Scaling Returns and Accounting Ratios in Cross-Sectional Regressions

Summary

The document considers how to express twelve-month stock returns and return on assets when using them in a cross-sectional regression alongside log market value. It compares fractional and percentage units for the return and ROA, and asks whether ROA should instead be logged. The central issue is how variable scaling affects coefficient interpretation and comparisons of estimated slopes over time.

The accepted response says either fractional units for returns and ratios, or percentage units for both, can support similar graphs and interpretations, provided the coefficient for the log variable is rescaled appropriately when expressing its effect for a one-percent change. Percentage figures may be easier to communicate. No regression data, estimates, or formal discussion of logging ROA are supplied, so the response is guidance about units and coefficient presentation rather than evidence that one specification is statistically preferable.

Key ideas

• Changing a variable from fractional units to percentage units rescales its regression coefficient.
• Returns and ratios can be represented consistently as fractions or percentages for plotting slopes.
• A coefficient on log market value needs appropriate rescaling when interpreted for a one-percent change.
• Percentage-form results may be more intuitive to report.
• The discussion does not establish that logging ROA is appropriate or compare model fit across specifications.

Tags

# Scaling variables (Fraction vs % vs log) when regressing twelve month returns


# Scaling variables (Fraction vs % vs log) when regressing twelve month returns












Dear Stack community,

My question is the following;

If my dependent variable is twelve month returns.

And as independent variables I have fiscal year variables like ROA and log variables like the log of the market value.

Where ROA = Net income / total assets

Should I scale ROA either as a fraction (e.g 0.05), a log (e.g -1.30) or a % (e.g 5%).

And similarly how should I scale returns?

I guess the anwser depends on what change I want to analyze.

However if I want to fit both slopes in a graph over time and interpret their coefficients for a cross section of stocks, would it make most sense to use a fraction, % or log for ROA?

Any clear intuition for this...?

## Answer by Julien Maas (score 1, accepted)

https://quant.stackexchange.com/a/77639

To graph them I can either keep both the dependent and independent variables as fractions with the log variable as a log. Or both the independent and dependent variables as a % and divide the beta of the log by 100 to get a 1% change interpretation.

Either way will give me a similar graph, and similar interpretations. It's just that when reporting the figures, % returns and % ratio's may be more intuitive.

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