Very often in seminars people compare the (biased because of endogeneity) results of their OLS estimation with those (unbiased) from an IV strategy estimation. Assuming everything is ok with the IV assumptions, my question focuses on what we can learn about the omitted variable bias from comparing the IV and the OLS estimate. I think that it is correct to say (but I'm not sure) that if the OLS estimate is bigger than the IV one it means that the omitted variable is positively (or negatively) correlated with both the outcome variable and the regressor of interest: so through that channel, the true effect of X on Y is amplified and the OLS cannot disentangle this. Viceversa, if the IV is bigger than the OLS it means that the omitted variable is correlated in opposite direction with X and Y (i.e. positively with X and negatively with Y, or the other way around). In this case the true effect of X on Y is attenuated by the omitted variable and again the OLS cannot see this. However, I was wondering whether this is true only if the true effect of X on Y has a positive sign. If it has a negative sign, should all this be the other way around?
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$\begingroup$ Except under very strong assumptions on your instrumental variable, $\beta_{IV}$ and $\beta_{OLS}$ do not target the same estimand, thus will differ for various reasons including (but importantly not limited to!) the omitted variable bias. $\endgroup$– DurdenCommented May 15 at 19:28
1 Answer
The first point to be made is that IV is biased, but consistent. Only with a large sample size can we treat IV estimates as the truth.
Your discussion is largely correct.
Let the OLS and IV estimates be $\widehat{\beta_{OLS}}$ and $\widehat{\beta_{IV}}$ and let us suppose $\widehat{\beta_{IV}}$ is "the truth".
If $\widehat{\beta_{IV}}>\widehat{\beta_{OLS}}$, then OLS has negative bias. If the bias is due to an omitted confounder, then it is because the confounder is positively correlated with the outcome and negatively correlated with the treatment (or vice-versa). To answer your question this is true regardless of the sign of $\widehat{\beta_{OLS}}$. As an example, $\widehat{\beta_{IV}} = 3>\widehat{\beta_{OLS}} = -2$ has OLS with negative bias and $\widehat{\beta_{IV}}=3>\widehat{\beta_{OLS}}=1$ has OLS with negative bias.
If $\widehat{\beta_{IV}}<\widehat{\beta_{OLS}}$, then OLS has positive bias. If the bias is due to an omitted confounder, then it is because the confounder has the same sign of correlation with both the outcome and treatment (the correlations can both be positive or both be negative). This is again true regardless of the sign of $\widehat{\beta_{OLS}}$.
