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, ...)
## 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

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?

Value

For ar and its methods a list of class "ar" with the following elements:
orderThe order of the fitted model. This is chosen by minimizing the AIC if aic = TRUE, otherwise it is order.max.
arEstimated autoregression coefficients for the fitted model.
var.predThe prediction variance: an estimate of the portion of the variance of the time series that is not explained by the autoregressive model.
x.meanThe 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.
aicThe differences in AIC between each model and the best-fitting model. Note that the latter can have an AIC of -Inf.
n.usedThe number of observations in the time series.
order.maxThe value of the order.max argument.
partialacfThe estimate of the partial autocorrelation function up to lag order.max.
residresiduals 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.
methodThe value of the method argument.
seriesThe name(s) of the time series.
frequencyThe frequency of the time series.
callThe 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.

concept

autoregression

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.

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.

Examples

Run this code

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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