Steady-State Covariance Matrix for the Observed Variables in the State Space Model

The steady-state covariance matrix for the observed variables in the state space model \(\mathrm{Cov} \left( \mathbf{y} \right)\) is given by $$ \mathrm{Cov} \left( \mathbf{y} \right) = \boldsymbol{\Lambda} \mathrm{Cov} \left( \boldsymbol{\eta} \right) \boldsymbol{\Lambda}^{\prime} + \boldsymbol{\Theta} $$ where \(\boldsymbol{\Lambda}\) is the matrix of factor loadings, \(\boldsymbol{\Theta}\) is the covariance matrix of the measurement error, and \(\mathrm{Cov} \left( \boldsymbol{\eta} \right)\) is the steady-state covariance matrix for the latent variables.

Usage

SSMCovY(lambda, theta, cov_eta)

Arguments

lambda

Numeric matrix. Factor loading matrix linking the latent variables to the observed variables (\(\boldsymbol{\Lambda}\)).

theta

Numeric matrix. The covariance matrix of the measurement error (\(\boldsymbol{\Theta}\)).

cov_eta

Numeric matrix. The steady-state covariance matrix for the latent variables in the state space model

See also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimCovDiagN(), SimCovN(), SimIotaN(), SimNuN(), SimPhiN(), SimPhiN2(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Author

Ivan Jacob Agaloos Pesigan

Examples

beta <- matrix(
  data = c(
    0.7, 0.5, -0.1,
    0.0, 0.6, 0.4,
    0.0, 0.0, 0.5
  ),
  nrow = 3
)
psi <- 0.1 * diag(3)
lambda <- diag(3)
theta <- diag(3)
cov_eta <- SSMCovEta(
  beta = beta,
  psi = psi
)
SSMCovY(
  lambda = lambda,
  theta = theta,
  cov_eta = cov_eta
)
#>            [,1]      [,2]       [,3]
#> [1,] 1.19607843 0.1183232 0.02985385
#> [2,] 0.11832319 1.3437711 0.13818551
#> [3,] 0.02985385 0.1381855 1.26638284