# in_paramspace_int: Determine whether the parameter vector lies in the parameter... In saviviro/gmvarkit: Estimate Gaussian or Student's t Mixture Vector Autoregressive Model

## Description

`in_paramspace_int` checks whether the parameter vector lies in the parameter space.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```in_paramspace_int( p, M, d, params, model = c("GMVAR", "StMVAR", "G-StMVAR"), all_boldA, alphas, all_Omega, W_constraints = NULL, stat_tol = 0.001, posdef_tol = 1e-08, df_tol = 1e-08 ) ```

## 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. `all_boldA` 3D array containing the ((dp)x(dp)) "bold A" matrices related to each mixture component VAR-process, obtained from `form_boldA`. Will be computed if not given. `alphas` (Mx1) vector containing all mixing weight parameters, obtained from `pick_alphas`. `all_Omega` 3D array containing all covariance matrices Ω_{m}, obtained from `pick_Omegas`. `W_constraints` set `NULL` for reduced form models. For structural models, this should be the constraint matrix W from the list of structural parameters. `stat_tol` 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. `posdef_tol` 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. `df_tol` the parameter vector is considered to be outside the parameter space if all degrees of freedom parameters are not larger than `2 + df_tol`.

## Details

The parameter vector in the argument `params` should be unconstrained and it is used for structural models only.

## Value

Returns `TRUE` if the given parameter values are in the parameter space and `FALSE` otherwise. This function does NOT consider the identifiability condition!

## References

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

• 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.

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