Question

What is the interpretation of the coefficient on the covariate of interest (the endogenous one) in an instrumental variable regression?

Answer 1

In regression analysis, an instrumental variable is used when there is/are endogenous variable/s in the regression model especially implying that the dependent variable in the model might be influenced by or associated with other variable/s through it's/their association with the explanatory or independent variable/s. In other words, instrumental variables or IVs can better explain the variation in the dependent variable in the regression model through their statistical association with the independent or explanatory variables. Therefore, IVs do not have any direct statistical association with the dependent variable but is related to the dependent variable though their statistical association or correlation with the explanatory or independent variables. The IV is uncorrelated with the error term and statistically correlated with endogenous explanatory or independent variable in the regression model. Hence, the coefficient estimate of the IV in any regression model essential shows the overall or subsequent statistical relationship between the IV and the dependent variable by taking into consideration or account the statistical association between the IV and the endogenous independent variable, controlling for the exogenous independent variables in the model. Thus, let's suppose that Y and X are the dependent and the endogenous independent variables in any regression model and Z be the IV in this case, controlling for other explanatory variables. Therefore, the coefficient causal estimate   of IV or (IV) in the model=(dY/dZ)/(dY/dX) which shows the the slope coefficient of Z with respect to Y over the coefficient estimate of X with respect to Y signifying that the coefficient estimate of IV basically shows the statistical relation between the IV and the dependent variable through the statistical association between the IV and X in any regression model.