In a regression, I'm using household income and household specific expenditures as independent variables, both with a natural logarithmic transformation, and I want to control them by the household size. I find in the literature that in that kind of cases, it is used the natural logarithm this last variable, but I don't get the logic. If I'm not wrong, the household size is the number of people living in the household, so i find that the interpretation would be very weird: an increase in 1% of the number of people leads to a x% in Y?
They use log of household size to adjust for the skewness in the data distribution for household size because some households are too small while others are too large
THE PURPOSE is to put your data in the same scale by taking the natural log. After the modeling is completed, you can covert back the number to get the original value scale. For example, if the family size is 5, the log LN(5) = 1.61 when you convert to the original value: exp(1.61) = 5.0028.
WHY using log? When dealing with income level or economic value, the numbers are generally large. The log has the effect of de-scaling effect of the large number for ease of handling. After the calculation or modeling is complete, the final result can be converted back to its original scale.
Clearly household size can have an impact on household size, but linear in logsize is a big simplification. I would consider doing separate regressions for each household size.
The previous answer was meant to start: "Clearly household size can have an impact on household income"
The use of logarithm in general is to keep all data at the same scale, in your case, households usually have big differences in the number of their members, which affect their income, but you can convert the logs back to their original numbers as mentioned by Paul Louangrath
Allow me to share another view on WHY THE Log/ln.
Take a household of 2, a household of 4, a household of 8, I am skipping cases in-between.
Look at their housing. For 2, you're ok with a bathroom, for 4 it's already much better to have two bathrooms/toilets, for 8 maybe 3 bathrooms/toilets.
So you can see that the reasonable and comfortable shared resource is roughly varying as the Log of the household size.
Think of something else, still very practical, the rectangular dining-room table.
Let me get something from the internet from a search on
"dining table for 8":
Dimensions: (W) 180CM x (H) 78CM x (D) 90CM
Surface = 1.62 m2
"dining table for 4"
Dimensions: (W) 90CM x (H) 78CM x (D) 90CM
Surface = 0.81 m2
"dining table for 2"
Dimensions: (W) 75CM x (H) 76CM x (D) 75CM
Surface = 0.56 m2
call these example of values v2, v4, v8
v4/v2=1.45
v8/v4=2
Here you see that the surface of the table as function of the number of seated people is non-linear and concave, roughly logarithmic
Hence it makes also sense to have a Log scale on the household for a linear scale on table surface.
We could continue by looking at house or apartment surface, etc...
More generally, the reasonable sharing (in a family or household) and expected economies of scale show that resources maybe evolve linearly as function of the Log of the household (rather than as a linear function of the direct size of the household).
This might also hold somewhat for food consumption, if you assume on average that a household of 2 is two adults eating each one full size meal, whereas a household of 4 is more likely to have two children and two adults, and assuming that the duration of the childhood age (eating on average half size meal) is much more significant than the duration of adolescence (where they eat more than their parents, maybe?), you get again a reasonable Log approximation... Funny isn't it?
It even works for some aspects of individual consumption.
Now the water consumption, the electricity consumption, etc, do not seem to be easily modeled in Log of household size...
I hope that the above examples (not proofs, just illustrative cases which are likely or at least make sense) show why a log model might make sense.
Note that we have done economics without introducing any money or income. This is deeper and more basic.
- Badges
- Science topic
- Similar topics
- Social Science