partialAR (version 1.0.12)

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

# NOT RUN {
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)
# }