# multiple regression with constraints of independent variables

I'm running a regression analysis with independent variables $$X_{1}, X_{2}, \cdots, X_{n}$$ and dependent variable $$Y$$. There is a constraint among some of the independent variables, say, $$X_{1} + X_{2} + X_{3} = 1$$. What kinds of regression models (or other data science techniques) could be used in this scenario?

• Maybe compositional data. – user2974951 Sep 2 '19 at 12:16
• With just one dependent variable, this is multiple rather than multivariate. No one has yet given the simplest response which is that if some of your predictors add to a total, then you can dispense with one of them. Anything more complicated is not essential on that ground alone. To see this, imagine a two-category classification yielding say fraction which are red and fraction which are not red. There is no more information in not red than in red, so choose one of the two variables. – Nick Cox Sep 2 '19 at 13:18
• @NickCox ok I think I got your point. But here to me, if you wish to interpret the constraint on predictors alone (not the predictors time the beta), it does not mean that for every sample you have the constraint holding, so that you can get rid of one predictor in the whole model. In my personal opinion (which we can debate clearly) it rather means that you regress iff the selected predictors satisfy that relationship. So you just consider the samples where such relationship holds. Is it correct Nick? Or maybe Am I missing your point? – Fr1 Sep 2 '19 at 13:30
• Thank you @NickCox, I changed the title to multiple regression. – waynelee1217 Sep 2 '19 at 13:42
• 'As @user2974951 hints briefly. data now often called compositional are the fractions (proportions, percents, whatever) of mutually exclusive categories, so constrained in principle to add to 1 (100%). . Examples that are common are: texture of materials; chemical composition of materials; categories of expenditure; etc. – Nick Cox Sep 2 '19 at 13:59

Maybe the constraint is on coefficients of such independent variables, or the sum of the products $$x_{1} \beta_{1} + x_{2} \beta_{2} + x_{3} \beta_{3}$$?
• Thank you @Fr1. I'm running a dataset with each row meets the constraint I mentioned above. I believe I'm not constraining $\beta_{1}X_{1}+\beta_{2}X_{2}+\beta_{3}X_{3}=1$. Or should I have no constraint and just interpret the values of $\beta_{1}, \beta_{2}, \beta_{3}$? – waynelee1217 Sep 2 '19 at 13:47