9.1 Is a straight line the right model?
A slope of \(-0.006\) km per litre per kilogram. Is that a lot?
Section 7.10 fitted fuel efficiency on weight for 74 cars and returned a slope of \(-0.00601\).
Nothing about that number tells you whether the effect is large.
cars <- read.csv("data/car-mileage.csv")
c(per_kg = coef(lm(mileage_kmpl ~ weight_kg, data = cars))[2],
per_tonne = coef(lm(mileage_kmpl ~ I(weight_kg / 1000), data = cars))[2])#> per_kg.weight_kg per_tonne.I(weight_kg/1000)
#> -0.006009 -6.008687The same relationship, the same data, the same fit — and two numbers differing by a factor of a thousand. Measure efficiency in litres per hundred kilometres instead and the sign flips.
Behind that awkwardness lies a deeper point, and it is the real reason economists reach for logarithms.
Most economic relationships are proportional rather than absolute.
Losing ₹10,000 is a catastrophe for a household earning ₹50,000 a year and a rounding error for one earning ₹50 lakh. The same absolute change means something entirely different at different points of the distribution, which is why economists instinctively discuss income, prices and output in percentages.
A level–level model denies this. It insists that a one-unit change in \(x\) produces the same absolute change in \(y\) everywhere — for the smallest car and the largest, the poorest household and the richest.
Two further problems follow from the same source.
Units are arbitrary. A coefficient in kilograms and one in tonnes describe the same relationship, but neither is easier to judge than the other. A proportional measure does not change when someone changes their mind about units.
Straight lines are not always the right shape. Section 6.2 showed that the conditional mean of GPA against attendance was not exactly linear, and took a line as an approximation. Sometimes a better approximation is available at no cost, by transforming a variable before fitting.
Taking logarithms addresses all three.