Please help me with this question, thank you! Show solutions please! Omitted var

Question

Please help me with this question, thank you! Show solutions please!

Omitted variable bias (OVB) is a very important issue to understand in econometrics. Take a look at this short video on OVB. https://www.youtube.com/watch?v=6I1tUM0RB6I It is useful and at an introductory level. Consider the regression we did in class and is in your textbook, looking at test scores across a large random sample of students. However, the effect that this regression is after is about the difference public vs. private schools have: T estScorei = 0 + 1P rivatei + ui .

Do you think that 1, suffers from OVB? Give an example of key variables that you think might be omitted. Pretend that only 1 variable is omitted, the F amilyIncome. Make an educated guess of the correlation between F amilyIncome and P rivate. Defend the rationale for your assumption.

Now,

Explanation / Answer

Yes, 1 suffers from OVB, since there are other key variables that might be impacting the value of 1. A great example would be the funding of the school, location of the school, family income of students (as you’ve already mentioned) etc.

The correlation between Family Income and Private is bound to be positive, i.e. higher the family income more likely the family has sent their child to a private school. This is because private school fees are a lot higher than public schools, hence it is natural to assume that wealthier families would send their kids to private schools.

Now let’s look at OVB

The regression equation that was given is:

Test scores = Yi

Private = 1 ( - 0, if public)

Equation after regression:

Yi = + 1 * Privatei + ui

Due to OVB, the actual true regression should be:

Yi = + b1 * Privatei + + b2 * FamilyIncomei + ui

We already know from OLS, that:

1 = Cov(Privatei , Yi) / V(Privatei)

1 = Cov(Privatei , + b1 * Privatei + + b2 * FamilyIncomei + ui) / V(Privatei)

1 = Cov(Privatei , ) + + b1*Cov( Privatei, Privatei ) + b2*Cov( FamilyIncomei , Privatei ) + Cov(Privatei, ui)

/ V(Privatei)

1 = 0 + b1 * Var( Privatei) + b2*Cov( FamilyIncomei , Privatei ) + 0 / V(Privatei)

1 =   b1 + b2*Cov( FamilyIncomei , Privatei ) / V(Privatei) ………………………………1

Correlation between Family Income and Private Schools = Family Income, Private

Family Income, Private = Cov( FamilyIncomei , Privatei ) / (V(Privatei) * V(FamilyIncomei))

Cov( FamilyIncomei , Privatei ) / V(Privatei) = Family Income, Private * V(FamilyIncomei) / (V(Privatei)

Hence, equation 1 can be written as….

1 =   b1 + b2* Family Income, Private * V(FamilyIncomei) / (V(Privatei)

Given that the correlation Family Income, Private is positive, the direction of bias is going to be positive.

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