# ar.ols

0th

Percentile

##### 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. If FALSE, the model of order order.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 $a$ 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 "ar" with the following elements:
order
The order of the fitted model. This is chosen by minimizing the AIC if aic = TRUE, otherwise it is order.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 if intercept 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 with ar.
resid
residuals from the fitted model, conditioning on the first order observations. The first order residuals are set to NA. If x is a time series, so is resid.
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.

##### References

Luetkepohl, H. (1991): Introduction to Multiple Time Series Analysis. Springer Verlag, NY, pp.\ifelse{latex}{\out{~}}{ } 368--370.

ar
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)