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Suppose, I estimate a simple linear probability model:

$P(Y=1)=\beta_0+ \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 \times X_2 + u$,

where $Y$, $X_1$, and $X_2$ are dummy variables. All standard OLS assumptions hold. Further assume that I can reject the hypotheses: $H_0: \beta_1 =0 $ and $H_0: \beta_3 =0$, using simple $t$-tests. My question: What is the correct way of calculating the predicted values for $E(Y|X_1=1, X_2=1)$? Should I take $\widehat{\beta}_2$ into account?

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Yes, you should. Just because you fail to reject, does not mean that $\beta_2$ is actually zero. The data is merely consistent with that, but it might just be hard to detect the effect, given the noise and the amount of data.

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