I am regressing a set of covariates on a dummy variable $Y_i$ taking the value of 1 if an event occurs and 0 otherwise. My linear probability model includes a triple interaction between two dummies $X_1,X_2$ and a continuous variable $X_3$.

It goes as follow:

$Y_i = \beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_3+\beta_4X_1X_2+\beta_5X_1X_3+\beta_6X_2X_3+\beta_7X_1X_2X_3$

I basically want to assess the impact of $X_3$ on $Y=1$ for different subpopulations. Result from the linear probability estimation gives:

$\beta_1 = 0.25^{***};$

$\beta_2 = 0.06;$

$\beta_3 = -0.002;$

$\beta_4 = 0.24;$

$\beta_5 = -0.06^{***};$

$\beta_6 = -0.02;$

$\beta_7 = -0.03;$

I am trying to understand the effect that the variable $X_3$ has on the probability that $Y=1$. From the result above I conclude that $X_3$ has no statistically significant impact on the probability that $Y=1$ for individuals for which $X_1=0$ and $X_2=0$, and for $X_1=0$ and $X_2=1$.

But, $X_3$ reduces the probability that $Y=1$ for the subpopulation for which $X_1=1$ and $X_2=0$ and for $X_1=0$ and $X_2=1$. Furthermore the impact of $X_3$ on the probability that $Y=1$ is similar for individuals for which $X_1=1$ regardless of whether $X_2=1$ or $X_2=0$.

I am wondering whether I should take into account the coefficients of $\beta_1,\beta_2, \beta_3$ in my interpretation and whether I am interpreting the results of my estimation in the right way.


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