Assume the following linear relationship: $Y_i = \beta_0 + \beta_1 X_i + u_i$, where $Y_i$ is the dependent variable, $X_i$ a single independent variable and $u_i$ the error term. According to Stock & Watson (Introduction to Econometrics; Chapter 4), the first least squares assumption is $E[u_i|X_i]=0$.
The following figure shows the estimated relationship:
Obviously, the relationship between $Y$ and $X$ is a nonlinear. As a result, the average of the estimated residuals $\widehat{u}_i$ is not necessarily zero for each $X_1$ value. I have two questions:
(i) Does this imply that $E[u_i|X_i] \neq 0$? If not, why is this the case?
(ii) Is the OLS estimator of $\beta_1$ always biased, if the population relationship is nonlinear. Why is this the case? What is the formal condition?