From: Ravi Varadhan <rvaradhan_at_jhmi.edu>

Date: Tue, 01 May 2012 13:51:07 +0000

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

R-devel_at_r-project.org mailing list

https://stat.ethz.ch/mailman/listinfo/r-devel Received on Tue 01 May 2012 - 13:55:40 GMT

Date: Tue, 01 May 2012 13:51:07 +0000

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.

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

> 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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