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

The steady-state covariance matrix 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 \(\mathbf{C} = \boldsymbol{\Sigma}\).

Usage

LinSDECov(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(), LinSDEMean(), SimBetaN(), SimPhiN(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), TestPhi(), TestStability(), TestStationarity()

Author

Ivan Jacob Agaloos Pesigan

Examples

phi <- matrix(
  data = c(
    -0.10,
    0.05,
    0.05,
    -0.10
  ),
  nrow = 2
)
sigma <- matrix(
  data = c(
    2.79,
    0.06,
    0.06,
    3.27
  ),
  nrow = 2
)
LinSDECov(phi = phi, sigma = sigma)
#>      [,1] [,2]
#> [1,] 19.2 10.5
#> [2,] 10.5 21.6