gmvarkit (version 1.1.1)

cond_moments: Compute conditional moments of a GMVAR model

Description

loglikelihood compute conditional regimewise means, conditional means, and conditional covariance matrices of a GMVAR model.

Usage

cond_moments(data, p, M, params, parametrization = c("intercept",
  "mean"), constraints = NULL, to_return = c("regime_cmeans",
  "total_cmeans", "total_ccovs"))

Arguments

data

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

p

a positive integer specifying the autoregressive order of the model.

M

a positive integer specifying the number of mixture components.

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).

parametrization

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

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.

to_return

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

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=="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).

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 Kalliovirta et al. (2016).

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.

See Also

Other moment functions: get_regime_autocovs, get_regime_means, uncond_moments

Examples

# NOT RUN {
data <- cbind(10*eurusd[,1], 100*eurusd[,2])
params222 <- c(-11.904, 154.684, 1.314, 0.145, 0.094, 1.292, -0.389,
 -0.070, -0.109, -0.281, 0.920, -0.025, 4.839, 11.633, 124.983, 1.248,
  0.077, -0.040, 1.266, -0.272, -0.074, 0.034, -0.313, 5.855, 3.570,
  9.838, 0.740)
cond_moments(data=data, p=2, M=2, params=params222, parametrization="mean",
  to_return="regime_cmeans")
cond_moments(data=data, p=2, M=2, params=params222, parametrization="mean",
  to_return="total_cmeans")
cond_moments(data=data, p=2, M=2, params=params222, parametrization="mean",
  to_return="total_ccovs")
# }