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

[Ravi Varadhan]

Comparing such disparate, non-nested models can be quite problematic.  I am not sure what AIC/BIC comparisons mean in such cases.  The issue of different constants should be the least of your worries.

Ravi

-----Original Message-----
From: r-devel-bounces at r-project.org [mailto:r-devel-bounces at r-project.org] On Behalf Of Jouni Helske
Sent: Tuesday, May 01, 2012 2:17 PM
To: r-devel at r-project.org
Subject: Re: [Rd] The constant part of the log-likelihood in StructTS

Ok, it seems that R's AIC and BIC functions warn about different constants, so that's probably enough. The constants are not irrelevant though, if you compute the log-likelihood of one model using StructTS, and then fit alternative model using other functions such as arima(), which do take account the constant term, and use those loglikelihoods for computing for example BIC, you get wrong results when checking which model gives lower BIC value. Hadn't though about it before, have to be more careful in future when checking results from different packages etc.

Jouni


On Tue, May 1, 2012 at 4:51 PM, Ravi Varadhan <rvaradhan at jhmi.edu> wrote:

> 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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