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Why is it that in many econometrics papers, especially the ones dealing with financial time-series data returns are modelled as logarithmic differences in prices $r_t=\ln⁡(p_t )-\ln⁡(p_{t-1} )$, what is the advantage/disadvantage of using instead the traditional percentage change formula $r_t=p_t/p_{t-1}-1$?

Johan Stax Jakobsen
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Hatori_Hanzo
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    I wouldn't refer to the second formula as "traditional". The first is the continuously compounded return and the second is the period return. There are technical reasons that make it simpler or more appropriate to use one or the other depending on the context. There are also contexts where it doesn't really matter (since they are approximately the same for "small" returns) and others where it does but authors automatically reach for one or the other without real justification. If you have a more specific context in mind, you should specify. – Chris Haug Feb 21 '22 at 16:32
  • The main point is explained by Nick Cox in this answer https://stats.stackexchange.com/a/545410/198058 – ColorStatistics Feb 21 '22 at 18:17
  • See https://fan.princeton.edu/fan/FinEcon/chap1.pdf, section 1.1.3: "One immediate convenience in using log returns is that the additivity in multiperiod log returns, i.e. the k period log return ... is the sum of the k one-period log returns" – Adrian Feb 21 '22 at 18:28
  • Some people get confused by percentages: increase something by $10\%$ and then decrease that by $10\%$ (or in the opposite order) and you end up with $99\%$ of what you started with, which is not particularly desirable; by contrast adding $0.1$ to the logarithm and then subtracting $0.1$ (or in the opposite order) obviously gets you back where you started. And percentages use multiplication or division while your linear regression techniques may assume addition. – Henry Feb 21 '22 at 18:28

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The use of linear returns (percentage change) and log returns are both used in financial applications. Two arguments for using log returns for time series modelling are

  • The distribution of log returns can unlike linear returns easily be project to any horizon
  • Log returns typically have a symmetric distribution which makes modelling easier (stock prices are often assumed to be log normally distributed - log-returns follow a normal distribution)
Johan Stax Jakobsen
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