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

## Description

Kalman gain matrix of the partially autoregressive model

## Usage

 `1` ```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

## Details

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

 ```1 2 3``` ``` [ 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

 ```1 2 3``` ``` [ 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

 ```1 2 3``` ``` [ 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.

## Value

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

## Author(s)

Matthew Clegg [email protected]

## 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

`fit.par`
 ```1 2 3``` ```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) ```