How to test whether OVB by examining two regressors (X_1, X_2) using hypothesis test with null hypothesis H0: corr(X_1,X_2) = 0 Suppose you have an i.i.d. sample ${(_i , _{1,i} , _{2,i} ):  = 1, ... , }$. You want to estimate the causal effect
of $_1$ on $$. You first run a regression $_i = _0 + _1_{1,i} + _i$ and get the following result:
where the numbers in parentheses are standard errors.
Now, suppose you worry about omitted variable bias (OVB), so you are considering whether to put $_{2,}$ into the regression. The only condition you can check is that whether $_1$ and $_2$ are correlated. Propose a test with the null $H_0: (_1_2) = 0$ based on regression. Describe what regression you would run and what the test procedure would be. Explain why the proposed test works.
I know that OVB occurs when omitting a regressor (putting the regressor in the error term $u_i$ instead of putting as a new regressor $X_2$) that can affect $Y$ or $X_1$ and $X_2$ (other omitted variable) are correlated. However, how does testing whether correlation = 0 helps to know whether this has OVB or not? I don't understand
** I encounter another question, which is how to test whether it is significant? How to obtain $SE((_1_2))$? I was thinking about obtaining t-statistics and compare with 1.96 since I want alpha = 0.05
 A: Cause omitted variables have two properties, they are correlated to the variable of the interest and they affect Y. Thus to see whether X_2 is an omitted variable, we are supposed to clarify whether corr(X_1,X_2) = 0.
A: In his usual definition given in econometric books, OVB occurs when two conditions appears:

*

*the excluded variables ($X_2$ in your example) is correlated with at least one included ($X_1$ in your example).


*the excluded variables ($X_2$) is correlated with error term of the short model ($u_1$ in your notation).
then check if $\rho(X_1,X_2)\neq0$ it is not enough.
The most used way in order to deal with OVB is to run the short and long regressions. If the coefficient of the variable of interest change significantly between short and long regression we can conclude that the short one suffer of OVB and long regression is better. In some case the change in sign is impressive. At the opposite if we observe certain coefficient stability, short regression is better. In your example $1,2$ his the value of interest.
However it was shown that this strategy has several drawbacks.
In my view the first thing tha we have to note is that authors that talk about OVB, usually have in mind causal inference; in particular a confounding bias example. Indeed even in your example it seems the case.
In order to use regression for causal inference I suggest to change perspective. This my reply and refs therein can help: Under which assumptions a regression can be interpreted causally?
