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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