R Force regression coefficients to add up to 1

Question

I'm trying to run a simple OLS regression with a restriction that the sum of the coefficients of two variables add up to 1.

I want:

Y = α + β1 * x1 + β2 * x2 + β3 * x3,
where β1 + β2 = 1

I have found how to make a relation between coefficients like:

β1 = 2* β2

But I haven't found how to make restrictions like:

β1 = 1 - β2

How would I do it in this simple example?

data <- data.frame(
  A = c(1,2,3,4),
  B = c(3,2,2,3),
  C = c(3,3,2,3),
  D = c(5,3,3,4)
)

lm(formula = 'D ~ A + B + C', data = data)

Answer 1

b1 + b2 = 1
Let us fit this to the model

result <- lm(Y ~  offset(x1) + I(x2 - x1) + x3, data = data_frame)

This means
Y = a +x1+b2*(x2-x1)+b3*x3

after substituting b1 = 1- b2
new_x = x2-x1 and the coefficient for x1 is 1
b1+b2+b3 = 1

result <-  lm(Y ~ offset(x1) + I(x2 - x1) + I(x3 - x1), data = data_frame)

Y = a +x1 + b2*(x2-x1)+b3*(x3-x1)

after substituting b1 =1-b2-b3

b1+b2+b3+..... = 1

I believe the pattern is obvious... all you have to do is subtract one variable, x1, from the other variables (x2, x3, etc.) and set the coefficient of that variable, x1, to 1.

Example:  b1 + b2 = 1

# Data
data_frame<- iris[, 1:4]
colnames(data_frame) <- c("Y", paste0("x", 1:3, collaapse=""))
# b1 + b2 = 1
result <- lm(Y ~  offset(x1) + I(x2 - x1) + x3, data = data_frame)
coeff_2 <- coef(result)
b_1 <- 1 - coeff_2[2]
b_2 <- coeff_2[2]