uncond_moments_int: Calculate the unconditional mean, variance, the first p autocovariances, and the first p autocorrelations of a GMVAR process

Description

uncond_moments_int calculates the unconditional mean, variance, the first p autocovariances, and the first p autocorrelations of the specified GMVAR process.

Usage

uncond_moments_int(
  p,
  M,
  d,
  params,
  parametrization = c("intercept", "mean"),
  constraints = NULL
)

Arguments

a positive integer specifying the autoregressive order of the model.

a positive integer specifying the number of mixture components.

the number of time series in the system.

params

a real valued vector specifying the parameter values.

For unconstrained models:

Should be size $((M(pd^2+d+d(d+1)/2+1)-1)x1)$ 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 notation is in line with the cited article by Kalliovirta, Meitz and Saikkonen (2016) introducing the GMVAR model.

parametrization

"mean" or "intercept" determining whether the model is parametrized with regime means $\mu_{m}$ or intercept parameters $\phi_{m,0}$, m=1,...,M.

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 a list with three components:

$uncond_mean: a length d vector containing the unconditional mean of the process.
$autocovs: an $(d x d x p+1)$ array containing the lag 0,1,...,p autocovariances of the process. The subset [, , j] contains the lag j-1 autocovariance matrix (lag zero for the variance).
$autocors: the autocovariance matrices scaled to autocorrelation matrices.

Details

The unconditional moments are based on the stationary distribution of the process.

References

Kalliovirta L., Meitz M. and Saikkonen P. 2016. Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498.
Lutkepohl H. 2005. New Introduction to Multiple Time Series Analysis, Springer.