[R] Arima

[Prof Brian Ripley]

On Mon, 17 Dec 2001 Gerard.Keogh at cso.ie wrote:

> On Sun, 16 Dec 2001, Pascal Grandeau wrote:
> >
> > Does anyone make a routine for regression with ARMA errors with least
> > squares ?
>
> Prof Brian Ripley replies:
>
> What does that mean? `with least squares' implies independent errors.
> arma() fits by so-called *conditional* least squares: that leaves out
> terms in the log-likelihood which can be important, especially near
> non-stationarity. I've never understood why anyone would want to do that,
> except as a poor man's computational approximation.

[End of quote]


> When considering the MA(1) model Harvey in "Time Series Models p60" says
> that assuming the initial disturbance to be fixed and equal to zero makes
> the problem of maximising the likelihood function equivalent to minimising
> the sum of squares of the errors - the result is then called the
> conditional sum of squares (CSS) estimate. The calculation of this
> "conditional likelihood function" is therefore simplified considerably and
> the resulting equations which are still nonlinear in the parameters are
> more readily optimised because analytic derivatives are available.
>
> Of course the exact likelihood function of any ARMA(p,q) model can be
> generated from Kalman recursions via the prediction error decomposition.
> Harvey's main argument for using the CSS estimate relies on the fact that
> maximising the likelihood is time consuming for large p+q (for myself, I
> take time consuming to mean that it's often very hard to find a solution to
> a nonlinear problem!). However, I suspect that with the computing power now
> available the time issue may be far less relevant.

Exactly, as I said.

> One final point though is that the CSS estimate may provide reasonable
> starting values for the optimisation of the exact likelihood.

Given that arima0() does the exact likelihood, and I've never had to wait
more than a few seconds for it to do so, I still don't see why
anyone would ask for conditional least squares instead, which was the
request.

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272860 (secr)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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