Omitted Variable Bias for linear regression - Sign of coefficient changes.

Question

My question relates to determining the direction of bias when the regression coefficient changes sign (from negative to positive) however the absolute value is smaller in the new estimate.

The original simple linear regression model gives a coefficient $\beta1 = -0.31$.

After including an omitted variable with coefficient $\beta2 = 0.07$, our original coefficient changes to $\beta1 = 0.12$.

We are asked to determine whether or not this change suggests a positive or negative correlation between our two explanatory variables.

Of course, we are given that $\beta2 > 0$. In order to determine whether the cov(x1,x2) is positive or negative, we must determine whether our original estimate was an overestimate (positive bias) or an underestimate (negative bias).

Since the absolute value of the estimator decreases after the introduction of the omitted variable, I am inclined to say that our original was an overestimate (i.e. positive bias) and that x1 and x2 are positively associated.

However, since the sign changed from negative to positive there is reason to believe that our original estimate was indeed an underestimate (i.e. negative bias) and that x1 and x2 are negatively associated.

Answer 1

I am not an econometrician, but assuming $\beta_{1}=0.12$ and $\beta_{2}=0.07$ is the correct model, in the incorrect model $\beta_{1}=-0.31$ was 'sucking up' some of the predictive power of $\beta_{2}$. If they were positively correlated, then the original $\beta_{1}>0.12$. If they were negatively correlated, then the original $\beta_{1}<0.12$.

Also, I am not sure what you mean by the original estimate being over- or under-estimate. You have $\beta_{1}=-0.30$ changing to $\beta_{1}=0.12$, so the original was underestimate?