Help to understand assumptions of logs

[Marcus Eklund]

Hi,

Can someone please help me understand the difference in the expectations assumptions if I:
1. predict a persons income based on her years in education vs..
2. predict a persons logged income based on her years in education?

I understand that in the first scenario I expect a linear relationship so that each year equals to the same raise in income, but what about the other? Do I then expect that the first education years leads to a bigger change in income than after many years or the otherway around?

Best,
Marcus

[Nick Cox]

Income as a function of year will be a straight line plotted against year if income increases by a constant each year e.g. USD 1 or USD 500.

Log income as a function of year will be a straight line if income increases by a constant percent each year e.g; 1% or 5%,

What you will see otherwise depends on what else is happening, a monumental tautology!

The assumption behind using logarithms is just you feed them with positive values.

I am not really sure what the question is about, but good if this helps a bit.

[Marcus Eklund]

Thank you! I understand the difference I think. So that means that in the logged version, income increases by a smaller percent at lower years and a higher percent when there are many years? so the step from 1 year to 2 year on logged income is smaller than from 7 to 8 years?

[Nick Cox]

You're thinking the wrong way round, I suspect. If the pattern is one in which the percent change decreases, you'd see curvature on a graph with log scale for income whereby the slope decreases. Taking logs itself doesn't change the underlying data, just how they appear.

[Joao Santos Silva]

Dear Marcus Eklund,

In addition to Nick's clear and helpful advice, please note that in general a predictor of log wages cannot be used to predict wages; you need reasonably strong assumptions to be able to obtain a predictor for y from a predictor of ln(y) -- this is the so-called retransformation problem.

Best wishes,

Joao