In econometrics, it is common to work with the difference-in-differences of conditional means. For example, let $Y$ denote a variable of interest and $X_{1}$ and $X_{2}$ denote binary regressors. The difference-in-differences is given by $\mathbb{E}\left ( Y|X_{1}=1,X_{2}=1 \right )-\mathbb{E}\left ( Y|X_{1}=0,X_{2}=1 \right )-(\mathbb{E}\left ( Y|X_{1}=1,X_{2}=0 \right )-\mathbb{E}\left ( Y|X_{1}=0,X_{2}=0 \right ))$.
Suppose there is a third binary regressor $X_{3}$. The triple-difference is given $\left \{ \mathbb{E}\left ( Y|X_{1}=1,X_{2}=1,X_{3}=1 \right )-\mathbb{E}\left ( Y|X_{1}=0,X_{2}=1,X_{3}=1 \right )-(\mathbb{E}\left ( Y|X_{1}=1,X_{2}=0,X_{3}=1 \right )-\mathbb{E}\left ( Y|X_{1}=0,X_{2}=0,X_{3}=1 \right )) \right \}\\-\left \{ \mathbb{E}\left ( Y|X_{1}=1,X_{2}=1,X_{3}=0 \right )-\mathbb{E}\left ( Y|X_{1}=0,X_{2}=1,X_{3}=0 \right )-(\mathbb{E}\left ( Y|X_{1}=1,X_{2}=0,X_{3}=0 \right )-\mathbb{E}\left ( Y|X_{1}=0,X_{2}=0,X_{3}=0 \right )) \right \}$
I am looking for a general expression for the $t$-th difference. So far I came up with
$\sum_{\textbf{x}_{1:t}\in\left \{ 0,1 \right \}^{t}}(-1)^{\prod_{i\in\left \{ 1,...,t \right \}}(x_{i}+1)}\mathbb{E}\left [ Y|\textbf{X}_{1:t}=\textbf{x}_{{1:t}} \right ]$. But I am still looking for a more simplified expression. Any suggestions?