1,...,M, contains the coefficient matrices and ψ (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. `same_means` 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"`. `structural_pars` 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 (λ_{2},..., λ_{M}) = C_{λ} γ where γ 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 λ_{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).

## Value

Returns an (d x d x p+1 x M) array containing the first p regimewise autocovariance matrices. The subset `[, , j, m]` contains the j-1:th lag autocovariance matrix of the m:th regime.

## References

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

• Lütkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.

• McElroy T. 2017. Computation of vector ARMA autocovariances. Statistics and Probability Letters, 124, 92-96.

• Virolainen S. 2020. Structural Gaussian mixture vector autoregressive model. Unpublished working paper, available as arXiv:2007.04713.

• Virolainen S. 2021. Gaussian and Student's t mixture vector autoregressive model. Unpublished working paper, available as arXiv:2109.13648.

saviviro/gmvarkit documentation built on Oct. 25, 2021, 2:14 a.m.