# Positive interaction term with one negative component?

Say I have an augmented growth regression, where my Y is GDP growth, and my Xs are the classical MRW variables, + international aid + corruption + the interaction term between the two.
Basically, I'm interested in seeing if lower corruption increases the effectiveness of international aid, which means I'd like to see a positive sign on my interaction term.
However, I also think I should see a negative sign on corruption, as the higher corruption, the lower should growth be.
Now, my questions:

1. Is this possible as I describe it? Can I see a negative component but a positive interaction term? In which case, how should I interpret it?
2. Or is there a reason the coefficient on corruption should be positive, and still indicate that higher "institutional quality" has a positive impact on growth by itself and also in interaction with aid?

Ignore the other variables. Let $g$ be growth, $C$ corruption and $A$ aid. You look at

$$g = \alpha C + \beta A + \gamma A\cdot C$$

You ask, "does lower corruption increases the effectiveness of foreign aid?"

For this you have indeed to examine the cross-partial derivative

$$\frac {\partial g}{\partial A\partial C} = \gamma$$

This is "how growth is affected at the margin by Aid, if corruption changes"... So if lower corruption increases the effectiveness of Aid, $\gamma$ should be negative, not positive...

...if your corruption index is mapped as "the higher the value of $C$ the higher the corruption".

If your "corruption index" is mapped in reverse (and so in reality it measures "non-corruption"), the anticipated sign reverses also.