Interpretation of interesting fixed effects results I have a model with two variables, $X_1$ and $X_2$. A regression of $Y$ on both of these yields significant positive coefficients with a larger effect for $X_1$ than $X_2$.
$X_1$ and $X_2$ are very likely to be bidirectionally related so I ran a two-way fixed effects model to identify the channels of causality. What I obtained surprised me:
$X_2$ predicts $X_1$ very well, with the contemporaneous variable and three lags being highly significant.
$X_1$ does not predict $X_2$ very well, with only one lag being weakly significant.
How is this possible in a purely econometric context? If $X_2$ (mostly) precedes/causes $X_1$, how would the effect for $X_1$ when Y is regressed on both be so much stronger/more significant?
Has anyone had experience with something like this?

Maybe this simple formula can help a bit. Let $y = b_0 + b_1x_1 + b_2x_2$ + e , therefore:

When doing multiple regression the magnitude and sign of the covariances can alter your results significantly. It can increase or decrease the coefficients in the multiple regression from the bivariate regression, and also alter the significance or redistribute it. This is very common in equations with high presence of multicollinearity. The results can indeed be counter-intuitive.
For an actual econometric interpretation, we might need to know which variables are you using.
