Here is my understanding of the linear regression model with single regressor: We suppose that the population regression function takes the form of $Y_i=\beta_0+\beta_1X_i+u_i$. Furthermore, for the parameters to have causality meanings, we assume that:
- $(X_i,Y_i)$ are i.i.d. for $i=1,...,n$
- Large outliers are unlikely
My first question is about the first assumption. It, combined with the assumption that $Y=\beta_0+\beta_1X$, gives $E(Y|X)=\beta_0+\beta_1X$. But isn't this equation the whole starting point of our regression analysis? If not, what are we trying to capture in the first place?
It seems like either $E(u_i|X_i)=0$ or $E(Y|X)=\beta_0+\beta_1X$ combined with the assumption that $Y_i=\beta_0+\beta_1X_i+u_i$ implies the other. Which comes first?
My second question is about the expectation of $u$. Do we assume that $E(u_i)=0$ or is it implied by other assumptions? Because in Wooldridge's Introduction to Econometrics, it's an assumption but in Stock and Watson's, it looks like it's implied. I'm confused here.