Coefficient of an Interaction Term when Regressors are Independent Say we have an OLS regression of $y$ on $x_1$ and $x_2$, where both $x_1$ and $x_2$ are independent from each other, and create the following regression:
$$
y= \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2
$$
How would one go about showing that the coefficient on interaction term, $\beta_3$, should be zero?
Regressing $x_1$ on $x_2$ will give a coefficient of 0, since $\hat{\beta}=cor(Yi,Xi)⋅\frac{SD(Yi)}{SD(Xi)}$, but I am not sure how to fit this in to what I am trying to show above, or if this is the right way of showing that.
 A: Next to Dave's +1 numerical example, consider an applied one, explaining wages as a function of covariates like ability, gender etc.
It is well-documented that ability and gender both have an influence on wages (i.e., $\beta_1,\beta_2$ both are nonzero), because being more able is good for your earnings and there is such a thing as wage discrinimation. Also, we may assume that gender and ability are independent - women are no more/less able than men in general/on average.
That, however, does not rule out that there is an interaction with ability ($\beta_3\neq0$). For example, there might be more wage discrimination for high-skilled women than for low-skilled women, where wages for the latter group may involve less individual negotiation due to wage-setting negotiated via unions and smaller glass-ceilings effects etc.
A: This is not correct, and it is possible to simulate a counterexample, such as with R code.
set.seed(2022)
x1 <- c(0,1,0,1,0,1,0,1)
x2 <- c(0,0,0,0,1,1,1,1)
y <- x1 + x2 + x1*x2 + rnorm(length(x1))
summary(lm(y ~ x1*x2))

This has independent $x_1$ and $x_2$ but also results in a nonzero coefficient on their interaction.
