Omited variable bias clarification

Question

Let’s say we have a true model for health that goes $h=c+bw+di+u$ where w is weight and i is income. Now this means that holding weight constant a one unit change in income on average causes a d unit change in health. Now let’s assume income and weight are correlated. We we regress health on only income we will get a larger coefficient estimate. Now this is generally referred to as omitted variable bias. But I don’t see where the bias is. Isn’t the coefficient just larger because we are not holding weight constant? Isn’t this model just as valid?

Edit: let me try to clarify my question. Let’s say due to leaving out weight in our regression our estimate for the income coefficient is too large due to OVB. Too large compared to what? The coefficient is correctly telling me what the difference in expected health is between two people if all I know is that they have a one unit difference in income? What is the correct coefficient supposed to tell me?

Answer 1

Note that omitted variable bias is only occuring when the regressors itself are correlated with eachother and the dependent variable. To answer your question: The coefficient is larger because the change in variation is unrightfully and indirectly attributed to the single regressor. I use the term indirectly, because the regression coefficient is inflated due to the fact that the omitted variable is correlated with the regressor.

Whether the model is valid, is based on interpretation and the objective of modelling. The model is valid if you are only interested in the impact of the single regressor. If not, the model suffers from omitted bias.