4.5 Decomposition

The linear regression is estimating the conditional mean. The conditional mean function is written as:

\[\begin{equation} E(Y_i|X_i) = \alpha + \beta X_i \tag{4.2} \end{equation}\]

The above equality directly comes from the underlying assumption \(E(u|X)=0\), or the independence of error term once conditioned on \(X\).

We can rewrite the regression specification as:

\[\begin{equation} Y_i = \underbrace{E(Y_i | X_i)}_{conditional \; expectation} + \underbrace{\epsilon_i}_{unobserved \; component} \tag{4.3} \end{equation}\]

Here, we’ve decomposed the regression specification into two parts: \(i)\) conditional mean; and \(ii)\) unobserved component.