[R] Estimation of AR(1) model by QML.

[Prof Brian Ripley]

On 30/12/2013 22:57, Simon Zehnder wrote:
> Why not using optim on the likelihood in a) with normally distributed standard errors and for b) optim with a likelihood with t(3)-distributed standard errors?

Because evaluating the likelihood is a tricky business here.

I don't know what is meant by 'QML' here (even if the Q means 'quasi', 
the term is used in several distinct senses, none common for time-series).

Suppose you observe (Y(1) ... Y(T)).  Note that Y(1) depends on the 
unobserved Y(0).  In the Gaussian case you can write down the joint 
multilvariate normal distribution and compute it with some matrix 
algebra (assuming stationarity, which means a constrained optimization). 
  But in case b) the joint distribution is not multivariate T.

An approach from 50+ years ago is to condition on Y(1), when this is 
just a regression.  For the Gaussian case, see ar.ols(): for the t(3) 
case you can use a robust regression function (but beware that the 
optimization can be tricky).

> Best
>
> Simon
>
> On 30 Dec 2013, at 21:19, Xuse Chuse <chuse22 at gmail.com> wrote:
>
>> Dear Users,
>>
>> I am trying to estimate a model Y(t)=alpha+rho*Y(t-1)+e(t) where i know
>> e(t)~t(3).
>>
>> a) I want to estimate (alpha, rho) by QML estimation assuming (wrongly)
>> that e(t)~N(0,sigma2) and calculate the standard errors.
>> b) Estimate (alpha, rho) by ML estimation assuming (correctly) e(t)~t(3)
>> and compute standard errors.
>>
>> Can anyone help me out to figure out how I could answer this question in R?
>> Thank you beforehand.
>>
>> Cheers,
>> Chuse

-- 
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 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595