Fit an autoregressive time series model to the data by ordinary least squares, by default selecting the complexity by AIC.
ar.ols(x, aic = TRUE, order.max = NULL, na.action = na.fail,
demean = TRUE, intercept = demean, series, …)A univariate or multivariate time series.
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.
Maximum order (or order) of model to fit. Defaults to \(10\log_{10}(N)\) where \(N\) is the number of observations.
function to be called to handle missing values.
should the AR model be for x minus its mean?
should a separate intercept term be fitted?
names for the series. Defaults to
deparse(substitute(x)).
further arguments to be passed to or from 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 (or zero if demean is false)
of the series used in fitting and for use in prediction.
The intercept in the model for
x - x.mean, or zero if intercept is false.
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.
The value of the order.max argument.
NULL. For compatibility with ar.
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 character string "Unconstrained LS".
The name(s) of the time series.
The frequency of the time series.
The matched call.
The asymptotic-theory standard errors of the coefficient estimates.
ar.ols fits the general AR model 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.
For definiteness, note that the AR coefficients have the sign in
$$x_t - \mu = a_0 + a_1(x_{t-1} - \mu) + \cdots + a_p(x_{t-p} - \mu) + e_t$$
where \(a_0\) is zero unless intercept is true, and
\(\mu\) is the sample mean if demean is true, zero
otherwise.
Order selection is done by AIC if aic is true. This is
problematic, as ar.ols does not perform
true maximum likelihood estimation. The AIC is computed as if
the variance estimate (computed from the variance matrix of the
residuals) 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.
Some care is needed if intercept is true and demean is
false. Only use this is the series are roughly centred on
zero. Otherwise the computations may be inaccurate or fail entirely.
Luetkepohl, H. (1991): Introduction to Multiple Time Series Analysis. Springer Verlag, NY, pp.368--370.
ar(lh, method = "burg")
ar.ols(lh)
ar.ols(lh, FALSE, 4) # fit ar(4)
ar.ols(ts.union(BJsales, BJsales.lead))
x <- diff(log(EuStockMarkets))
ar.ols(x, order.max = 6, demean = FALSE, intercept = TRUE)
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