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Steady-State Covariance Matrix for the Latent Variables in the Linear Stochastic Differential Equation Model

Source: R/simStateSpace-lin-sde-cov-eta.R
LinSDECovEta.Rd

The steady-state covariance matrix for the latent variables in the linear stochastic differential equation model \(\mathrm{Cov} \left( \boldsymbol{\eta} \right)\) is the solution to the Sylvester equation, i.e. $$ \mathbf{A} \mathbf{X} + \mathbf{X} \mathbf{B} + \mathbf{C} = \mathbf{0} , $$ where \(\mathbf{X}\) is unknown, \(\mathbf{A} = \boldsymbol{\Phi}\), \(\mathbf{B} = \boldsymbol{\Phi}^{\prime}\), and \(\mathbf{C} = \boldsymbol{\Sigma}\) where \(\boldsymbol{\Phi}\) is the drift matrix and \(\boldsymbol{\Sigma}\) is the covariance matrix of volatility or randomness in the process.

Usage

LinSDECovEta(phi, sigma)

Arguments

phi

Numeric matrix. The drift matrix which represents the rate of change of the solution in the absence of any random fluctuations (\(\boldsymbol{\Phi}\)).

sigma

Numeric matrix. The covariance matrix of volatility or randomness in the process (\(\boldsymbol{\Sigma}\)).

See also

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

Author

Ivan Jacob Agaloos Pesigan

Examples

phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0.0, 0.0, -0.693
  ),
  nrow = 3
)
sigma <- matrix(
  data = c(
    0.24455556, 0.02201587, -0.05004762,
    0.02201587, 0.07067800, 0.01539456,
    -0.05004762, 0.01539456, 0.07553061
  ),
  nrow = 3
)
LinSDECovEta(
  phi = phi,
  sigma = sigma
)
#>           [,1]      [,2]      [,3]
#> [1,] 0.3425148 0.3296023 0.0343817
#> [2,] 0.3296023 0.5664625 0.2545955
#> [3,] 0.0343817 0.2545955 0.2999909