[nerdponx]

That the linear model is "correct" is only assumption if you're trying to draw probabilistic inferences.

There's nothing stopping you from using it as a "best fit line", even when you have no reason to believe those assumptions. But then it's just a best-fit line. It tells you the direction and magnitude of linear trend, nothing more. That's never wrong in any sense, it's just that sometimes it's not very useful.

[smu3l]

From the semiparametric perspective, you can still make correct inferences about the estimated parameters even if the model is not correctly specified, as long as you use the so-called robust estimator of the variance.

[nonbel]

This is impossible. If the model is incorrectly specified (does not include all and only the relevant parameters and interactions), it doesn't matter much what games are played with the math. Changing the model will change the estimates...

Edit: For example, see here where making arbitrary choices of how to code categorical variables will change the estimates: https://news.ycombinator.com/item?id=16719754

If you change the model the meaning of all the coefficients changes.

[smu3l]

Oops I did not see your response until now.

I agree, changing the model changes the estimates, because the parameters you are estimating change.

However, given one misspecified model, the parameters of that model are still well defined, though they may not have the interpretation they would if the model was correctly specified. As OP called it, this is the "best fit line", and is a projection of the truth onto your model. E.g. for a simple linear regression of Y on X, where the true conditional mean of Y given X is not linear, there is still some "true" best line. This line depends also on the distribution of X, though it would not if the model was correct. Estimates from linear regression will converge to the parameters of this line, though using the usual standard errors will be wrong.

There's a very general theorem or corollary that covers this in Asymptotic Statistics by van der Vaart. I think in the chapter about M estimators, right around where MLEs are covered, but I don't have it in front of me.

[nonbel]

There are multiple inference levels here.

First, there is the statistical level, at which we are drawing some conclusion about the model parameter. This may work even for a misspecified model.

Then there is the level at which you want to draw some conclusion about reality, call it the "scientific level". If the model is misspecified, the parameters/coefficients may or may not correspond to the thing of interest. Perhaps the model is a close enough approximation for those values to be meaningful, perhaps not...

I think it is the second ("scientific level") of inference that most people are concerned about. The rigor of the proofs/theorems that may work at the statistical level does not extend to the scientific level.

Afaict, the majority of erroneous inference occurs at the scientific level and statistical error/uncertainty is a sort of minimum error/uncertainty.

[gmfawcett]

Yes, well put!