uncond_moments_int {gmvarkit}

Calculate the unconditional mean, variance, the first p autocovariances, and the first p autocorrelations of a GMVAR, StMVAR, or G-StMVAR process

Description

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

Usage

uncond_moments_int(
  p,
  M,
  d,
  params,
  model = c("GMVAR", "StMVAR", "G-StMVAR"),
  parametrization = c("intercept", "mean"),
  constraints = NULL,
  same_means = NULL,
  weight_constraints = NULL,
  structural_pars = NULL
)

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 \theta = (\upsilon_{1}, ...,\upsilon_{M}, \alpha_{1},...,\alpha_{M-1},\nu), 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,

• \nu=(\nu_{M1+1},...,\nu_{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 \theta = (\phi_{1,0},...,\phi_{M,0},\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), where

• \psi (qx1) satisfies (\phi_{1},..., \phi_{M}) = C \psi where C is a (Mpd^2xq) constraint matrix.

For same_means models:

Should have the form \theta = (\mu,\psi, \sigma_{1},...,\sigma_{M},\alpha_{1},...,\alpha_{M-1},\nu), 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 models with weight_constraints:

Drop \alpha_1,...,\alpha_{M-1} from the parameter vector.

For structural models:

Reduced form models can be directly used as recursively identified structural models. If the structural model is identified by conditional heteroskedasticity, the parameter vector 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},\nu), where

• \lambda_{m}=(\lambda_{m1},...,\lambda_{md}) contains the eigenvalues of the mth 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 via C_lambda:

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.

If \lambda_{mi} are constrained via fixed_lambdas:

Drop \lambda_{2},..., \lambda_{M} from the parameter vector.

Above, \phi_{m,0} is the intercept parameter, A_{m,i} denotes the ith coefficient matrix of the mth 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, 2022). 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.

In the GMVAR model, M1=M and \nu 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 \nu 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.

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".

weight_constraints

a numeric vector of length M-1 specifying fixed parameter values for the mixing weight parameters \alpha_m, \ m=1,...,M-1. Each element should be strictly between zero and one, and the sum of all the elements should be strictly less than one.

structural_pars

If NULL a reduced form model is considered. Reduced models can be used directly as recursively identified structural models. For a structural model identified by conditional heteroskedasticity, should be a list containing at least the first one of 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.

• fixed_lambdas - a length d(M-1) numeric vector (\lambda_{2},..., \lambda_{M}) with elements strictly larger than zero specifying the fixed parameter values for the parameters \lambda_{mi} should be constrained to. This constraint is alternative C_lambda. Ignore (or set to NULL) if the eigenvalues \lambda_{mi} should not be constrained.

See Virolainen (forthcoming) 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).

Details

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

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.

References

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

• Lütkepohl 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. (forthcoming). A statistically identified structural vector autoregression with endogenously switching volatility regime. Journal of Business & Economic Statistics.

• Virolainen S. 2022. Gaussian and Student's t mixture vector autoregressive model with application to the asymmetric effects of monetary policy shocks in the Euro area. Unpublished working paper, available as arXiv:2109.13648.