ar Fit Autoregressive Models to Time Series

Description

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

Usage

ar(x, aic = TRUE, order.max = NULL,
   method = c("yule-walker", "burg", "ols", "mle", "yw"),
   na.action, series, ...)

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

ar.yw(x, ...)
## Default S3 method:
ar.yw(x, aic = TRUE, order.max = NULL,
      na.action = na.fail, demean = TRUE, series, ...)
## S3 method for class '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 class '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*log10(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 deparse1(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] - m = a[1]*(x[t-1] - m) + … + a[p]*(x[t-p] - m) + 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, including missing.
n.obs	The number of non-missing 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.

If x contains missing values, see NA, also consider using arima(), possibly with method = "ML".

Author(s)

Martyn Plummer. Univariate case of ar.yw, ar.mle and C code for univariate case of ar.burg by B. D. Ripley.

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. doi: 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

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")