Correlation between two interaction terms I'm working on a project involving regression models with multiple interaction terms. An example regression would be:
$Y$ ~ $A$ + $B$ + $C$ + $D$ + $AB$ + $CD$
where $AB$ = $A*B$ and $CD$ = $C*D$
All variables are continuous, normally distributed, and symmetric, with a mean of 0 and sd of 1.
I know the correlations between every term and $Y$, the correlations between $A$,$B$,$C$, and $D$, as well as the correlations between the two interaction terms ($AB$ and  $CD$) and $A$,$B$,$C$, and $D$.
My intuition is that using this information I can figure out the correlation between $AB$ and $CD$ without first simulating them, but I'm not sure how to do it. Does anyone have any ideas?
 A: I don't think you would need the conditions related to the response $Y$, because what you are interested is completely determined by the joint distribution of the $4$-dimensional random vector $(A, B, C, D)$.  In fact,
\begin{align}
\operatorname{Corr}(AB, CD) = \frac{\operatorname{Cov}(AB, CD)}{\sqrt{\operatorname{Var}(AB)\operatorname{Var}(CD)}},
\end{align}
where
\begin{align}
& \operatorname{Var}(AB) = E[A^2B^2] - (E[AB])^2, \\
& \operatorname{Var}(CD) = E[C^2D^2] - (E[CD])^2, \\
& \operatorname{Cov}(AB, CD) = E[ABCD] - E[AB]E[CD]. 
\end{align}
With the conditions you stated, $E[AB], E[CD]$ can be determined.  However, higher-order moments $E[A^2B^2], E[C^2D^2], E[ABCD]$ cannot be determined.
For example, consider $(A_1, B_1, C_1, D_1) \sim N_4(0, I_{(4)})$ and $(A_2, B_2) \sim N_2(0, I_{(2)})$, $C_2 = (2\xi - 1)A_2, D_2 = (2\xi - 1)B_2$, where $\xi \sim \text{Bin}(1, 1/2)$ is independent of $(A_2, B_2)$.  It is then easy to verify that

*

*$A_i \sim N(0, 1), B_i \sim N(0, 1), C_i \sim N(0, 1), D_i \sim (0, 1), i = 1, 2$.

*All pairwise correlations of $(A_i, B_i, C_i, D_i)$ are $0$, $i = 1, 2$.

*All pairwise correlations of $(A_iB_i, A_i, B_i, C_i, D_i)$ and
$(C_iD_i, A_i, B_i, C_i, D_i)$ are $0$, $i = 1, 2$.

Therefore both $(A_1, B_1, C_1, D_1)$ and $(A_2, B_2, C_2, D_2)$ give exactly the same information in your question.  However,
\begin{align}
& E[A_1B_1C_1D_1] = E[A_1]E[B_1]E[C_1]E[D_1] = 0, \\
& E[A_2B_2C_2D_2] = E[(2\xi - 1)^2A_2^2B_2^2] = 1.  
\end{align}
This shows $\operatorname{Corr}(A_1B_1, C_1D_1) = 0$, while $\operatorname{Corr}(A_2B_2, C_2D_2) = 1$, i.e., the correlation between interaction terms cannot be uniquely determined given the conditions presented.
