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.
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).
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 Poisson regression (see here When to use an offset in a Poisson regression?)
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:
$$Y_1=\alpha_0 + \alpha_{X} X + \beta_0 Y_0+\beta_X X Y_0 + \epsilon Y_0$$
The intercept $\alpha_0$ and linear term $\alpha_{X}$ 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 $Y_0^{-1}$
The answer by Firebug is very helpful and it shows you some of your modelling options. Of these options, I strongly recommend measuring change using the logarithmic difference and modelling using some variation of the core model form:
$$\log Y_t = \log Y_{t-1} + \beta X_t + \varepsilon_t.$$
Using the logarithmic difference to express percentage changes has a number of mathematical and interpretive advantages (see related questions here and here for explanation). In particular, the logarithmic difference expresses changes in terms of the "force of growth" which is a natural method of measurement. Moreover, the range of possible values for the logarithmic difference is unbounded, whereas the percentage or proportion change is bounded from below by the fact that there is a maximum negative growth of 100% of the previous value. The absence of a lower bound on the measurement of logarithmic difference makes it much easier to model using traditional regression methods.
This type of model using the logarithmic difference expression of changes is commonly used in finance and areas of time-series analysis. It is generally considered to be the most useful way to measure and model proportionate changes in values over time. In the above model form the term $\log Y_{t-1}$ forms an offset (with no accompanying coefficient) which means that the growth rate is assumed to be independent of the size of the previous value. If you want to relax this assumption you can use the extended model form:
$$\log Y_t = (1+\alpha) \log Y_{t-1} + \beta X_t + \varepsilon_t,$$
in which case the parameter $\alpha$ serves to estimate whether there is any relationship between the growth rate and the size of the previous value in the time-series.
