arrob: Robust Fitting of Autoregressive Models
In robts: Robust Time Series Analysis

Description Usage Arguments Details Value Author(s) References See Also Examples

Robust fit of an autoregressive model to a time series. Its order is either optimized with respect to a robustified AIC or specified by the user.

arrob(x, aic = TRUE, order.max, method = c("yw", "regression", "gm", "filter"),
      na.action = na.fail, series = deparse(substitute(x)),asyvar=FALSE,bootpar=list(num=100,blockl=floor(2*length(x)^(1/2))), ...)

arrob.yw(x, order.max, aic = TRUE, aicpenalty = function(p) 2*p,
      na.action = na.fail, series = deparse(substitute(x)),
      acf.approach = c("GK", "median", "multi", "partrank", "RA", "rank",
      "filter", "trim", "bireg"), locfn = median, scalefn = Qn, ...)
      
arrob.regression(x, order.max, aic = TRUE, aicpenalty = function(p) 2*p,
      na.action = na.fail, series = deparse(substitute(x)),
      intercept = TRUE, scalefn = Qn, ...)
      
arrob.filter(x, order.max, aic = TRUE, aicpenalty = function(p) 2*p,
      na.action = na.fail, series = deparse(substitute(x)),
      psi.l = 2, psi.0 = 3)

`x`	numeric vector of a univariate time series.
`aic`	logical indicating whether the AR order should be estimated by robust AIC criteria considering orders up to `order.max` (if `aic = TRUE`) or should be equal to `order.max` (if `aic = FALSE`).
`order.max`	integer value determining the (maximal) order of the AR fit. If missing, this value is chosen to be `floor(min(c((n - 1) / 4, 10 * log(n, base = 10))))`, where `n` is the number of observations.
`method`	character string naming the estimation method to be used, see Details.
`na.action`	function to be called to handle missing values. Default is `na.fail`. Other reasonable possibilities are `na.omit` (omits all missing values at the beginning and the end of the time series and fails if there are more missing values), `na.contiguous` (uses longest contiguous stretch of non-missing observations) and `na.extremify` (experimental, sets missing values to very extreme values in the expectation that they have only little influence on the outcome of robust estimation methods).
`series`	the time series name.
`asyvar`	logical indicating whether the covariance matrix of the fitted AR coefficients should be estimated by a block bootstrap. The default is `FALSE` since it can take some time.
`bootpar`	list containing parameters of the blockbootstrap which is used to estimate the variance of the fitted AR coefficients, see Details.
`aicpenalty`	function of the model order defining the penalty term of the model selection criteria. The default results in a robust AIC.
`...`	further arguments to be passed to internal function `arrob.yw`, `arrob.regression`, `arrob.filter` or `arrob.gm`. See below or on the respective help page.
`acf.approach`	character string naming the function to calculate the autocorrelation function. See `acfrob` for details. It is not recommended to use `acf.approach = "filter"` since one can get an AR fit more directly by `method = "filter"`.
`locfn`	function which calculates a location estimator. Its argument must be a vector of the data.
`scalefn`	function which calculates a scale estimator. Its argument must be a vector of the data. One could for example use one of the functions `Qn` (the default), `Sn`, `scaleTau2` (all from the package `robustbase`), `mad` or (not robust) `sd`.
`intercept`	logical whether the regression model includes an intercept.
`psi.l`	numeric value determining the psi function, see the help page of `ARfilter` for details.
`psi.0`	numeric value determining the psi function, see the help page of `ARfilter` for details.

There are many possibilities for robust estimation of AR models. The argument method specifies which of the following methods is to be used, where method = "yw" is the default.

"yw": One approach is to use a roust estimation of the autocorrelation function. The corresponding AR coefficients can be then either computed by solving the Yule-Walker equations method = "yule-walker" employing the Durbin-Levinson algorithm. The acf estimator can be determined by acf.fun. See the help page of acfrob for an overview of implemented procedures for estimation of the acf.
"regression": The AR coefficients are computed by a robust regression. Here the dependent variable is the time series x and the independent variables are the up to order.max lagged time series. The regression is done by MM estimation with the function lmrob, see Rousseeuw et al. (2014).
"filter": The regression coefficients are computed via robust filtering as described in Chapter 8.6 of Maronna et al. (2006). See the help page of ARfilter for details.
"gm": Generalized M estimates are used for obtaining an AR fit as described in Maronna et al. (2006). See the help page of arrob.gm for details and further arguments which are passed to that function by the ... argument.

If aic = TRUE, AR models for every order up to order.max are estimated and the one with the smallest model selection criterion returned. The criterion is the sum of the logarithm of a robust variance estimation of the residuals and a penalty term pen(p) divided by n-p depending on the AR order p, i.e.,

log(sigma_hat^2) + (pen(p))/(n-p)

The penalty term can be set by the argument aicpenalty.

If asyvar=TRUE the variance of the fitted AR coefficients is estimated by a moving block bootstrap, see Lahiri (1999) for a detailed description of this method. The blocklength can be set by the blockl argument and thr number of bootstrap repetitions by num. Note that there are no theoretical justifications that this estimation is consistent for robust methods.

Object of classes "arrob" and "ar". This is a list which includes all elements of an object of class "ar" (see ar for details) plus the following additional element:

`x`	the original time series `x`.

Note that list element aic gives the value of the robust information criterion and not its difference with the lowest information criterion of all considered models, as it is returned by the function ar.

Alexander Dürre, Tobias Liboschik and Jonathan Rathjens

Lahiri, S.N. (1999): Theoretical Comparisons of Block Bootstrap Methods, The annals of Statistics, vol. 27, 386–404.

Maronna, R. A., Martin, R. D., and Yohai, V. J. (2006): Robust Statistics: Theory and Methods, Wiley, chapter 8, doi: 10.1002/0470010940.

Rousseeuw, P., Croux, C., Todorov, V., Ruckstuhl, A., Salibian-Barrera, M., Verbeke, T., Koller, M. and Maechler, M. (2014): robustbase: Basic Robust Statistics. R package version 0.91-1. URL http://cran.r-project.org/package=robustbase.

Classical, nonrobust fitting is provided by the function ar.

S3 methods residuals.arrob, fitted.arrob, filtered.arrob and predict.arrob for objects of class "arrob".

set.seed(1066)
tss <- arima.sim(model = list(ar = 0.3, ma = 0.5), n = 100)
arrob(tss, method = "yw")
arrob(tss, method = "regression")
arrob(tss, method = "filter")
arrob(tss, method = "gm")