ar
Fit Autoregressive Models to Time Series
Fit an autoregressive time series model to the data, by default selecting the complexity by AIC.
 Keywords
 ts
Usage
ar(x, aic = TRUE, order.max = NULL, method = c("yulewalker", "burg", "ols", "mle", "yw"), na.action, series, ...)
ar.burg(x, ...)
"ar.burg"(x, aic = TRUE, order.max = NULL, na.action = na.fail, demean = TRUE, series, var.method = 1, ...)
"ar.burg"(x, aic = TRUE, order.max = NULL, na.action = na.fail, demean = TRUE, series, var.method = 1, ...)
ar.yw(x, ...)
"ar.yw"(x, aic = TRUE, order.max = NULL, na.action = na.fail, demean = TRUE, series, ...)
"ar.yw"(x, aic = TRUE, order.max = NULL, na.action = na.fail, demean = TRUE, series, var.method = 1, ...)
ar.mle(x, aic = TRUE, order.max = NULL, na.action = na.fail, demean = TRUE, series, ...)
"predict"(object, newdata, n.ahead = 1, se.fit = TRUE, ...)
Arguments
 x
 A univariate or multivariate time series.
 aic
 Logical flag. If
TRUE
then the Akaike Information Criterion is used to choose the order of the autoregressive model. IfFALSE
, the model of orderorder.max
is fitted.  order.max
 Maximum order (or order) of model to fit. Defaults
to the smaller of $N1$ and $10*log10(N)$
where $N$ is the number of observations
except for
method = "mle"
where it is the minimum of this quantity and 12.  method
 Character string giving the method used to fit the
model. Must be one of the strings in the default argument
(the first few characters are sufficient). Defaults to
"yulewalker"
.  na.action
 function to be called to handle missing values.
 demean
 should a mean be estimated during fitting?
 series
 names for the series. Defaults to
deparse(substitute(x))
.  var.method
 the method to estimate the innovations variance (see ‘Details’).
 ...
 additional arguments for specific methods.
 object
 a fit from
ar
.  newdata
 data to which to apply the prediction.
 n.ahead
 number of steps ahead at which to predict.
 se.fit
 logical: return estimated standard errors of the prediction error?
Details
For definiteness, note that the AR coefficients have the sign in
$$x_t  \mu = a_1(x_{t1}  \mu) + \cdots + a_p(x_{tp}  \mu) + e_t$$
ar
is just a wrapper for the functions ar.yw
,
ar.burg
, ar.ols
and ar.mle
.
Order selection is done by AIC if aic
is true. This is
problematic, as of the methods here only ar.mle
performs
true maximum likelihood estimation. The AIC is computed as if the variance
estimate were the MLE, omitting the determinant term from the
likelihood. Note that this is not the same as the Gaussian likelihood
evaluated at the estimated parameter values. In ar.yw
the
variance matrix of the innovations is computed from the fitted
coefficients and the autocovariance of x
.
ar.burg
allows two methods to estimate the innovations
variance and hence AIC. Method 1 is to use the update given by
the LevinsonDurbin recursion (Brockwell and Davis, 1991, (8.2.6)
on page 242), and follows SPLUS. Method 2 is the mean of the sum
of squares of the forward and backward prediction errors
(as in Brockwell and Davis, 1996, page 145). Percival and Walden
(1998) discuss both. In the multivariate case the estimated
coefficients will depend (slightly) on the variance estimation method.
Remember that ar
includes by default a constant in the model, by
removing the overall mean of x
before fitting the AR model,
or (ar.mle
) estimating a constant to subtract.
Value

For
 order
 The order of the fitted model. This is chosen by
minimizing the AIC if
aic = TRUE
, otherwise it isorder.max
.  ar
 Estimated autoregression coefficients for the fitted model.
 var.pred
 The prediction variance: an estimate of the portion of the variance of the time series that is not explained by the autoregressive model.
 x.mean
 The estimated mean of the series used in fitting and for use in prediction.
 x.intercept
 (
ar.ols
only.) The intercept in the model forx  x.mean
.  aic
 The differences in AIC between each model and the
bestfitting model. Note that the latter can have an AIC of
Inf
.  n.used
 The number of observations in the time series.
 order.max
 The value of the
order.max
argument.  partialacf
 The estimate of the partial autocorrelation function
up to lag
order.max
.  resid
 residuals from the fitted model, conditioning on the
first
order
observations. The firstorder
residuals are set toNA
. Ifx
is a time series, so isresid
.  method
 The value of the
method
argument.  series
 The name(s) of the time series.
 frequency
 The frequency of the time series.
 call
 The matched call.
 asy.var.coef
 (univariate case,
order > 0
.) The asymptotictheory variance matrix of the coefficient estimates. For
ar
and its methods a list of class "ar"
with
the following elements:
predict.ar
, a time series of predictions, or if
se.fit = TRUE
, a list with components pred
, the
predictions, and se
, the estimated standard errors. Both
components are time series.
Note
Only the univariate case of ar.mle
is implemented.
Fitting by method="mle"
to long series can be very slow.
References
Brockwell, P. J. and Davis, R. A. (1991) Time Series and Forecasting Methods. Second edition. Springer, New York. Section 11.4.
Brockwell, P. J. and Davis, R. A. (1996) Introduction to Time Series and Forecasting. Springer, New York. Sections 5.1 and 7.6.
Percival, D. P. and Walden, A. T. (1998) Spectral Analysis for Physical Applications. Cambridge University Press.
Whittle, P. (1963) On the fitting of multivariate autoregressions and the approximate canonical factorization of a spectral density matrix. Biometrika 40, 129134.
See Also
ar.ols
, arima
for ARMA models;
acf2AR
, for AR construction from the ACF.
arima.sim
for simulation of AR processes.
Examples
library(stats)
ar(lh)
ar(lh, method = "burg")
ar(lh, method = "ols")
ar(lh, FALSE, 4) # fit ar(4)
(sunspot.ar < ar(sunspot.year))
predict(sunspot.ar, n.ahead = 25)
## try the other methods too
ar(ts.union(BJsales, BJsales.lead))
## Burg is quite different here, as is OLS (see ar.ols)
ar(ts.union(BJsales, BJsales.lead), method = "burg")