The central problem with all statistical information criteria is really the central problem with statistical models in general:
They aren't dynamical models.
If one admits that dynamical models are necessary(even if only implicitly*) for time series datacausal inference, one is immediately drawn to algorithmic information criteria. The gold standard algorithmic information criterion is approximating the Kolmogorov complexity of the data.
There are two bad habits in sociology that contribute to the failure to recognize this:
- Focusing on a single outcome variable.
- Failure to adopt the single-dataset multiple analyst approach to contain the worst practices of sociology.
#1 blocks one from considering dynamical models because single variable dynamical systems are practically useless.
#2 is related to #1 in that even when a single-dataset is provided to multiple analysts, it is generally within the context of predicting a single variable of interest. But if, rather than constructing a new dataset for each new analysis, the community were to curate a "single dataset" of high quality, involving many inter-related social measures, it would become apparent that dynamical models were the only way to predict not just single measures, but all measures of interest.
That this is still not recognized in sociology, despite the emergence of state space models in machine learning, is a clear indictment of its academic culture.
* Although dynamical systems are dependent on time series data, that is too-restrictive since causal inference is often if not usually attempted without explicit time series data. In causal analyses time dependence is assumed even if not imputed as a latent variable. The inferred causal processes generating the data under algorithmic information (embodied, say, in a state space model) must, of course, be explicit.