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

[Ravi Varadhan]

This is not a problem at all.  The log likelihood function is a function of the model parameters and the data, but it is defined up to an additive arbitrary constant, i.e. L(\theta) and L(\theta) + k are completely equivalent, for any k. This does not affect model comparisons or hypothesis tests.

Ravi
________________________________________
From: r-devel-bounces at r-project.org [r-devel-bounces at r-project.org] on behalf of Jouni Helske [jounihelske at gmail.com]
Sent: Monday, April 30, 2012 7:37 AM
To: r-devel at r-project.org
Subject: [Rd] The constant part of the log-likelihood in StructTS

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

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