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, …)
a univariate or multivariate time series.
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.
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.
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".
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.
should a mean be estimated during fitting?
names for the series. Defaults to
deparse(substitute(x)).
the method to estimate the innovations variance (see ‘Details’).
additional arguments for specific methods.
a fit from ar().
data to which to apply the prediction.
number of steps ahead at which to predict.
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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