Can someone explain more clearly what is a measurement error and how is it different from omitted variable. I know the theoretical implications, but I don't really know how to tell which problem I'm dealing with, in a context, for example, using education to explain wage. We can have ability, which is correlated with education, but it can't be measured. Is this a measurement problem? Why not omitted variable? Thank you!
I don't know if I'm the best to explain this but I'll give it a go. A measurement error is, for instance, when the person was supposed to code you as LFG in a dataset, but coded you as GFL. So then the data does not reflect the true 'value'.
An omitted variable, in a regression is a variable that is important to the equation, but that you've left out. Say you think gender affects wage. If you ran a regression, and omitted (left out) education, you would have an omitted variable bias problem in your estimation.
Granted there are many things in this kind of wage equation that are omitted, perhaps age, your line of work, where you live, etc. Often omitted variable bias refers to something that you can't really quantify. Say motivation also has an impact on your wage. But how would you add motivation? In your example, ability is something that can't be measured. You didn't measure it, so there was no measurement error. Running the regression to find out what effect gender and education have on wage, without including unobservable variables (motivation and ability) leads to omitted variable bias.