# 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 class 'default':
ar.burg(x, aic = TRUE, order.max = NULL,
na.action = na.fail, demean = TRUE, series,
var.method = 1, \dots)
## S3 method for class 'mts':
ar.burg(x, aic = TRUE, order.max = NULL,
na.action = na.fail, demean = TRUE, series,
var.method = 1, \dots)

ar.yw(x, ...)
## S3 method for class 'default':
ar.yw(x, aic = TRUE, order.max = NULL,
na.action = na.fail, demean = TRUE, series, \dots)
## S3 method for class 'mts':
ar.yw(x, aic = TRUE, order.max = NULL,
na.action = na.fail, demean = TRUE, series,
var.method = 1, \dots)

ar.mle(x, aic = TRUE, order.max = NULL, na.action = na.fail,
demean = TRUE, series, ...)

## S3 method for class 'ar':
predict(object, newdata, n.ahead = 1, se.fit = TRUE, \dots)

##### 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. 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 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
`"yule-walker"`

. - 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

$$x_t - \mu = a_1(x_{t-1} - \mu) + \cdots + a_p(x_{t-p} - \mu) + e_t$$

`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: order The order of the fitted model. This is chosen by minimizing the AIC if `aic = TRUE`

, otherwise it is`order.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 for`x - x.mean`

.aic The differences in AIC between each model and the best-fitting 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 first`order`

residuals are set to`NA`

. If`x`

is a time series, so is`resid`

.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 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.

##### concept

autoregression

##### 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.

##### 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")
```

*Documentation reproduced from package stats, version 3.3, License: Part of R 3.3*