Interpreting coefficient in a log-log model

[Sussy Bruno]

Hi everyone,

I am currently working with a log log model specification. To my understanding, the correct interpretation of the coefficients is 1% increase in x is associated with a ß1% increase in y
The issue that I am having is, my explanatory variable is in percentage (0.01-0.99), how does the interpretation change in this case?

I checked this in the Wooldridge book but so far, couldn't clarify my doubt.
My guess so far is:

1 percentage point increase in x leads to a ß1% increase in y, is this correct?
I have seen some people using the means to interpret this kind of relationships, could I leave it in a simple way, or is it always necessary to calculate this at the mean, or any specific value?

[Clyde Schechter]

There is no simple straightforward interpretation for this. And a 1 percentage point increase in x does not lead to a

ß*100 percent increase in y. In fact, it does not lead to any consistent, predictable change in y: it depends on what value of x you start from. That is why you see people doing this at the mean of x: that provides a defined starting point. Let's say,. for the sake of illustration that the mean value of x is 17%. Then a 1 percentage point increase in x changes it to 18%. So log x goes up by log(18%) - log(17%) = log(18/17) = 0.057 to 3 decimal places. Since log x has increased by about 0.057, log y will increase by 0.057*ß. If, again for demonstration purposes, ß = 0.5, that means that log y increases by 0.057*0.5 = 0.0285. That, in turn means that y increases by a factor of exp(0.0285) = 1.029. So y has increased by 2.9%.

[Jeff Wooldridge]

When you have a log-log model, ß is an elasticity no matter what are y and x. It's just that a "one percent increase in x" has to be understood properly. This is where the difference between a "one percent increase" and a "one percentage point increase" is important. If the unemployment rate goes from 5.0 to 6.0, or 0.05 to 0.06 in decimal format, that is a one percentage point increase but a 20 percent increase. The problem with using log(x) when x is a percent or a proportion is that you are measuring a one percent increase in something that is a percent. This may be intended but it's probably not the most common way of measuring effects.

I would suggest using log(y) and x. Then, ß is the percent change in y when x increases by 0.01, which means one percentage point (if x is a proportion as in the original statement, not a percent).

I think I talk a bit about this in my book, but it's perhaps buried in Chapter 6.

[Sussy Bruno]

Thanks to both.

To Mr. Wooldridge. The percentage has been transformed on purpose. In that case, I still do not have clear whether it should be interpreted as a 1 percentage point increase or a 1% increase on x.
For example, in the estatement by Clyde I have a one percentage point increas (17% to 18%) whereas in general terms a 1% increase would be (17%*(1.01)) from 17% to 17,17%.

This is where my confusion lays, mainly in understanding if it should be 1% increase (even when the variable is already in percentage) or 1 percentage point.

Thanks

[Hans Nostiz]

What would change in the interpretation of #2 in the case that the outcome variable is also stored as a percentage in the log-log framework? In other words, the response and the explanatory variable are already in percentages (0.01 - 0.99) and additionally log-transformed.

[Nick Cox]

In my book values like 0.01 to 0.99 could be proportions or fractions (or even probabilities) while values like 1 to 99 are not proportions, but could possibly be percentages.

One way of squaring the two is to consider that % in 99% is just notation for /100 so (as you're implying) they're the same kind of beast really and -- in anachronistic terms that have only been used in recent years -- use of % is just use of a display format.

Stata has no way of knowing, however, whenever what it sees, stores and shows as 99 is really equal or equivalent to 0.99 so far as you're concerned, so watch out.

I don't know what kind of answer is being sought in #5, but with students or colleagues I always draw graphs. Here is a start with x^b for b > 0 and then a x^b for a > 0 just changes the numbers on the y axis. For negative b there are different graphs.

ssc inst mycolours
mycolours

local both aspect(1) legend(order(1 "1/3" 2 "3") ring(0) pos(5) col(1)) ytitle(y, orient(horiz))  ra(0.01 0.99)

twoway function y1 = x^(1/3), lc("`ora'") ra(0.01 0.99) lp(dash) || ///
function y2 = x^3 ,  `both' lc("`blu'") yla(, ang(h)) ra(0.01 0.99)  name(G1, replace)

twoway function y1 = x^(1/3), lc("`ora'")  ra(0.01 0.99) lp(dash) || ///
function y2 = x^3 , `both' lc("`blu'")  yla(.01 .02 .05 .1 .2 .5 1, ang(h))  ysc(log) xsc(log) xla(.01 .02 .05 .1 .2 .5 1) name(G2, replace)

graph combine G1 G2 , title({it:y} = {it:x{sup:b}})

[Hans Nostiz]

First of all, many thanks to Nick for his prompt answer. The question I mentioned in #2, however, is how the interpretation changes in case of both the explanatory variable and the response variable are already proportions (0.01-0.99) and in addition log-transformed (not just one of them as in #1).
I suppose we could now directly view the coefficients in a regression table as percentage point changes rather than percentage changes.

[Nick Cox]

I read #5 as being about two variables. but -- indeed -- once you have other variables it's hard to reduce everything to one graph.