# Restricted Weighted Linear Regression in R

I have the following issue. I would like to run a linear regression imposing a constraint on the weighted coefficients. Let me construct an example:

Consider the following cross-sectional regression

$$r_{i} = \alpha + \beta_{1} I_{i, 1} + \beta_{2} I_{i, 2} + \beta_{3} I_{i, 3} + \epsilon_{i}$$

where $$r_{i}$$ is the return of stock $$i$$ and $$I_{i,1}$$ represents a dummy variable which is one if the company belongs to a certain industry and zero otherwise (here I have three industries). The regression should be a weighted linear regression, using the value weights of industry j in the value-weighted market, as weights. In R it would look something like this:

lm(r ~ I1 + I2 + I3, weights = w, data = data)


Now I would like to constrain the regression such that the weighted coefficients for the dummies to add up to zero:

$$\sum_{j=1}^{3} w_{j} \beta_{j} = 0$$

In a second step, the regression will be extended to continuous variables (i.e. risk factor returns). Those variables should be weighted with the stocks' market capitalizations. So I have one regression for which I have different weights for the continuous variables and the dummy variables. Once market capitalizations and once the weights of the industries in the market. Additionally, I have the mentioned condition for the weighted dummy coefficients.

Does anyone know how I can achieve that in R?

If anyone is interested, I would like to reproduce the results of the paper:

Jose Menchero (2010) - Characteristics of Factor Portfolios

which relies on results of

Steven L. Heston, K. Geert Rouwenhorst (1994) - Does industrial structure explain the benefits of international diversification?

Thank you very much in advance.

The easiest way to solve this problem is to algebraically eliminate one of the weights by solving for one of the regression weights in

$$\sum_{j=1}^{3} w_{j} \beta_{j} = 0.$$

For example, if we solve for $$\beta_1$$ we get

$$(1) \quad \beta_1=-\dfrac{w_2}{w_1}\beta_2-\dfrac{w_3}{w_1}\beta_3.$$

Now, replace $$\beta_1$$ in the regression equation

$$r_{i} = \alpha + \beta_{1} I_{i, 1} + \beta_{2} I_{i, 2} + \beta_{3} I_{i, 3} + \epsilon_{i}$$ $$= \alpha + \left[-\dfrac{w_2}{w_1}\beta_2-\dfrac{w_3}{w_1}\beta_3\right] I_{i, 1} + \beta_{2} I_{i, 2} + \beta_{3} I_{i, 3} + \epsilon_{i}$$ $$\implies r_i= \alpha + \beta_{2} \left[I_{i, 2}-\dfrac{w_2}{w_1}I_{i,1}\right] + \beta_{3} \left[I_{i, 3} -\dfrac{w_3}{w_1}I_{i,1}\right] + \epsilon_{i}$$

The terms in the brackets are your new predictors. This problem has now turned to the unconstrained linear regression problem. As soon as you have determined the coefficients $$\beta_2$$ and $$\beta_3$$ you can use equation $$(1)$$ to determine the last regression weight $$\beta_1$$.