Generalized Autoregressive Score (GAS) models, also known as Dynamic Conditional Score (DCS) models, are an important development. They extend significantly the scope of observation-driven models, with their simple closed-form likelihoods, in contrast to parameter-driven models whose estimation and inference require heavy simulation.

Many talented people are pushing things forward. Most notable are the Amsterdam group (Siem Jan Koopman et al.; see the GAS site) and the Cambridge group (Andrew Harvey et al., see Andrew's interesting new book). The GAS site is very informative, with background description, a catalog of GAS papers, code in Ox and R, conference information, etc. The key paper is Creal, Koopman and Lucas (2008). (It was eventually published in 2012 in

*Journal of Applied Econometrics,*proving once again that the better the paper, the longer it takes to publish.)

The GAS idea is simple. Just use a conditional observation density \(p(y_t |f_t)\) whose time-varying parameter \(f_t\) follows the recursion

\begin{equation}f_{t+1} = ω + β f_t + α S(f_t) \left [ \frac{∂logp(y_t | f_t)}{∂ f_t} \right ],~~~~~~~(1) \end{equation} where \(S(f_t)\) is a scaling function. Note in particular that the scaled score drives \(f_t\). The resulting GAS models retain observation-driven simplicity yet are quite flexible. In the volatility context, for example, GAS can be significantly more flexible than GARCH, as Harvey emphasizes.

Well, the GAS idea

*seems*simple. At least it's simple to implement if taken at face value. But I'm not sure that I understand it fully. In particular, I'm hungry for a theorem that tells me in what sense (1) is the "right" thing to do. That is, I can imagine other ways of updating \(f_t\), so why should I necessarily adopt (1)? It would be great, for example, if (1) were the provably unique solution to an optimal approximation problem for non-linear non-Gaussian state space models.

*Is*it? (It sure looks like a first-order approximation to

*something*.) And if so, might we want to acknowledge that in doing the econometrics, instead of treating (1) as if it were the DGP? And could we somehow

*improve*the approximation?

To the best of my knowledge, the GAS/DCS literature is silent on such fundamental issues. But based on my experience with the fine scholarship of Creal, Harvey, Koopman, Lucas, and their co-authors and students, I predict that answers will arrive soon.