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A standard, even if not strictly mandatory, assumption in OLS regression is that the error term is Gaussian.

$$y = X\beta + \epsilon$$

$$\epsilon \sim N(0, \Sigma) $$

We also talk about the regression as predicting the mean of some distribution.

What restrictions are there on the error term for us to be estimating a conditional mean?

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  • $\begingroup$ One simple answer is this: When you apply least squares to simple one-way ANOVA data (regardless of any assumptions), the ordinary sample averages for each group are obtained. $\endgroup$
    – BigBendRegion
    Commented Sep 9, 2020 at 19:29

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Applied on a data sample, a linear regression estimated by OLS minimizes the sum of squared errors. In population, the sum of squared errors is minimized by the conditional expectation. Under assumptions that guarantee predictive consistency of the OLS estimator, OLS regression will be estimating the conditional mean. The assumptions can be found in this answer by Michael.

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