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Suppose you have an i.i.d. sample ${(π‘Œ_i , 𝑋_{1,i} , 𝑋_{2,i} ): 𝑖 = 1, ... , 𝑛}$. You want to estimate the causal effect of $𝑋_1$ on $π‘Œ$. You first run a regression $π‘Œ_i = 𝛼_0 + 𝛼_1𝑋_{1,i} + 𝑒_i$ and get the following result:enter image description here

where the numbers in parentheses are standard errors.

Now, suppose you worry about omitted variable bias (OVB), so you are considering whether to put $𝑋_{2,𝑖}$ into the regression. The only condition you can check is that whether $𝑋_1$ and $𝑋_2$ are correlated. Propose a test with the null $H_0: 𝜌(𝑋_1𝑋_2) = 0$ 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 $SE(𝜌(𝑋_1𝑋_2))$? I was thinking about obtaining t-statistics and compare with 1.96 since I want alpha = 0.05

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    $\begingroup$ Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. $\endgroup$ Dec 7, 2020 at 14:45
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    $\begingroup$ Thank you, I have just edited. I will make sure to do that before posting next time. $\endgroup$
    – gggg
    Dec 7, 2020 at 15:43

2 Answers 2

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In his usual definition given in econometric books, OVB occurs when two conditions appears:

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

  2. 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?

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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.

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  • $\begingroup$ I understand that part, but I encounter another question, which is how to test whether it is significant? How to obtain SE(corr(X1,X2))? I was thinking about obtaining t-statistics and compare with 1.96 since I want alpha = 0.05 $\endgroup$
    – gggg
    Dec 9, 2020 at 7:06

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