[R] Arima

Gerard.Keogh@cso.ie Gerard.Keogh at cso.ie
Mon Dec 17 11:10:24 CET 2001

                    Prof Brian Ripley                                                                                 
                    <ripley at stats.ox.ac.        To:     Pascal Grandeau <pgrandeau at free.fr>                           
                    uk>                         cc:     r-help at stat.math.ethz.ch                                      
                    Sent by:                    Subject:     Re: [R] Arima                                            
                    owner-r-help at stat.ma                                                                              
                    16/12/01 21:45                                                                                    

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.

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.

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

Gerard Keogh

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