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I understand that coefficient correlations can arise in the presence of correlated covariates, essentially indicating that our inference for those parameters is coming from information that cannot be disentangled, which in turns alters the shape of our confidence regions.

In the classic linear regression model I'm also aware that the intercept and the coefficient have a correlation that is bounded given the data, as mentioned in this question.

Therefore parameter correlations would seem to be a byproduct of the information-sharing restrictions imposed by a combination of the data and the model we're using, but have no relationship to any "real" feature of the phenomenon being studied? Is there any situation where inference on those correlations would be of primary interest?

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