[R] Re: Arima

Adrian Trapletti a.trapletti at bluewin.ch
Tue Dec 18 09:49:19 CET 2001

> Date: Mon, 17 Dec 2001 13:42:16 +0000 (GMT)
> From: Prof Brian Ripley <ripley at stats.ox.ac.uk>
> Subject: Re: [R] Arima
> 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.

Let me just say a few words why I believe that optimizing the CSS might still be useful in some cases (or at
least have the possibility to choose between the CSS and the exact likelihood). As we all know arima models
are at best good approximations to "real" data. Therefore, from my point of view a discussion about
optimizing exact likelihood or not is rather academic. Except in a simulation setup, both the CSS and the
exact likelihood are only approximations to the "real" likelihood. And it is not at all clear which of both
provides a better approximation to the "real world". An example:

On the intraday level financial price time series may be approximated rather well by the random walk plus
noise model. However, the innovations are far from being Gaussian (have much much fatter tails, ARCH effects,
structural breaks etc). Hence, an MA(1) model with non-Gaussian (and maybe non iid) innovations is a good
candidate model for the return time series. However, explicitely modelling the innovation process is often
not a good idea (too complex, still missspecified due to structural breaks, etc). Therefore, just using a
Gaussian MA(1) model might be the first choice. The results from a small MC simulation:

DGP: MA(1) with a coefficient of -0.5, rt innovations with df = 2, number of Obs. = 100, number of simulated
paths = 5000:
Model: Gaussian MA(1)

> sqrt(mean((ma1+0.5)^2))
[1] 0.09509874
> sqrt(mean((ma2+0.5)^2))
[1] 0.09313703
> mean(ma1)
[1] -0.5223137
> mean(ma2)
[1] -0.5196037
> sd(ma1)
[1] 0.0924531
> sd(ma2)
[1] 0.09105966

where ma1 is the vector containing the estimated MA(1) coefficients from arima0 (exact likelihood) and ma2
from arma (CSS). Hence, it seems that the CSS provides in this example a marginally better estimator.


Dr. Adrian Trapletti        Phone:     +41 (0)1 994 56 31
Wildsbergstrasse 31         Fax  :     +41 (0)1 994 56 33
CH-8610 Uster               Email: a.trapletti at bluewin.ch
Switzerland                 WWW  :   trapletti.homeip.net

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