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 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
$m$ 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
best-fitting 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 asymptotic-theory 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{~}}{ } 368--370.
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)