| % File src/library/stats/man/ar.ols.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2014 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{ar.ols} |
| \alias{ar.ols} |
| \title{Fit Autoregressive Models to Time Series by OLS} |
| \usage{ |
| ar.ols(x, aic = TRUE, order.max = NULL, na.action = na.fail, |
| demean = TRUE, intercept = demean, series, \dots) |
| } |
| \arguments{ |
| \item{x}{A univariate or multivariate time series.} |
| |
| \item{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{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{na.action}{function to be called to handle missing values.} |
| |
| \item{demean}{should the AR model be for \code{x} minus its mean?} |
| |
| \item{intercept}{should a separate intercept term be fitted?} |
| |
| \item{series}{names for the series. Defaults to |
| \code{deparse(substitute(x))}.} |
| |
| \item{\dots}{further arguments to be passed to or from methods.} |
| } |
| \description{ |
| Fit an autoregressive time series model to the data by ordinary |
| least squares, by default selecting the complexity by AIC. |
| } |
| \details{ |
| \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}{(x[t] - m) = a[0] + a[1]*(x[t-1] - m) + \dots + 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. |
| } |
| \value{ |
| A list of class \code{"ar"} with the following elements: |
| \item{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{ar}{Estimated autoregression coefficients for the fitted |
| model.} |
| \item{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{x.mean}{The estimated mean (or zero if \code{demean} is false) |
| of the series used in fitting and for use in prediction.} |
| \item{x.intercept}{The intercept in the model for |
| \code{x - x.mean}, or zero if \code{intercept} is false.} |
| \item{aic}{The differences in AIC between each model and the |
| best-fitting model. Note that the latter can have an AIC of \code{-Inf}.} |
| \item{n.used}{The number of observations in the time series.} |
| \item{order.max}{The value of the \code{order.max} argument.} |
| \item{partialacf}{\code{NULL}. For compatibility with \code{ar}.} |
| \item{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{method}{The character string \code{"Unconstrained LS"}.} |
| \item{series}{The name(s) of the time series.} |
| \item{frequency}{The frequency of the time series.} |
| \item{call}{The matched call.} |
| \item{asy.se.coef}{The asymptotic-theory standard errors of the |
| coefficient estimates.} |
| } |
| \author{Adrian Trapletti, Brian Ripley.} |
| \seealso{ |
| \code{\link{ar}} |
| } |
| \references{ |
| Luetkepohl, H. (1991): \emph{Introduction to Multiple Time Series |
| Analysis.} Springer Verlag, NY, pp.\sspace{}368--370. |
| } |
| \examples{ |
| 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) |
| } |
| \keyword{ts} |