# How do I run a regression with the restriction that one set of parameters are proportional to another?

I'm running a regression with a set of 3 dummy variables (for four categories of a variable) and these 3 dummies interacted with a continuous variable. I want to impose the restriction that the vector of the coefficients of the interacted dummies is proportional to the coefficients of the dummies, e.g. each interaction term has a coefficient 2x larger than the coefficient on it's non-interacted dummy.

Is it possible to impose these restrictions with a linear regression model? or will I have to use non-linear least squares?

(P.S. the proportionality is a ratio to be estimated, so that would be another free parameter in the model)

• The question is a bit unclear. Perhaps write out the model you're working with (before the "proportionality constraint") and it might help shed some light on what you're trying to do. – heh Dec 2 '19 at 21:55
• You can estimate the linear model with whatever restriction you want---by minimizing sum of squared residuals subject to the restriction. This is a standard constrained optimization problem in economics/econometrics. The usual OLS is a special case where the SSR is minimized with no restriction. – Michael Dec 4 '19 at 11:14

Let's say you have a regression $$Y = \beta_1 X_1 + \beta_2 X_2 + \varepsilon$$ and you want to impose $$\beta_1 = 2\beta_2$$.
This is equivalent to saying $$Y = 2 \beta_2 X_1 + \beta_2 X_2 + \varepsilon = \beta_2 (2 X_1 + X_2) + \varepsilon$$. That is, you can generate a new variable $$X_3 = 2X_1 + X_2$$ and regress $$Y$$ on $$X_3$$.