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\name{arma}
\alias{arma}
\title{Fit ARMA Models to Time Series}
\description{
Fit an ARMA model to a univariate time series by conditional least
squares. For exact maximum likelihood estimation see
\code{\link{arima0}}.
}
\usage{
arma(x, order = c(1, 1), lag = NULL, coef = NULL,
include.intercept = TRUE, series = NULL, qr.tol = 1e-07, \dots)
}
\arguments{
\item{x}{a numeric vector or time series.}
\item{order}{a two dimensional integer vector giving the orders of the
model to fit. \code{order[1]} corresponds to the AR part and
\code{order[2]} to the MA part.}
\item{lag}{a list with components \code{ar} and \code{ma}. Each
component is an integer vector, specifying the AR and MA lags that are
included in the model. If both, \code{order} and \code{lag}, are
given, only the specification from \code{lag} is used.}
\item{coef}{If given this numeric vector is used as the initial estimate
of the ARMA coefficients. The preliminary estimator suggested in
Hannan and Rissanen (1982) is used for the default initialization.}
\item{include.intercept}{Should the model contain an intercept?}
\item{series}{name for the series. Defaults to
\code{deparse(substitute(x))}.}
\item{qr.tol}{the \code{tol} argument for \code{\link{qr}} when computing
the asymptotic standard errors of \code{coef}.}
\item{\dots}{additional arguments for \code{\link{optim}} when fitting
the model.}
}
\details{
The following parametrization is used for the ARMA(p,q) model:
\deqn{y[t] = a[0] + a[1]y[t-1] + \dots + a[p]y[t-p] + b[1]e[t-1] +
\dots + b[q]e[t-q] + e[t],}
where \eqn{a[0]} is set to zero if no intercept is included. By using
the argument \code{lag}, it is possible to fit a parsimonious submodel
by setting arbitrary \eqn{a[i]} and \eqn{b[i]} to zero.
\code{arma} uses \code{\link{optim}} to minimize the conditional
sum-of-squared errors. The gradient is computed, if it is needed, by
a finite-difference approximation. Default initialization is done by
fitting a pure high-order AR model (see \code{\link{ar.ols}}).
The estimated residuals are then used for computing a least squares
estimator of the full ARMA model. See Hannan and Rissanen (1982) for
details.
}
\value{
A list of class \code{"arma"} with the following elements:
\item{lag}{the lag specification of the fitted model.}
\item{coef}{estimated ARMA coefficients for the fitted model.}
\item{css}{the conditional sum-of-squared errors.}
\item{n.used}{the number of observations of \code{x}.}
\item{residuals}{the series of residuals.}
\item{fitted.values}{the fitted series.}
\item{series}{the name of the series \code{x}.}
\item{frequency}{the frequency of the series \code{x}.}
\item{call}{the call of the \code{arma} function.}
\item{vcov}{estimate of the asymptotic-theory covariance matrix for the
coefficient estimates.}
\item{convergence}{The \code{convergence} integer code from
\code{\link{optim}}.}
\item{include.intercept}{Does the model contain an intercept?}
}
\references{
E. J. Hannan and J. Rissanen (1982):
Recursive Estimation of Mixed Autoregressive-Moving Average
Order.
\emph{Biometrika} \bold{69}, 81--94.
}
\author{
A. Trapletti
}
\seealso{
\code{\link{summary.arma}} for summarizing ARMA model fits;
\code{\link{arma-methods}} for further methods;
\code{\link{arima0}}, \code{\link{ar}}.
}
\examples{
data(tcm)
r <- diff(tcm10y)
summary(r.arma <- arma(r, order = c(1, 0)))
summary(r.arma <- arma(r, order = c(2, 0)))
summary(r.arma <- arma(r, order = c(0, 1)))
summary(r.arma <- arma(r, order = c(0, 2)))
summary(r.arma <- arma(r, order = c(1, 1)))
plot(r.arma)
data(nino)
s <- nino3.4
summary(s.arma <- arma(s, order=c(20,0)))
summary(s.arma
<- arma(s, lag=list(ar=c(1,3,7,10,12,13,16,17,19),ma=NULL)))
acf(residuals(s.arma), na.action=na.remove)
pacf(residuals(s.arma), na.action=na.remove)
summary(s.arma
<- arma(s, lag=list(ar=c(1,3,7,10,12,13,16,17,19),ma=12)))
summary(s.arma
<- arma(s, lag=list(ar=c(1,3,7,10,12,13,16,17),ma=12)))
plot(s.arma)
}
\keyword{ts}
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