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)

  • $\begingroup$ 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. $\endgroup$ – heh Dec 2 '19 at 21:55
  • $\begingroup$ 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. $\endgroup$ – Michael Dec 4 '19 at 11:14

Just to make things simple I'll show how to do it for a simpler case.

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 $X_3$.

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