\HeaderA{ar.ols}{Fit Autoregressive Models to Time Series by OLS}{ar.ols}
\keyword{ts}{ar.ols}
\begin{Description}\relax
Fit an autoregressive time series model to the data by ordinary
least squares, by default selecting the complexity by AIC.
\end{Description}
\begin{Usage}
\begin{verbatim}
ar.ols(x, aic = TRUE, order.max = NULL, na.action = na.fail,
       demean = TRUE, intercept = demean, series, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] A univariate or multivariate time series.
\item[\code{aic}] Logical flag.  If \code{TRUE} then the Akaike Information
Criterion is used to choose the order of the autoregressive
model. If \code{FALSE}, the model of order \code{order.max} is
fitted.
\item[\code{order.max}] Maximum order (or order) of model to fit. Defaults
to \eqn{10\log_{10}(N)}{10*log10(N)} where \eqn{N}{} is the number
of observations.
\item[\code{na.action}] function to be called to handle missing values.
\item[\code{demean}] should the AR model be for \code{x} minus its mean?
\item[\code{intercept}] should a separate intercept term be fitted?
\item[\code{series}] names for the series.  Defaults to
\code{deparse(substitute(x))}.
\item[\code{...}] further arguments to be passed to or from methods.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
\code{ar.ols} fits the general AR model to a possibly non-stationary
and/or multivariate system of series \code{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

\deqn{x_t - \mu = a_0 + a_1(x_{t-1} - \mu) + \cdots + a_p(x_{t-p} - \mu) + e_t}{\code{(x[t] - m) = a[0] + a[1]*(x[t-1] - m) + ... +  a[p]*(x[t-p] - m) + e[t]}}

where \eqn{a_0}{a[0]} is zero unless \code{intercept} is true, and
\eqn{\mu}{m} is the sample mean if \code{demean} is true, zero
otherwise.

Order selection is done by AIC if \code{aic} is true. This is
problematic, as \code{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 \code{intercept} is true and \code{demean} is
false. Only use this is the series are roughly centred on
zero. Otherwise the computations may be inaccurate or fail entirely.
\end{Details}
\begin{Value}
A list of class \code{"ar"} with the following elements:
\begin{ldescription}
\item[\code{order}] The order of the fitted model.  This is chosen by
minimizing the AIC if \code{aic=TRUE}, otherwise it is
\code{order.max}.
\item[\code{ar}] Estimated autoregression coefficients for the fitted
model.
\item[\code{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.
\item[\code{x.mean}] The estimated mean (or zero if \code{demean} is false)
of the series used in fitting and for use in prediction.
\item[\code{x.intercept}] The intercept in the model for
\code{x - x.mean}, or zero if \code{intercept} is false.
\item[\code{aic}] The value of the \code{aic} argument.
\item[\code{n.used}] The number of observations in the time series.
\item[\code{order.max}] The value of the \code{order.max} argument.
\item[\code{partialacf}] \code{NULL}.  For compatibility with \code{ar}.
\item[\code{resid}] residuals from the fitted model, conditioning on the
first \code{order} observations.  The first \code{order} residuals
are set to \code{NA}. If \code{x} is a time series, so is
\code{resid}.
\item[\code{method}] The character string \code{"Unconstrained LS"}.
\item[\code{series}] The name(s) of the time series.
\item[\code{asy.se.coef}] The asymptotic-theory standard errors of the
coefficient estimates.
\end{ldescription}
\end{Value}
\begin{Author}\relax
Adrian Trapletti, Brian Ripley.
\end{Author}
\begin{References}\relax
Luetkepohl, H. (1991): \emph{Introduction to Multiple Time Series
Analysis.} Springer Verlag, NY, pp. 368--370.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{ar}{ar}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
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)
\end{ExampleCode}
\end{Examples}

