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("yule-walker", "burg", "ols", "mle", "yw"),
na.action, series, …)ar.burg(x, …)
# S3 method for default
ar.burg(x, aic = TRUE, order.max = NULL,
na.action = na.fail, demean = TRUE, series,
var.method = 1, …)
# S3 method for mts
ar.burg(x, aic = TRUE, order.max = NULL,
na.action = na.fail, demean = TRUE, series,
var.method = 1, …)
ar.yw(x, …)
# S3 method for default
ar.yw(x, aic = TRUE, order.max = NULL,
na.action = na.fail, demean = TRUE, series, …)
# S3 method for mts
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, …)
# S3 method for ar
predict(object, newdata, n.ahead = 1, se.fit = TRUE, …)
Arguments
- x
a univariate or multivariate time series.
- aic
logical
. IfTRUE
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 \(N-1\) and \(10\log_{10}(N)\) where \(N\) is the number of non-missing observations except for
method = "mle"
where it is the minimum of this quantity and 12.- method
character string specifying the method to fit the model. Must be one of the strings in the default argument (the first few characters are sufficient). Defaults to
"yule-walker"
.- na.action
function to be called to handle missing values. Currently, via
na.action = na.pass
, only Yule-Walker method can handle missing values which must be consistent within a time point: either all variables must be missing or none.- 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
$$x_t - \mu = a_1(x_{t-1} - \mu) + \cdots + a_p(x_{t-p} - \mu) + e_t$$
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 Levinson-Durbin recursion (Brockwell and Davis, 1991, (8.2.6)
on page 242), and follows S-PLUS. 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 ar
and its methods a list of class "ar"
with
the following elements:
The order of the fitted model. This is chosen by
minimizing the AIC if aic = TRUE
, otherwise it is order.max
.
Estimated autoregression coefficients for the fitted model.
The prediction variance: an estimate of the portion of the variance of the time series that is not explained by the autoregressive model.
The estimated mean of the series used in fitting and for use in prediction.
(ar.ols
only.) The intercept in the model for
x - x.mean
.
The differences in AIC between each model and the
best-fitting model. Note that the latter can have an AIC of -Inf
.
The number of observations in the time series, including missing.
The number of non-missing observations in the time series.
The value of the order.max
argument.
The estimate of the partial autocorrelation function
up to lag order.max
.
residuals from the fitted model, conditioning on the
first order
observations. The first order
residuals
are set to NA
. If x
is a time series, so is resid
.
The value of the method
argument.
The name(s) of the time series.
The frequency of the time series.
The matched call.
(univariate case, order > 0
.)
The asymptotic-theory variance matrix of the coefficient estimates.
For 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.
If x
contains missing values, see NA
, also consider
using arima()
, possibly with method = "ML"
.
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, 129--134. 10.2307/2333753.
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)
# NOT RUN {
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")
# }