gmvarkit (version 1.1.1)

in_paramspace: Determine whether the parameter vector lies in the parameter space or not

Description

in_paramspace checks whether the given parameter vector belongs to the parameter space or not. Does NOT consider the identifiability condition!

Usage

in_paramspace(p, M, d, params, constraints = NULL)

Arguments

p

a positive integer specifying the autoregressive order of the model.

M

a positive integer specifying the number of mixture components.

d

number of time series in the system.

params

a real valued vector specifying the parameter values.

For regular models:

Should be size \(((M(pd^2+d+d(d+1)/2+1)-1)x1)\) and have form \(\theta\)\( = \)(\(\upsilon\)\(_{1}\), ...,\(\upsilon\)\(_{M}\), \(\alpha_{1},...,\alpha_{M-1}\)), 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.

For constrained models:

Should be size \(((M(d+d(d+1)/2+1)+q-1)x1)\) and have form \(\theta\)\( = (\phi_{1,0},...,\phi_{M,0},\)\(\psi\) \(,\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1})\), where:

• \(\psi\) \((qx1)\) satisfies (\(\phi\)\(_{1}\)\(,...,\) \(\phi\)\(_{M}) =\) \(C \psi\). Here \(C\) is \((Mpd^2xq)\) constraint matrix.

Above \(\phi_{m,0}\) is the intercept parameter, \(A_{m,i}\) denotes the \(i\):th coefficient matrix of the \(m\):th mixture component, \(\Omega_{m}\) denotes the error term covariance matrix of the \(m\):th mixture component and \(\alpha_{m}\) is the mixing weight parameter. 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. The notations are in line with the cited article by Kalliovirta, Meitz and Saikkonen (2016).

constraints

a size \((Mpd^2 x q)\) constraint matrix \(C\) specifying general linear constraints to the autoregressive parameters. We consider constraints of form (\(\phi\)\(_{1}\)\(,...,\)\(\phi\)\(_{M}) = \)\(C \psi\), where \(\phi\)\(_{m}\)\( = (vec(A_{m,1}),...,vec(A_{m,p}) (pd^2 x 1), m=1,...,M\) contains the coefficient matrices and \(\psi\) \((q x 1)\) contains the constrained 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.

Value

Returns TRUE if the given parameter vector lies in the parameter space and FALSE otherwise.

References

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

Examples

# NOT RUN {
# GMVAR(1,1), d=2 model:
params112 <- c(1.07, 127.71, 0.99, 0.00, -0.01, 0.99, 4.05,
  2.22, 8.87)
in_paramspace(p=1, M=1, d=2, params=params112)

# GMVAR(2,2), d=2 model:
params222 <- c(1.39, -0.77, 1.31, 0.14, 0.09, 1.29, -0.39,
 -0.07, -0.11, -0.28, 0.92, -0.03, 4.84, 1.01, 5.93, 1.25,
  0.08, -0.04, 1.27, -0.27, -0.07, 0.03, -0.31, 5.85, 3.57,
  9.84, 0.74)
in_paramspace(p=2, M=2, d=2, params=params222)

# GMVAR(2,2), d=2 model with AR-parameters restricted to be
# the same for both regimes:
C_mat <- rbind(diag(2*2^2), diag(2*2^2))
params222c <- c(1.03, 2.36, 1.79, 3.00, 1.25, 0.06,0.04,
 1.34, -0.29, -0.08, -0.05, -0.36, 0.93, -0.15, 5.20,
 5.88, 3.56, 9.80, 0.37)
in_paramspace(p=2, M=2, d=2, params=params222c, constraints=C_mat)
# }