gmvarkit (version 1.5.0)

in_paramspace_int: Determine whether the parameter vector lies in the parameter space

Description

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

Usage

in_paramspace_int(
  p,
  M,
  d,
  params,
  all_boldA,
  alphas,
  all_Omega,
  W_constraints = NULL,
  stat_tol = 0.001,
  posdef_tol = 1e-08
)

Arguments

p

a positive integer specifying the autoregressive order of the model.

M

a positive integer specifying the number of mixture components.

d

the number of time series in the system.

params

a real valued vector specifying the parameter values.

For unconstrained models:

Should be size $((M(p{d}^2+d+d(d+1)/2+1)-1)x1)$ and have the form $\theta$$=$($\upsilon$${}_1$, ...,$\upsilon$${}_M$, ${\alpha }_1,...,{\alpha }_{M-1}$), where

• $\upsilon$${}_m$ $=({\varphi }_{m,0},$$\varphi$${}_m$$,{\sigma }_m)$

• $\varphi$${}_m$$=(vec(A_{m,1}),...,vec(A_{m,p})$

• and ${\sigma }_m=vech({\mathrm{\Omega }}_m)$, m=1,...,M.

For constrained models:

Should be size $((M(d+d(d+1)/2+1)+q-1)x1)$ and have the form $\theta$$=({\varphi }_{1,0},...,{\varphi }_{M,0},$$\psi$, ${\sigma }_1,...,{\sigma }_M,{\alpha }_1,...,{\alpha }_{M-1})$, where

• $\psi$ $(qx1)$ satisfies ($\varphi$${}_1$$,...,$ $\varphi$${}_M)=$ $C\psi$ where $C$ is $(Mp{d}^{2}xq)$ constraint matrix.

For same_means models:

Should have the form $\theta$$=($$\mu$,$\psi$, ${\sigma }_1,...,{\sigma }_M,{\alpha }_1,...,{\alpha }_{M-1})$, 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$$=$ ($\varphi$${}_1$$,...,$$\varphi$${}_M)$.

For structural GMVAR model:

Should have the form $\theta$$=({\varphi }_{1,0},...,{\varphi }_{M,0},$$\varphi$${}_1,...,$$\varphi$${}_M,vec(W),$$\lambda$${}_2,...,$$\lambda$${}_M,{\alpha }_1,...,{\alpha }_{M-1})$, where

• $\lambda$${}_m=({\lambda }_{m1},...,{\lambda }_{md})$ contains the eigenvalues of the $m$th mixture component.

If AR parameters are constrained:

Replace $\varphi$${}_1$$,...,$ $\varphi$${}_M$ with $\psi$ $(qx1)$ that satisfies ($\varphi$${}_1$$,...,$ $\varphi$${}_M)=$ $C\psi$, as above.

If same_means:

Replace $({\varphi }_{1,0},...,{\varphi }_{M,0})$ 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:

Replace $\lambda$${}_2,...,$$\lambda$${}_M$ with $\gamma$ $(rx1)$ that satisfies ($\lambda$${}_2$$,...,$ $\lambda$${}_M)=$ $C_{\lambda }\gamma$ where $C_{\lambda }$ is a $(d(M-1)xr)$ constraint matrix.

Above, ${\varphi }_{m,0}$ is the intercept parameter, $A_{m,i}$ denotes the $i$th coefficient matrix of the $m$th mixture component, ${\mathrm{\Omega }}_m$ 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, 2020). If $M=1$, ${\alpha }_m$ and ${\lambda }_{mi}$ are dropped. If parametrization=="mean", just replace each ${\varphi }_{m,0}$ 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. The notation is in line with the cited article by Kalliovirta, Meitz and Saikkonen (2016) introducing the GMVAR model.

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 ${\mathrm{\Omega }}_m$, 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.

Value

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

Details

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

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.

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