ENH: oulier-influence for misspecified models ?

[josef-pkt]

For now just a question: How does model misspecification affect outlier-influence measures?

possible case, scale misspecification
based on #7951 (comment)

Under full MLE, scale=1, e.g. Poisson or HetModels. Any dispersion/variance parameters are part of the estimated parameters.
However, the model could be misspecified, either in scale, e.g. with over dispersion, or in any of the robust cov_types, HC, ....

GLM, OLS have scale as auxiliary parameter in continuous endog case, but it's not part of params and needs to be treated separately.

GLM allows excess dispersion, scale != 1, also in the full MLE case like Poisson, Binomial.
This affects the studentized residuals in the Influence classes.

more general:
How should we treat misspecified models, models that use robust cov_type or excess scale for wald inference?

Currently. there is no systematic treatment of sandwich cov_params or other misspecification in the Influence classes.

Related: Should we add a cov_type="excess-scale" generically. It applies only to models that does not estimate scale directly.

[josef-pkt]

possible case for looking into this: overdispersed Poisson

Compare GLM or MLE influence under two scale assumptions

• MLE: scale fixed at 1, resid_pearson are not scaled (are scaled by 1)
• Pseudo-MLE: with pearson_chi2 scale,
  • resid_pearson will be fully scaled.
  • parts that rely on bse/cov_params will be scaled.
  • parts that rely on hessian, or score will not be scaled