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I think the best way how to explain this is to first quickly explain what identification actually is. As mentioned in this thread: For example, in the John Stachurski, a primer in econometric theory the identification is a process of finding out if the parameters are identifiable and identifiability is defined as “Identifiability means that the parameter ...

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Yes, it is because they are estimated jointly and the point estimates are conditional on other point estimates and also because of omitted variable bias. To be more specific, contrast an example of univariate OLS with bivariate OLS, both of which are estimated by minimizing the sum of squared residuals, but resulting formulas for estimated coefficients are ...

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If you want to explain the fraction of variance explained, use $R^2$. If you want to compare $R^2$'s but adjust (very weakly) for model size, use adjusted $R^2$'s. If you want to compare these models, use an $F$-test since Model 1 is nested in Model 2.

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For the above equation is differentiable, and hence the partial derivatives will exist: $\frac{\partial Y}{\partial X_2}= 2\beta_2X_2$. Adding a dummy variable should not affect the above derivative for all practical purposes - the function will be differentiable a.e. (note that you can't differentiate with respect to a dummy variable though) Edit: After the ...

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