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

<p>Let's say you have a regression <span class="math-container">$Y = \beta_1 X_1 + \beta_2 X_2 + \varepsilon$</span> and you want to impose <span class="math-container">$\beta_1 = 2\beta_2$</span>.</p>

<p>This is equivalent to saying <span class="math-container">$Y = 2 \beta_2 X_1 + \beta_2 X_2 + \varepsilon = \beta_2 (2 X_1 + X_2) + \varepsilon$</span>. That is, you can generate a new variable <span class="math-container">$X_3 = 2X_1 + X_2$</span> and regress <span class="math-container">$Y$</span> on <span class="math-container">$X_3$</span>.</p>

CC BY-SA 4.0
2019-12-04T08:30:06.453Z
2019-12-04T08:30:06.453Z
Art
<p>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. </p>

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

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

CC BY-SA 4.0
2019-12-02T21:55:51.097Z
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
CC BY-SA 4.0
2019-12-04T11:14:38.887Z
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
<econometrics><regression>
How do I run a regression with the restriction that one set of parameters are proportional to another?
Michael Tallent
# How do I run a regression with the restriction that one set of parameters are proportional to another?

(Asked by: username_0 on 2019-12-02)

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)

**Tags:** econometrics, regression

> **Comments:**
> **username_1**: 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.  
> **username_2**: 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.  

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## 1 answers

### Answer by username_3 on 2019-12-04. Score: 3

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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