Fit an autoregressive time series model to the data, by default selecting the complexity by AIC.

```
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, …)

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?

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.

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.

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.

`ar.ols`

, `arima`

for ARMA models;
`acf2AR`

, for AR construction from the ACF.

`arima.sim`

for simulation of AR processes.

```
# 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")
# }
```

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