I am interested in using OLS regression to model the relationship between two predictors (X1 and X2) and a response variable (Y). However, for theoretical reasons I know that when X1 = 0, Y must also equal 0. Furthermore, for theoretical reasons I also know Y must be positively related to X1. So I expect a positive correlation between X1 and Y, but I don’t know the value for the slope, or the strength of the relationship. Indeed, what I am most interested in is whether X2 is related to Y, once the variation explained by X1 is taken into account. So I have been considering fixing the intercept to run through the origin. I know from reading other posts that fixing the intercept in this way leads to a number of issues (e.g. high r2 values). However, because I know that the regression line must run through the origin, a model which doesn't do this seems unsatisfactory.
This leads me to my main query – my limited understanding of regression would have me believe that in fixing the intercept at zero, I am also specifying that the intercept for the relationship between Y and X2 is zero (R code: lm(Y ~ 0 + X1 + X2)). Is this right? If so, then then the approach seems flawed, because I don’t have prior knowledge about the relationship between Y and X2. Could people suggest an alternative approach?
Thank you very much in advance for any help, and I hope my question was clear/made sense.
NB The response variable y is bounded between 0 and 1. But it is not the product of trials with discrete outcomes (actually it is the value for an index), so I have been considering using linear regression with angular transformation of the response, rather than a binomial glm.