add.ar: AR(p) state component

Description

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

Usage

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

Arguments

state.specification

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

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 NULL, in which case the stationary distribution of the AR(p) process will be used as the initial state distribution.

sdy

The sample standard deviation of the time series to be modeled. Used to scale the prior distribution. This can be omitted if y is supplied.

Value

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

Details

The model is

$α_{t} = ϕ_{1} α_{i, t - 1} + \dots + ϕ_{p} α_{t - p} + ϵ_{t - 1} ϵ_{t} \sim N (0, σ^{2})$

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.

References

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.

Examples

Run this code

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.phi), lag.max = 30))
legend("topright", leg = c("empirical", "truth", "MCMC"), pch = c(NA, "+", "o"))

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