Regression: How to use it for causal inference?

1 The property of $\widehat{\beta}$

The following is always true: $$\widehat{\beta}_{k} =\frac{Cov(Y,\hat{x}_k)}{Var(\hat{x}_{k})}$$ where $\hat{x}_k$ is the residual of $x_{k}$ regressed on all the other covariates.

Particularly, for univariate regression $Y=\alpha+\beta x+\varepsilon$, we have $$\widehat{\beta}=\frac{Cov(Y,x)}{Var(x)}$$ However, we can’t guarantee $\widehat{\beta}=\beta$ in this general case.

2 When does $\widehat{\beta}=\beta$ ?

The key condition is $E(X\varepsilon)=0$, that is, $\varepsilon$ and $X$ are uncorrelated.

When $E(X\varepsilon)=0$ is met, not only do we have $\widehat{\beta}=\beta$, we can also guarantee:

To satisfy $E(X\varepsilon)=0$, we have several choices. If any of the following conditions are met, then we can guarantee $E(X\varepsilon)=0$:

• No omitted variables
• Even if we have omitted variables, prove that $X$ and the omitted variables are uncorrelated.
• $X’\beta=E(Y|X)$