My question is that does endogeneity exists if there is correlation between dependent variable and error term, but not in between error term and independent variable. So for Ex, we know there should be a positive correlation between x1 and y because of previous studies, but because, we didn't take a critical variable x2 into our regression model, the correlation between x1 and y is coming negative, this could be there because x2 is overpowering the effect of x1 to a great extent and hence the coefficients of x1 in our simple regression model would be biased but this doesn't imply that there is a correlation between x1 and x2, our 2 independent variables. This is a grave threat. So does this claim endogeneity?I have read that if the OLS Assumptions are satisfied, then it is the best estimator(both unbaised and minimum variance) of the coefficient of the independent variable. But then if you say this is not endogeneity, then all assumptions are satisfied for this regression model, and then too i have a biased coefficient in my regression model.
OLS Assumption-No correlation should be there between error term and independent variable and error term and dependent variable
No correlation between independent variable and error term is admissible definition, even if mean independence is more correct, but no correlation between dependent variable and error term is completely incorrect assumption. It is an absurd request. Error term is, by construction, a part of dependent variable, then, they must be dependent.
My question is that does endogeneity exists if there is correlation between dependent variable and error term, but not in between error term and independent variable
exactly the opposite is true.
The rest of the story sound like omitted variables problem. For discussion read here: https://en.wikipedia.org/wiki/Omitted-variable_bias
note that in this situation a correlation between $x_1$ and $x_2$, your notation, is implied. Maybe you are interested in endogeneity problems. For discussion read here:
finally note that endogeneity not always is crucial assumption in regression; read here: