Does it make sense to interact two uncorrelated independent variables in linear regression? Should two continuous variables be correlated to include their interaction? I.e. does y = x1 + x2 + x1x2 make sense if correlation between x1 and x2 is weak at -0.1, and their coefficients in the regression have opposite signs (x1 β1 close to +1 and x2 β2 close to -1)?
And if the estimated coefficient on x1x2 is then +3, does this effectively mean the total effect of x2 is actually positive (with β3 = +3 offsetting β2 = -1), thus both variables have a net positive association with y?
Lastly, if trying to represent this with scatter plots like pairs(y x1 x2 x1x2) in r, wouldn't individual and time fixed effects (panel data) make any relationship impossible to see visually?
Thanks!
 A: To slightly add to Dave's +1 answer, recall that omitting a regressor from a regression model will not affect the remaining coefficients (something that I would argue is more relevant and less trivial than a "better" (higher) $R^2$, which will always obtain with additional regressors - although Dave of course also points to adjusted $R^2$) if it's uncorrelated (orthogonal) with the remaining regressors.
To see this, write the OLS estimate containing two blocks of regressors $X_1$ and $X_2$ (in our example, $X_1$ could consist of x^2 and $X_2$ of x1 and x2) as
$$
\hat\beta=\begin{pmatrix}X_1'X_1&X_1'X_2\\
X_2'X_1&X_2'X_2\end{pmatrix}^{-1}\begin{pmatrix}X_1'y\\
X_2'y\end{pmatrix}
$$
That is, if the two blocks are orthogonal ($X_1'X_2=0$), we obtain
\begin{eqnarray*}
\hat\beta&=&\begin{pmatrix}X_1'X_1&0\\
0&X_2'X_2\end{pmatrix}^{-1}\begin{pmatrix}X_1'y\\
X_2'y\end{pmatrix}\\
&=&\begin{pmatrix}(X_1'X_1)^{-1}&0\\
0&(X_2'X_2)\end{pmatrix}\begin{pmatrix}X_1'y\\
X_2'y\end{pmatrix}\\
&=&\begin{pmatrix}(X_1'X_1)^{-1}X_1'y\\
(X_2'X_2)^{-1}X_2'y\end{pmatrix},
\end{eqnarray*}
i.e. the coefficients of two separate regressions.
Now, in Dave's example, the interaction has its own partial effect (equal to 5), and, while x1 and x2 are uncorrelated, x12 is not uncorrelated with x1 nor x2. I slightly extend Dave's example:
set.seed(2022)
N <- 10
x1 <- rep(c(0, 1), N) # (0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1)
x2 <- c(rep(0, N), rep(1, N)) # (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
x12 <- x1 * x2 # interaction between x1 and x2
#
y <- x1 - x2 + 5*x12 + rnorm(length(x1))

lm.int <- lm(y ~ x1 + x2 + x12)
lm.wo.int <- lm(y ~ x1 + x2)
summary(lm.int)
summary(lm.wo.int)

cor(x1, x2)  # zero
cor(x12, x2) # not zero
cor(x12, x1) # not zero

A: Definitely!
set.seed(2022)
N <- 10
x1 <- rep(c(0, 1), N) # (0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1)
x2 <- c(rep(0, N), rep(1, N)) # (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
x12 <- x1 * x2 # interaction between x1 and x2
#
y <- x1 - x2 + 5*x12 + rnorm(length(x1))

In this example, $x_1$ and $x_2$ are independent (not just uncorrelated) and have opposite signs in the regression model. However, the interaction term is important. Compare the $R^2$ or adjusted $R^2$ your models have when you regress on just $x_1$ and $x_2$ versus regressing on those variables plus their interaction.
