kalman.gain.par: Kalman gain matrix of the partially autoregressive model
In partialAR: Partial Autoregression

Description Usage Arguments Details Value Author(s) References See Also Examples

Kalman gain matrix of the partially autoregressive model

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

`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

The state space representation of the partially autoregressive model is given as

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    [ 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

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    [ 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

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    [ 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.

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

Matthew Clegg matthewcleggphd@gmail.com

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

fit.par

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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)