I am currently writing a registered report for data which will be clustered within eight countries. Since that is too few to do a multilevel model with random effects (McNeish & Stapleton, 2016), I've chosen a fixed effects approach, where the country clusters are modelled as fixed effects (dummy variables) and the standard error is is multiplied by the square root of the unconditional design effect (McNeish & Kelley, 2019; McNeish & Stapleton, 2016). I've already set up a Monte Carlo simulation to do a power analysis for such a model.

However, one there is one problem. One analysis in my study will be to check for the effect of one variable while controlling for another variable, and there is reason to assume both would be correlated. Some degree of multicollinearity is of course fine, but I'm worried about what to do if it gets too substantial. I cannot drop one of the variables, since they both need to be there for theoretical reasons. In regular multiple regression models, a typical recommendation is ridge regression - but to do my fixed effects model, I would have to obtain p values by first multiplying the standard error with the square root of the unconditional design effect, and standard error estimates in penalised models seem to be an issue, with statistical software often not containing functions to obtain them because they wouldn't be very meaningful anyway. So, I'm a bit stuck on how to proceed here - do I use ridge regression and obtain the standard error estimates anyway (and how do I then obtain p values from this)? Is there another approach to dealing with multicollinearity that's more suitable to this use case than ridge regression? Or should I give up and accept that a ridge regression fixed effects model simply doesn't work?

McNeish, D. M., & Kelley, K. (2019). Fixed effects models versus mixed effects models for clustered data: Reviewing the approaches, disentangling the differences, and making recommendations. Psychological Methods, 24(1), 20–35. https://doi.org/10.1037/met0000182

McNeish, D. M., & Stapleton, L. M. (2016). Modeling clustered data with very few clusters. Multivariate Behavioral Research, 51(4), 495–518. https://doi.org/10.1080/00273171.2016.1167008