

   FFiitt AAuuttoorreeggrreessssiivvee MMooddeellss ttoo TTiimmee SSeerriieess

        ar(x, aic = TRUE, order.max = NULL,
           method=c("yule-walker", "burg", "ols", "mle"), na.action, series)
        ar.burg(x, aic = TRUE, order.max = NULL, na.action, demean = TRUE, series,
                var.method = 1)
        ar.yw(x, aic = TRUE, order.max = NULL, na.action, demean = TRUE, series)
        ar.ols(x, aic = TRUE, order.max = NULL, na.action, demean = TRUE, series)
        ar.mle(x, aic = TRUE, order.max = NULL, na.action, demean = TRUE, series)
        predict(ar.obj, newdata, n.ahead = 1, se.fit = TRUE)

   AArrgguummeennttss::

          x: A univariate or multivariate time series.

        aic: Logical flag.  If `TRUE' then the Akaike Informa-
             tion 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 except for `method="mle"' where it is
             the minimum of this quantity and 12.

     method: Character string giving the method used to fit the
             model.  Must be one of the strings in the default
             argument (the first few characters are suffi-
             cient).  Defaults to `"yule-walker"'.

   na.action: function to be called to handle missing values.

     demean: should mean be removed before fitting?

     series: name for the series.  Defaults to `deparse(substi-
             tute(x))'.

   var.method: the method to estimate the innovations variance
             (see Details).

     ar.obj: a fit from `ar'.

    newdata: data to which to apply the prediction.

    n.ahead: number of steps ahead at which to predict.

     se.fit: logical: return estimated standard errors of the
             prediction error?

   DDeessccrriippttiioonn::

        Fit an autoregressive time series model to the data, by
        default selecting the complexity by AIC.

   DDeettaaiillss::

        `ar' is just a wrapper for the functions `ar.yw',
        `ar.burg', `ar.ols' and `ar.mle'.

        Order selection is done by AIC if `aic' is true. This
        is problematic, as of the methods here only `ar.mle'
        performs true maximum likelihood estimation. The AIC is
        computed as if the variance estimate 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. In `ar.yw'
        the variance matrix of the innovations is computed from
        the fitted coefficients and the autocovariance of `x'
        and in `ar.ols' from the variance matrix of the residu-
        als.

        `ar.burg' allows two methods to estimate the innova-
        tions variance and hence AIC. Method 1 is to use the
        update given by the Levinson-Durbin recursion (Brock-
        well and Davis, 1991, (8.2.6) on page 242), and follows
        S-PLUS. Method 2 is the mean of the sum of squares of
        the forward and backward prediction errors (as in
        Brockwell and Davis, 1996, page 145). Percival and
        Walden (1998) discuss both.

        Remember that `ar' includes by default a constant in
        the model, by removing the overall mean of `x' before
        fitting the AR model, or (`ar.ols') estimating an addi-
        tive constant.

        `ar.ols' fits the general AR model (containing an
        intercept if `demean = TRUE') to a possibly non-sta-
        tionary and/or multivariate system of series `x'. The
        resulting unconstrained least squares estimates are
        consistent, even if some of the series are non-station-
        ary and/or co-integrated.

   VVaalluuee::

        For `ar' and its methods 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 fit-
             ted model.

   var.pred: The prediction variance: an estimate of the por-
             tion of the variance of the time series that is
             not explained by the autoregressive model.

     x.mean: The estimated mean of the series use in fitting
             and for use in prediction.

        aic: The value of the `aic' argument.

     n.used: The number of observations in the time series.

   order.max: The value of the `order.max' argument.

   partialacf: The estimate of the partial autocorrelation
             function up to lag `order.max'.

      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 value of the `method' argument.

     series: The name(s) of the time series.

   asy.var.coef: (univariate, not `ar.ols') The asymptotic-the-
             ory variance matrix of the coefficient estimates.

   asy.se.coef: (`ar.ols' only.) The asymptotic-theory standard
             errors of the coefficient estimates.

             For `predict.ar', a time series of predictions, or
             if `se.fit = TRUE', a list with components `pred',
             the predictions, and `se', the estimated standard
             errors. Both components are time series.

   NNoottee::

        Only the univariate cases of `ar.burg' and `ar.mle' are
        implemented.

        Fitting by `method="mle"' to long series can be very
        slow.

   AAuutthhoorr((ss))::

        Martyn Plummer, univariate case of `ar.yw', `ar.mle'
        and C code for `ar.burg' by B.D. Ripley, `ar.ols' by
        Adrian Trapletti.

   RReeffeerreenncceess::

        Brockwell, P. J. and Davis, R. A. (1991) Time Series
        and Forecasting Methods.  Second edition. Springer, New
        York. Section 11.4.

        Brockwell, P. J. and Davis, R. A. (1996) Introduction
        to Time Series and Forecasting. Springer, New York.
        Sections 5.1 and 7.6.

        Luetkepohl, H. (1991): Introduction to Multiple Time
        Series Analysis. Springer Verlag, NY, pp. 368-370.

        Percival, D. P. and Walden, A. T. (1998) Spectral Anal-
        ysis for Physical Applications. Cambridge University
        Press.

        Whittle, P. (1963) On the fitting of multivariate
        autoregressions and the approximate canonical factor-
        ization of a spectral density matrix. Biometrika 40,
        129-134.

   EExxaammpplleess::

        data(lh)
        ar(lh)
        ar(lh, method="burg")
        ar(lh, method="ols")

        data(LakeHuron)
        ar(LakeHuron)
        ar(LakeHuron, method="burg")
        ar(LakeHuron, method="ols")

        data(sunspot)
        sunspot.ar <- ar(sunspot.year)
        sunspot.ar
        ar(x = sunspot.year, method = "burg")
        ar(x = sunspot.year, method = "ols")
        ## next is slow and may have convergence problems,
        ## as it cares about invertibility
        ar(x = sunspot.year, method = "mle")

        predict(sunspot.ar, n.ahead=25)

        data(BJsales)
        ar(ts.union(BJsales, BJsales.lead))

        data(EuStockMarkets)
        x <- diff(log(EuStockMarkets))
        ar.ols(x, order.max=6)

