When to Use Log Returns in Regression Analysis

Summary

This note addresses whether log returns or simple returns are more appropriate as the dependent variable in a regression studying factors such as dividend yield and stock index returns. It corrects a common misunderstanding about regression assumptions: the response variable itself does not need to be independent and identically distributed or normally distributed; those assumptions, when used, concern the regression errors. The note also points out that modeling assumptions can be difficult to satisfy in practice.

It gives a practical reason to choose log returns: they add across time and make compounded returns easier to calculate. The discussion does not establish that log returns are generally superior for regression, nor does it compare estimates from the two return definitions. The appropriate choice therefore depends on the analysis and interpretation sought; using log returns does not by itself ensure that a regression’s assumptions are met.

Key ideas

• The dependent variable in a regression does not itself need to be independent and identically distributed.
• Distributional assumptions in standard regression analysis generally concern the residuals.
• Log returns are additive over time, which simplifies compounding calculations.
• Choosing log or simple returns is not settled solely by assumptions about the dependent variable.

Tags

# Should log returns be used in multilinear regressions?


# Should log returns be used in multilinear regressions?












As the title already says, should log returns, instead of simple returns, be used in regression analysis? In this case, I want to analyse the impact of specific factors (Dividend yield etc.) on the return of the Dow Jones stock index.

I know that Fama-French have developed their 3 factor model using only simple returns, but I often hear quants using log returns because those are closer to iid properties than simple returns and, after all, the dependent variable in a regression needs to be iid.

## Answer by madilyn (score 3, accepted)

https://quant.stackexchange.com/a/39588

As you pointed out, not necessarily:

> I know that Fama-French have developed their 3 factor model using only simple returns

That's because of a very common misconception:

> and, after all, the dependent variable in a regression needs to be iid.

In fact, the dependent variable does not have to be normally distributed or i.i.d. for that matter. That assumption only applies to the residuals. And even despite this, in practice, it's very difficult to meet those modeling assumptions.

The primary reason for using log returns is that they are time additive and easier to compute compound returns on.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.