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When you want to test whether explanatory variable x is endogenous (with y as dependent variable) , a test is to carry out an OLS regression with x as dependent variable and exogenous instruments z as independent variables. You then store the residuals v, and the carry out another regression with v as explanatory variable, alongside x, and y as dependent variable. When the coefficient of z is significant, this shows that x is endogenous. I don't understand why. What we want to see is whether cov(x, e) =/= 0. The regression described above only verifies that cov(y, v) =/= 0.
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