  For model-specification purpose, we study asymptotic behavior of the marginal
quasi-log likelihood associated with a family of locally asymptotically
quadratic (LAQ) statistical experiments. Our result entails a far-reaching
extension of applicable scope of the classical approximate Bayesian model
comparison due to Schwarz, with frequentist-view theoretical foundation. In
particular, the proposed statistics can deal with both ergodic and non-ergodic
stochastic-process models, where the corresponding $M$-estimator is of
multi-scaling type and the asymptotic quasi-information matrix is random.
Focusing on the ergodic diffusion model, we also deduce the consistency of the
multistage optimal-model selection where we may select an optimal sub-model
structure step by step, so that computational cost can be much reduced. We
illustrate the proposed method by the Gaussian quasi-likelihood for
diffusion-type models in details, together with several numerical experiments.
