# For OLS to be unbiased, do we need $x_i$ to be uncorrelated with $\epsilon_i$ or with $\epsilon_s$ for all $s$?

In some textbooks I've read, it is said that an assumption for OLS to be unbiased in the standard cross-sectional model $y_i=\alpha + \beta \cdot x_i +\epsilon_i$, we can use the assumption $E(\epsilon_i|x_1,...x_n)=0$.

Do we need this, or is OLS already unbiased if just $E(\epsilon_i|x_i)=0$ holds (in addition to the other required assumptions)?

If so, is there a particular reason why the textbooks would want to talk about the stronger assumption $E(\epsilon_i|x_1,...x_n)=0$?

• See stats.stackexchange.com/questions/323695/… or stats.stackexchange.com/questions/240383/… (so in fact you asked the question before). Apr 27, 2018 at 14:35
• @ChristophHanck, no that was a different question. This is specifically about correlation between the error and $x$ of different observations Apr 27, 2018 at 15:06
• After your edit, yes, sort of, but not before...and still not in the title of the question, because what you now state is mean independence, not uncorrelatedness. Apr 27, 2018 at 15:08
• @ChristophHanck, the question is about the difference between the relation between the error and the regressor of different observations vs the same observation, not specifically about the difference between correlation and mean independence (I understand the latter already, due to that previous question). Apr 27, 2018 at 15:21

## 1 Answer

If one takes your statement of a "standard cross-sectional model" to imply that we can take a random sample from the underlying population, then $$E(\epsilon_i|x_1,\ldots,x_n)=E(\epsilon_i|x_i)$$ because the random sampling/iid assumption implies that $\epsilon_i$ and $x_j$ are independent for $i\neq j$. Hence, $E(\epsilon_i|x_i)=0$ is enough to guarantee unbiasedness when coupled with an iid assumption.

As explained in the answers the comments link to, the iid assumption fails in, e.g., time series applications so that, in general, $E(\epsilon_i|x_i)=0$ is not enough to guarantee unbiasedness: e.g., in an AR(1) model, $E(\epsilon_i|x_i)$ corresponds to $E(\epsilon_i|y_{i-1})$. Under standard assumptions (e.g., that of $\epsilon_i$ a pure "innovation" independent of what happened in the past), $E(\epsilon_i|y_{i-1})=0$ and yet OLS is not unbiased.

• So if we assume a time series instead, would $E(\epsilon_i|x_1,...x_n)=0$ be enough, or would we additionally need that they are $i.i.d.$? Apr 27, 2018 at 15:22
• The first assumption yields unbiasedness, but as my other answer tries to explain, that assumption cannot work in a "meaningful" time series framework where the past affects the present and the present hence affects the future. When you assume iidness in a time series framework, you essentially do not have a time series setup anymore, as the different time periods are unrelated to each other. Apr 27, 2018 at 15:33
• There may be data that is ordered over time (like stock returns) that does not exhibit any dependence (in levels - for higher moments, see e.g. GARCH models), but this very fact immediately makes it uninteresting to fit a, say, AR(1) model to stock returns via OLS (or some other method). Apr 27, 2018 at 15:35