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We need to take some care with the notation because the models differ. Let the first (correct) model be $$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon\tag{1}$$ where the $\varepsilon_i$ have a common variance and zero means; and write the second model (which governs the very same variables $Y$, so no need to change their name) as $$Y = \alpha_0 +... 2 A reasonable approach for your problem is forward selection, a type of Stepwise Regression. You could fit a regression with just the A variables, and then compare this to the regression with the A,B, A,C, and A,D variables via F-test where, for example$$F_{A \; \text{vs.} \, A,B}=\frac{(SSE_A-SSE_{A,B})/(df_{A}-df_{A,B})}{SSE_{A,B}/df_{A,B}}. ...