Regarding the starting point and assumptions of linear regression model

Question

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:

1. $E(u_i|X_i)=0$
2. $(X_i,Y_i)$ are i.i.d. for $i=1,...,n$
3. 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.

Answer 1

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?

The two are exchangeable. You can either start from: $$ Y_i = \beta_0 + \beta_1X_i + u_i \tag{1} $$ and use this together with $\mathbb{E}(u_i|X_i)$ to get $$ \mathbb{E}(Y_i|X_i) = \beta_0 + \beta_1 X_i. \tag{2} $$ Vice versa if you start from $(2)$ you can define: $$ u_i = Y_i - \mathbb{E}(Y_i|X_i). $$ such that $Y_i = \beta_0 + \beta_1 X_i + u_i$ and: $$ \begin{align*} \mathbb{E}(u_i|X_i) &= \mathbb{E}(Y_i|X_i) - \mathbb{E}(\mathbb{E}(Y_i|X_i)|X_i),\\ &= 0 \end{align*} $$ So it does not matter. Either you say that the expected mean of $Y$ conditional on $X$ is linear, or you say that $Y$ is linear in $X$ and $u_i$ together with the assumption that the mean of $u_i$ conditional on $X$ is zero.

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.

In order to see what happens if we do not impose it, notice that in this case, we may write: $$ u_i = \mathbb{E}(u_i) + \underbrace{(u_i - \mathbb{E}(u_i))}_{\delta_i} $$ Now, $\delta_i$ has mean zero, $$ \mathbb{E}(\delta_i) = \mathbb{E}(u_i) - \mathbb{E}(u_i) = 0. $$ We can now write: $$ \begin{align*} Y_i &= \beta_0 + \beta_1 X_i + u_i,\\ &= \left(\beta_0 + \mathbb{E}(u_i)\right) + \beta_1 X_i + \delta_i \end{align*} $$ So if the mean of $u_i$ is not zero, it simply get's added to the intercept $\beta_0$. In other words, if you add an intercept to the regression, assuming that $\mathbb{E}(u_i) = 0$ is without loss of generality.