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I have regression results about the effect of a treatment on two outcome variables, $N$ and $D$.

The coefficient on $N$ is positive. The coefficient on $D$ is negative.

These results suggest that the effect on $\frac{N}{D}$ should be positive. The treatment increases the numerator, and decreases the denominator.

However, when I create a new variable $\frac{N}{D}$ and use this as an outcome variable, I get an negative coefficient on $\frac{N}{D}$.

I don't understand why this is happening. Full setup below.


I have an experiment in a panel data setting. All subjects are viewed twice (once pre- and once post- treatment). Between the two periods, randomly selected subjects are treated with a binary variable. The control group remains untreated.

I have two outcome variables, $N_{i,t}$ and $D_{i,t}$. Both underlying $N$ and $D$ variables are strictly positive.

I'm estimating regressions as:

$Outcome_{i,t}=\beta_{T}\times Treatment + TimeFEs + SubjectFEs+\epsilon$

Where:

  • TimeFEs = Time Fixed Effects (binaries variables for before/after)
  • SubjectFEs = Subject Fixed Effects (binaries for each subject).

The treatment coefficient ($\beta_{T}$) is positive for $N$, and negative for $D$.

However, when I create a new variable for each observation (call it "$R_{i,t}$" = $\frac{N_{i,t}}{D_{i,t}}$) and use it as the outcome, I get a negative treatment coefficient.

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