# 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`

. If`TRUE`

then the Akaike Information Criterion is used to choose the order of the autoregressive model. If`FALSE`

, the model of order`order.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")
# }
```

*Documentation reproduced from package stats, version 3.6.0, License: Part of R 3.6.0*