The Least Squares Assumption $E[u_i|X_i] =0$

Question

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?

Answer 1

If the true population regression function is nonlinear but the estimated regression is linear, then this functional form misspecification makes the OLS estimator biased. This bias is a type of omitted variable bias in which the omitted variables are the terms that reflect the missing nonlinear aspects of the regression function. For example, if the population regression function is a quadratic polynomial, then a regression that omits the square of the independent variable would suffer from omitted variable bias. To sum up, the estimator of the partial effect of a change in one of the variables will, in general, be biased.

Answer 2

Transformation of the original variables may be in order. What one wants to see ideally is not only linearity, but also either homoscedasticity and/or normal distribution of residuals. Try taking logarithms of the IV, other transformations include in general both or either axis taking of logarithms, roots and powers (e.g., square roots), exponentiation, reciprocals, and offsets of these ($x-k$, or $y-k$ where k is some constant).

Answer 3

The modeling assumption about $u_i$ in linear regression is $E(u_i)=0$ (you don't have to condition on $X_i$). Now the scatter plot you showed indicated that the function might involve a quadratic term in $X_i$ and hence the model that is linear in $X$ is misspecified. $E(U_iX_i)$ maybe 0 or it could be some other value when the model is misspecified. Because of the missing term(s), the slope parameter for $X$ is biased. I would suggest including a quadratic term in $X$ and seeing how the model and residuals look.