Continuous independent variables that sum up to 100%

[Cisse abs]

Dear Statalist Users,
Please I need help.
I am studying the effect of immigrant skills on redistribution using a logit model. Immigrants' skills are measured by 03 independent variables: the share of low-skilled, share of medium-skilled and the share of high-skilled within a country. The sum of these three independent variable equal 100%. The DV is a binary taking values 0 or 1.
Should I drop one variable from the regression or use it as a reference ? If yes which of them ?
How to interpret the result of the regression ?

Kind regards!

[Nick Cox]

You should find that Stata will omit one of the variables automatically, Best to do that yourself choosing on some other grounds, such as

* the least interesting or the least important substantively

* the most difficult to interpret

* the most problematic in terms of measurement or other errors.

Thus if my predictors in a US context were vote percents as measured % Democratic %Republican %other I would probably omit the last, which is nothing to do with disrespect for any choice or respect for any other choice.

[Cisse abs]

Thank you very much Dear Nick Cox. Your response is very helpful.
Suppose that I omit the share of low-skilled migrants, how can I interpret the coefficient of the 02 other variables in a logit model ?
For example, after computing the odds ratio, the coefficient of the share of medium-skilled migrants equal 1,5. Should I say that: "for 1% increases in the share of medium-skilled migrants relative to the share of low-skilled, the odds of being in favor of redistribution increase by a factor of 1,5" ?

[Nick Cox]

That is a harder question, at least for me, because an increase in one of those variables automatically implies a decrease in the sum of other two, and vice versa. Worse, omitting one doesn't solve the problem that an increase in one might be yoked with an increase in one and a compensating decrease in the third, and so forth.

There isn't, in short, an easy way in which you can interpret each effect separately, even with a ceteris paribus clause.

Other people may have a smarter answer. There is a very dense literature on compositional data analysis that says much more, but I am not familiar with it in detail.

[Cisse abs]

For lack of having a suitable way to interpret results, is there a transformation method to use in order to avoid this problem ?

[Jeff Wooldridge]

If x1 + x2 + x3 = 100, and you omit x3, which means estimating

y = a + b1*x1 + b2*x2 + ...,

then b1 is interpreted as the change in y when x1 increases by one percentage point holding x2 fixed. In other words, it's an increase in x1 of one percentage point and, necessarily, a decrease in x3 by one percentage point. Generally, it's the omitted category that drops to compensate any increase in an included variable. The other included shares are held fixe. So if x1 + x2 + ... + xK = 100 and xK is omitted, b1 is the change in y when x1 increases by one perc point while xK decreases by one perc point, with x2 ... xK_1 held fixed.

By the way, note the correct language is x1 increases by "one percentage point," not by "on percent."

With a logit, the interpretation is similar except now use the margins command to get the effects on the probability that y = 1. Or, use the odds ratio interpretation.

[Cisse abs]

Thank you very much dear professor Jeff Wooldridge and dear professor Nick Cox. Yours answers help me a lot.

[Maarten Buis]

Also see the last section of http://www.maartenbuis.nl/publications/prop.html for a graphical interpretation of such variables.

[Cisse abs]

Thank you very much Dear Maarten Buis.