cond_moments {gmvarkit}

Compute conditional moments of a GMVAR, StMVAR, or G-StMVAR model

Description

cond_moments compute conditional regimewise means, conditional means, and conditional covariance matrices of a GMVAR, StMVAR, or G-StMVAR model.

Usage

cond_moments(
  data,
  p,
  M,
  params,
  model = c("GMVAR", "StMVAR", "G-StMVAR"),
  parametrization = c("intercept", "mean"),
  constraints = NULL,
  same_means = NULL,
  weight_constraints = NULL,
  structural_pars = NULL,
  to_return = c("regime_cmeans", "regime_ccovs", "total_cmeans", "total_ccovs",
    "arch_scalars"),
  minval = NA,
  stat_tol = 0.001,
  posdef_tol = 1e-08,
  df_tol = 1e-08
)

Arguments

data

a matrix or class 'ts' object with d>1 columns. Each column is taken to represent a univariate time series. NA values are not supported.

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.

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).

to_return

should the regimewise conditional means, total conditional means, or total conditional covariance matrices be returned?

minval

the value that will be returned if the parameter vector does not lie in the parameter space (excluding the identification condition).

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.

df_tol

the parameter vector is considered to be outside the parameter space if all degrees of freedom parameters are not larger than 2 + df_tol.

Details

The first p values are used as the initial values, and by conditional we mean conditioning on the past. Formulas for the conditional means and covariance matrices are given in equations (3) and (4) of KMS (2016).

Value

If to_return=="regime_cmeans":

an [T-p, d, M] array containing the regimewise conditional means (the first p values are used as the initial values).

If to_return=="regime_ccovs":

an [d, d, T-p, M] array containing the regimewise conditional covariance matrices (the first p values are used as the initial values). The index [ , , t, m] gives the time t conditional covariance matrix for the regime m.

If to_return=="total_cmeans":

a [T-p, d] matrix containing the conditional means of the process (the first p values are used as the initial values).

If to_return=="total_ccov":

an [d, d, T-p] array containing the conditional covariance matrices of the process (the first p values are used as the initial values).

If to_return=="arch_scalars":

a [T-p, M] matrix containing the regimewise arch scalars multiplying error term covariance matrix in the conditional covariance matrix of the regime. For GMVAR type regimes, these are all ones (the first p values are used as the initial values).

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.

See Also

Other moment functions: get_regime_autocovs(), get_regime_means(), uncond_moments()

Examples

# GMVAR(2, 2), d=2 model;
params22 <- c(0.36, 0.121, 0.223, 0.059, -0.151, 0.395, 0.406, -0.005,
 0.083, 0.299, 0.215, 0.002, 0.03, 0.484, 0.072, 0.218, 0.02, -0.119,
 0.722, 0.093, 0.032, 0.044, 0.191, 1.101, -0.004, 0.105, 0.58)
cond_moments(data=gdpdef, p=2, M=2, params=params22, to_return="regime_cmeans")
cond_moments(data=gdpdef, p=2, M=2, params=params22, to_return="total_cmeans")
cond_moments(data=gdpdef, p=2, M=2, params=params22, to_return="total_ccovs")