partialAR (version 1.0.3)

kalman.gain.par: Kalman gain matrix of the partially autoregressive model

Description

Kalman gain matrix of the partially autoregressive model

Usage

kalman.gain.par(rho, sigma_M, sigma_R)

Arguments

rho

The coefficient of mean reversion

sigma_M

The standard deviation of the innovations of the mean-reverting component

sigma_R

The standard deviation of the innovations of the random walk component

Value

• Returns a two-component vector (K_M, K_R) representing the Kalman gain matrix.

Details

The state space representation of the partially autoregressive model is given as [ M[t] ] [ rho 0 ] [ M[t-1] ] [ epsilon_M[t] ] [ ] = [ ] [ ] + [ ] [ R[t] ] [ 0 1 ] [ R[t-1] ] [ epsilon_R[t] ] where the innovations epsilon_M[t] and epsilon_R[t] have the covariance matrix [ epsilon_M[t] ] [ sigma_M^2 0 ] [ ] ~ [ ] [ epsilon_R[t] ] [ 0 sigma_R^2 ] The steady state Kalman gain matrix is given by the matrix [ K_M ] [ ] [ K_R ] where $$K_M = 2 sigma_M^2 / (sigma_R * ( sqrt((rho + 1)^2 sigma_R^2 + 4 sigma_M^2) + (rho + 1) sigma_R ) + 2 sigma_M^2)$$ and $K_R = 1 - K_M$.

References

Clegg, Matthew. Modeling Time Series with Both Permanent and Transient Components using the Partially Autoregressive Model. Available at SSRN: http://ssrn.com/abstract=2556957

See Also

fit.par

Examples

kalman.gain.par(0, 1, 0)  # -> c(1, 0)   (pure AR(1))
kalman.gain.par(0, 0, 1)  # -> c(0, 1)   (pure random walk)
kalman.gain.par(0.5, 1, 1)  # -> c(0.3333, 0.6667)