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The linearity assumption says that the dependent variable is linear in parameters. When Gauss-Markov assumptions hold, OLS is BLUE, meaning smallest variance amongst Linear Unbiased estimators, where Linear means the OLS estimators, which produce estimates of the parameters for each sample, are linear functions of the dependent variable.

Does linearity assumption (y linear in parameters) restrict focus to linear estimators (parameter estimators linear in y)? Or the linearity assumption does not restrict focus to linear estimators, but rather a separate decision is made to look only at linear estimators?

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The L stands for linear estimators in $y$, i.e., estimators that may be written as $$ \tilde \beta=\sum_iw_iy_i $$ for some choice of $w_i$.

That said, if the underlying true model is nonlinear in the parameters (as, say, in a probit/logit model where the dependent variable is either 0 or 1), we can be pretty sure that OLS itself will not be unbiased, so that OLS does not even meet the "U"-requirement of (B)LU estimators.

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They're different.

It is valid to estimate the coefficients by using absolute loss: $L(y, \hat y) = \sum_i \vert y_i - \hat y_i\vert$. This results in a nonlinear estimate of $\beta$, yet the model is linear.

(This explicitly models the median of the conditional distribution, not the mean, but empirical median could be a perfectly reasonable way to estimate the population mean.)

If we have a nonlinear model, like a logistic regression, nothing keeps us from estimating the parameters linearly, though I have a hard time believing that any such estimate would be reasonable e.g., calculate the logistic regression $\hat\beta$ as you would in OLS: $\hat\beta = (X^TX)^{-1}X^Ty$. Your predictions might not be so good, but that is a linear estimator of $\beta$.

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