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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.

Nick Cox
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Harry Harrington
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    Hard to say without seeing any data. But it is possible to guess that if loss is the response, then some losses will be negative (i.e. gains), and conversely, so responses are likely to be both positive and negative, which is a constraint on what makes sense. I would consider working with log firm size. Why not the data and show us? – Nick Cox Oct 08 '20 at 23:10
  • % change from one quarter to next = outcome = dependent variable Company size = independent variable If you are looking at a whole lot of companies i presume you are comparing average % change of small companies in comparison to large companies? Or how many categories of company size have you got? – Gale Oct 08 '20 at 22:50

1 Answers1

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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 $Y_0^{-1}$

Firebug
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