[Rd] The constant part of the log-likelihood in StructTS

[Prof Brian Ripley]

On 30/04/2012 12:37, Jouni Helske wrote:
> Dear all,
>
> I'd like to discuss about a possible bug in function StructTS of stats
> package. It seems that the function returns wrong value of the
> log-likelihood, as the added constant to the relevant part of the
> log-likelihood is misspecified. Here is an simple example:
>
>> data(Nile)
>> fit<- StructTS(Nile, type = "level")
>> fit$loglik
> [1] -367.5194
>
> When computing the log-likelihood with other packages such as KFAS and FKF,
> the loglikelihood value is around -645.
>
> For the local level model, the likelihood is defined by -0.5*n*log(2*pi) -
> 0.5*sum(log(F_t) + v_t^2/sqrt(F_t)) (see for example  Durbin and Koopman
> (2001, page 30). But in StructTS, the likelihood is computed like this:
>
> loglik<- -length(y) * res$value + length(y) * log(2 * pi),
>
> where the first part coincides with the last part of the definition, but
> the constant part has wrong sign and it is not multiplied by 0.5. Also in
> case of missing observations, I think there should be sum(!is.na(y))
> instead of length(y) in the constant term, as the likelihood is only
> computed for those y which are observed.
>
> This does not affect in estimation of model parameters, but it could have
> effects in model comparison or some other cases.
>
> Is there some reason for this kind of constant, or is it just a bug?
>
> Best regards,
>
> Jouni Helske
> PhD student in Statistics
> University of Jyväskylä
> Finland

I think you missed the following on the help page:

   loglik: the maximized log-likelihood.  Note that as all these models
           are non-stationary this includes a diffuse prior for some
           observations and hence is not comparable with ‘arima’ nor
           different types of structural models.

It is explicitly not valid for almost all model comparisons, and those 
few that are valid will use the differences of the quoted values.

Yes, it was an error, but the constant in log-likelihoods is always 
arbitrary and here there is even more indeterminancy.

-- 
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