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What is the reason why we use natural logarithm (ln) rather than log to base 10 in specifying functions in econometrics?

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migrated from economics.stackexchange.com May 3 '12 at 18:22

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6 Answers

In the context of linear regression in the social sciences, Gelman and Hill write[1]:

We prefer natural logs (that is, logarithms base $e$) because, as described above, coefficients on the natural-log scale are directly interpretable as approximate proportional differences: with a coefficient of 0.06, a difference of 1 in $x$ corresponds to an approximate 6% difference in $y$, and so forth.

[1] Andrew Gelman and Jennifer Hill (2007). Data Analysis using Regression and Multilevel/Hierarchical Models. Cambridge University Press: Cambridge; New York, pp. 60-61.

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+1: For concrete reason to prefer the natural logarithm. –  Neil G Aug 6 '12 at 6:02
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There is no very strong reason for preferring natural logarithms. Suppose we are estimating the model:

ln Y = a + b ln X

The relation between natural (ln) and base 10 (log) logarithms is ln X = 2.303 log X (source). Hence the model is equivalent to:

2.303 log Y = a + 2.303b log X

or, putting a / 2.303 = a*:

log Y = a* + b log X

Either form of the model could be estimated, with equivalent results.

A slight advantage of natural logarithms is that their first differential is simpler: d(ln X)/dX = 1/X, while d(log X)/dX = 1 / ((ln 10)X) (source).

For a source in an econometrics textbook saying that either form of logarithms could be used, see Gujarati, Essentials of Econometrics 3rd edition 2006 p 288.

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The natural log is also useful in a semi-log time series regression since the estimated coefficients can be interpreted as continuously compounded growth rates. –  Jason B Mar 27 '12 at 18:20
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I think that the natural logarithm is used because the exponential is often used when doing interest/growth calculation.

If you are in continuous time and that you are compounding interests, you will end up having a future value of a certain sum equal to $F(t)=N.e^{rt}$ (where r is the interest rate and N the nominal amount of the sum).

Since you end up with exponential in the calculus, the best way to get rid of it is by using the natural logarithm and if you do the inverse operation, the natural log will give you the time needed to reach a certain growth.

Also, the good thing about logarithms (be it natural or not) is the fact that you can turn multiplications into additions.

As for mathematical explanations of why we end up using an exponential when compounding interest, you can find it here:http://en.wikipedia.org/wiki/Continuously_compounded_interest#Periodic_compounding

Basically, you need to take the limit to have an infinite number of interest rate payment, which ends up being the definition of exponential

Even thought, continuous time is not widely used in real life (you pay your mortgages with monthly payments, not every seconds..), that kind of calculation is often used by quantitative analysts.

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I probably would have given an answer like this. The point that it doesn't matter in modeling is a good one too. We could just as easily use base 2. The difference is only a constant factor –  Michael Chernick May 3 '12 at 18:44
Surely, you could write it $Nr^t$ (more intuitively), and so are you arguing that the base doesn't matter as Adam Bailey does? –  Neil G Aug 6 '12 at 6:16
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Is this unique to economics? The standard normal distribution features an $e^{-{1\over2}x^2}$ in it, and the normal distribution is only one of the large family of exponential distributions that cover a huge swath of statistics. (See GLM's.) It seems like the natural log would be useful in these cases.

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Yes, but the variance term of the normal distribution is equivalent to a change of base of the exponent. Even the way you've drawn it, the base could be $(\sqrt{e})^{-x^2}$ –  Neil G Aug 6 '12 at 5:59
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An additional reason why economists like to use regressions with logarithmic functional forms is an economic one: Coefficients can be understood as elasticities of a Cobb-Douglas function. This function is probably the most common one used among economists to analyze issues regarding microeconomic behaviour (consumers´preferences, technology, production functions) and macroeconomic issues (economic growth). The elasticity term is used to describe the degree of response of a change of a variable with respect to another.

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There is a good reason to use the log transformation of the variable if you think that the inverse function of logarithm is the exponential function which is a continuous version of conpounding. The economic variable which is growing around 10% at a time can be transformed to the variable with its mean around 10 (plus a constant). You cannot do that with the transformation of logarithm of different base.

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