# Coefficient of Ratio + Ratio of Coefficients: Inconsistency

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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