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,
  same_means = NULL,
  structural_pars = 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 the 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 the 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}) =$ $C \psi$ where $C$ is $(Mpd^2xq)$ constraint matrix.

For same_means models:

Should have the form $\theta$$ = ($$\mu$,$\psi$, $\sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1})$, where

$\mu$$= (\mu_{1},...,\mu_{g})$ where $\mu_{i}$ is the mean parameter for group $i$ and $g$ is the number of groups.
If AR constraints are employed, $\psi$ is as for constrained models, and if AR constraints are not employed, $\psi$$ = $ ($\phi$$_{1}$$,...,$$\phi$$_{M})$.

For structural GMVAR model:

Should have the form $\theta$$ = (\phi_{1,0},...,\phi_{M,0},$$\phi$$_{1},...,$$\phi$$_{M}, vec(W),$$\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 $\phi$$_{1}$$,...,$ $\phi$$_{M}$ with $\psi$ $(qx1)$ that satisfies ($\phi$$_{1}$$,...,$ $\phi$$_{M}) =$ $C \psi$, as above.
If same_means:: Replace $(\phi_{1,0},...,\phi_{M,0})$ 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 $(d(M-1) x r)$ 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. 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 $\phi_{m,0}$ 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

"intercept" or "mean" determining whether the model is parametrized with intercept parameters $\phi_{m,0}$ or regime means $\mu_{m}$, 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 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 ($\lambda$$_{2}$$,...,$ $\lambda$$_{M}) =$ $C_{\lambda} \gamma$ where $\gamma$ 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 $\lambda_{mi}$ 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 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.
L<U+00FC>tkepohl 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.