add.ar | R Documentation |

Add an AR(p) state component to the state specification.

```
AddAr(state.specification,
y,
lags = 1,
sigma.prior,
initial.state.prior = NULL,
sdy)
```

`state.specification` |
A list of state components. If omitted, an empty list is assumed. |

`y` |
A numeric vector. The time series to be modeled. |

`lags` |
The number of lags ("p") in the AR(p) process. |

`sigma.prior` |
An object created by SdPrior. The prior for the standard deviation of the process increments. |

`initial.state.prior` |
An object of class MvnPrior describing the
values of the state at time 0. This argument can be |

`sdy` |
The sample standard deviation of the time series to be
modeled. Used to scale the prior distribution. This can be omitted
if |

The model is

```
\alpha_{t} = \phi_1\alpha_{i, t-1} + \cdots + \phi_p
\alpha_{t-p} + \epsilon_{t-1} \qquad
\epsilon_t \sim \mathcal{N}(0, \sigma^2)
```

The state consists of the last `p`

lags of `alpha`

. The
state transition matrix has `phi`

in its first row, ones along
its first subdiagonal, and zeros elsewhere. The state variance matrix
has `sigma^2`

in its upper left corner and is zero elsewhere.
The observation matrix has 1 in its first element and is zero
otherwise.

Returns `state.specification`

with an AR(p) state component
added to the end.

Steven L. Scott steve.the.bayesian@gmail.com

Harvey (1990), "Forecasting, structural time series, and the Kalman filter", Cambridge University Press.

Durbin and Koopman (2001), "Time series analysis by state space methods", Oxford University Press.

`bsts`

.
`SdPrior`

```
n <- 100
residual.sd <- .001
# Actual values of the AR coefficients
true.phi <- c(-.7, .3, .15)
ar <- arima.sim(model = list(ar = true.phi),
n = n,
sd = 3)
## Layer some noise on top of the AR process.
y <- ar + rnorm(n, 0, residual.sd)
ss <- AddAr(list(), lags = 3, sigma.prior = SdPrior(3.0, 1.0))
# Fit the model with knowledge with residual.sd essentially fixed at the
# true value.
model <- bsts(y, state.specification=ss, niter = 500, prior = SdPrior(residual.sd, 100000))
# Now compare the empirical ACF to the true ACF.
acf(y, lag.max = 30)
points(0:30, ARMAacf(ar = true.phi, lag.max = 30), pch = "+")
points(0:30, ARMAacf(ar = colMeans(model$AR3.coefficients), lag.max = 30))
legend("topright", leg = c("empirical", "truth", "MCMC"), pch = c(NA, "+", "o"))
```

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