ar(x, aic = TRUE, order.max = NULL, method=c("yule-walker", "burg", "ols", "mle"), na.action, series) ar.burg(x, aic = TRUE, order.max = NULL, na.action, demean = TRUE, series, var.method = 1) ar.yw(x, aic = TRUE, order.max = NULL, na.action, demean = TRUE, series) ar.ols(x, aic = TRUE, order.max = NULL, na.action, demean = TRUE, series) ar.mle(x, aic = TRUE, order.max = NULL, na.action, demean = TRUE, series) predict(ar.obj, newdata, n.ahead = 1, se.fit = TRUE)
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 10*log10(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 mean be removed before fitting? |
series
|
name for the series. Defaults to
deparse(substitute(x)) .
|
var.method
| the method to estimate the innovations variance (see Details). |
ar.obj
|
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? |
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
and in ar.ols
from the variance matrix of the residuals.
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.
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.ols
) estimating an additive constant.
ar.ols
fits the general AR model (containing an intercept
if demean = TRUE
) to a possibly non-stationary and/or
multivariate system of series x
. The resulting
unconstrained least squares estimates are consistent, even if
some of the series are non-stationary and/or co-integrated.
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 use in fitting and for use in prediction. |
aic
|
The value of the aic argument.
|
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. |
asy.var.coef
|
(univariate, not ar.ols ) The asymptotic-theory
variance matrix of the coefficient estimates.
|
asy.se.coef
|
(ar.ols only.) The asymptotic-theory
standard errors 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.
ar.burg
and ar.mle
are
implemented.
Fitting by method="mle"
to long series can be very slow.
ar.yw
, ar.mle
and C code for ar.burg
by B.D. Ripley,
ar.ols
by Adrian Trapletti.Brockwell, P. J. and Davis, R. A. (1996) Introduction to Time Series and Forecasting. Springer, New York. Sections 5.1 and 7.6.
Luetkepohl, H. (1991): Introduction to Multiple Time Series Analysis. Springer Verlag, NY, pp. 368-370.
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.
data(lh) ar(lh) ar(lh, method="burg") ar(lh, method="ols") data(LakeHuron) ar(LakeHuron) ar(LakeHuron, method="burg") ar(LakeHuron, method="ols") data(sunspot) sunspot.ar <- ar(sunspot.year) sunspot.ar ar(x = sunspot.year, method = "burg") ar(x = sunspot.year, method = "ols") ## next is slow and may have convergence problems, ## as it cares about invertibility ar(x = sunspot.year, method = "mle") predict(sunspot.ar, n.ahead=25) data(BJsales) ar(ts.union(BJsales, BJsales.lead)) data(EuStockMarkets) x <- diff(log(EuStockMarkets)) ar.ols(x, order.max=6)