mutiple linear regression, omitted variable bias and causal effect

Question

1. I am struggling with the meaning of MLR, from my original understanding, the MLR intended to include more independent variables to make better predictions(independent variables could be either factor or continuous variables). Whereas from the Econometrics textbook, they call solving omitted variable bias by grouping data, this is adding the factor to the model. So isn't continuous independent variables resolve omitted variable bias?

2. In detecting the causal effect model, how does it differ from ACOVA? Suppose we investigate the $Sex$(factor with 2 levels) affecting the $Earnings$, with proper randomisation assigned

$$ Earnings =\beta_0+\beta_1Sex+\beta_2Edu+\sum_{i=1}\beta_ivariables_i $$

Isn't the $\beta_1$ capture the causal effect of $Sex$?

Answer 1

1. So isn't continuous independent variables resolve omitted variable bias?

Not necessary. If all the variables that you added, either continuous or categorical, are the all possible confounders - then yes, you "control" for all possible confounders and the OLS of $\beta_1$ is asymptotically unbiased. Otherwise - no. In practice, you can never know what are the all possible confounders.

2. Isn't the $β_1$ capture the causal effect of Sex?

As was mentioned in the comments - no. You cannot randomized sex as in RCTs. Even if we assume that the sex of the child is something random, the sex can still effect various other variables (including education), namely its effect maybe mediated through other variables. Therefore, $\beta_1$ is not the causal effect of sex on earnings. If there are no omitted confounders, then it maybe the direct effect of sex on earnings. Otherwise (with omitted variables) it is just a measure of association in the estimated model.