# Interpreting coefficients after transforming Y (dependent) variable

I have been working on linear regression for forecasting purposes.

I have a model where I have used a BoxCox transformation on the $$Y$$ variable (Sales) with $$\lambda = 0.3$$. The model is as follows:

$$Y = 13 - 0.2 X_1 + 3 X_2$$

How can I interpret the effect of sales from each variable, given the transformation?

For instance, assume $$X_1=5$$ and $$X_2=0.3$$ then we get:

$$Y = 13 - 0.2 \times 5 + 3 \times 0.3 = 12.9$$

Reverse BoxCox of 12.9 with $$\lambda = 0.3$$ is 196. So it forecasts 196 sales. But if we increase $$X_2$$ from 0.3 to 1 and increase $$X_1$$ from 5 to 10 then we get:

$$Y = 13 - 0.2 \times 10 + 3 \times 1 = 14$$

Reverse BoxCox then gives 244 sales forecasted sales. An increase of 48 sales. But how much of that is down to $$X_1$$ and $$X_2$$?

EDIT: to make the above point clear:

If $$X_1$$ went from 5 to 10 but $$X_2$$ doesn't change:

$$Y = 13 - 0.2 \times 10 + 3 \times 0.3 = 11.9$$, transform to 159 sales. A -37 decrease from the starting position (196-159)

However, if $$X_2$$ went from 0.3 to 1 but $$X_1$$ doesn't change: $$Y = 13 - 0.2 \times 5 + 3 \times 1 = 15$$, transforms to 294 sales. A +98 increase in sales (294-196).

But if $$X_2$$ increases sales by 98 and $$X_1$$ decreases sales by 37 you would expect the net increase to be +61 when you include both changes. But from earlier we found that by applying both we actually get a +48 increase in sales!

How can we work out the actual sales change resulting in changes in $$X_1$$ and $$X_2$$?