Using percentage change as a dependent variable? If I use percentage change as my dependent variable what is the correct modeling method?
I am trying to see how the size of a company affects its losses due to COVID-19. I use quarter over quarter change, and size as one of the independent variables.
 A: Assuming that you are modeling the difference of the previous quarter earning $Y_0$ and the next one $Y_1$ as
$$Z = \frac{\Delta Y}{Y_0}= \frac{Y_1-Y_0}{Y_0}=\frac{Y_1}{Y_0}-1$$
$Z$ can be negative (quarter-on-quarter loss), but that only happens due to that $-1$.
You can instead model solely the ratio between the quarterly earnings without loss of information (since it's simply $Z+1$)
$$Z'=\frac{Y_1}{Y_0}$$
Now, you have a few alternatives (see Linear Regression with a Dependent Variable that is a Ratio).
1. The logarithm
If you model $\log Z' = \log Y_1 - \log Y_0$, then you can offset your regression with \log Y_0:
$$\log Y_1=\log Y_0+\beta X + \epsilon$$
Since $Y_1$ is strictly positive, then you are good to go.
This offset is very common in Poison regression (see here When to use an offset in a Poisson regression?)
2. Modelling ratios
You can instead opt to model the ratio itself.
$$\frac{Y_1}{Y_0}=\beta_0+\beta_X X + \epsilon$$
According to Kronmal a direct regression may incur spurious correlations due to that ratio.
Instead, recognizing that the ratio is a multiplicative interaction, Kronmal proposes the study of the augmented model.
If you "multiply" the denominator to the right hand side, you get instead:
$$Y_1=\alpha_0 + \alpha_{Y_0} Y_0 + \beta_0 Y_0+\beta_X X Y_0 + \epsilon Y_0$$
The intercept $\alpha_0$ and linear term $\alpha_{Y_0}$ main effects are added to make the model complete (see why here: Including the interaction but not the main effects in a model).
You also have to weight your samples by $Y_0^{-1}$
