ar.ols
Fit Autoregressive Models to Time Series by OLS
Fit an autoregressive time series model to the data by ordinary least squares, by default selecting the complexity by AIC.
 Keywords
 ts
Usage
ar.ols(x, aic = TRUE, order.max = NULL, na.action = na.fail, demean = TRUE, intercept = demean, series, ...)
Arguments
 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. IfFALSE
, the model of orderorder.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.
 na.action
 function to be called to handle missing values.
 demean
 should the AR model be for
x
minus its mean?  intercept
 should a separate intercept term be fitted?
 series
 names for the series. Defaults to
deparse(substitute(x))
.  ...
 further arguments to be passed to or from methods.
Details
ar.ols
fits the general AR model to a possibly nonstationary
and/or multivariate system of series x
. The resulting
unconstrained least squares estimates are consistent, even if
some of the series are nonstationary and/or cointegrated.
For definiteness, note that the AR coefficients have the sign in
$$x_t  \mu = a_0 + a_1(x_{t1}  \mu) + \cdots + a_p(x_{tp}  \mu) + e_t$$
where $a[0]$ is zero unless intercept
is true, and
$m$ 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.
Value

A list of class
 order
 The order of the fitted model. This is chosen by
minimizing the AIC if
aic = TRUE
, otherwise it isorder.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 (or zero if
demean
is false) of the series used in fitting and for use in prediction.  x.intercept
 The intercept in the model for
x  x.mean
, or zero ifintercept
is false.  aic
 The differences in AIC between each model and the
bestfitting model. Note that the latter can have an AIC of
Inf
.  n.used
 The number of observations in the time series.
 order.max
 The value of the
order.max
argument.  partialacf
NULL
. For compatibility withar
. resid
 residuals from the fitted model, conditioning on the
first
order
observations. The firstorder
residuals are set toNA
. Ifx
is a time series, so isresid
.  method
 The character string
"Unconstrained LS"
.  series
 The name(s) of the time series.
 frequency
 The frequency of the time series.
 call
 The matched call.
 asy.se.coef
 The asymptotictheory standard errors of the coefficient estimates.
"ar"
with the following elements:
References
Luetkepohl, H. (1991): Introduction to Multiple Time Series Analysis. Springer Verlag, NY, pp.\ifelse{latex}{\out{~}}{ } 368370.
See Also
Examples
library(stats)
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)