Linear Regressions with constraint coefficients

Hi Gabor,

Thanks a lot for your help!

I tried to implement your nonlinear least squares solver on my data set. I was just wondering about the argument start. If I would like to force all my coefficients to be inside an interval, let?s say, between 0 and 1, what kind of starting values are normally recommended for the start argument (e.g. Using a 4 factor model with b1, b2, b3 and b4, I tried start = list(b1 = 0.5, b2 = 0.5, b3 = 0.5, b4 = 0.5))? I also tried other starting values ... Hence, the outputs are very sensitive to that start argument?     

Thanks a lot for your answer in advance!

Kind regards,
Aljosa

Aljosa Aleksandrovic, FRM, CAIA
Quantitative Analyst - Convertibles
aljosa.aleksandrovic at man.com
Tel +41 55 417 76 03

Man Investments (CH) AG
Huobstrasse 3 | 8808 Pf?ffikon SZ | Switzerland

-----Original Message-----
From: Gabor Grothendieck [mailto:ggrothendieck at gmail.com] 
Sent: Dienstag, 26. April 2016 17:59
To: Aleksandrovic, Aljosa (Pfaeffikon)
Cc: r-help at r-project.org
Subject: Re: [R] Linear Regressions with constraint coefficients

This is a quadratic programming problem that you can solve using either a quadratic programming solver with constraints or a general nonlinear solver with constraints.  See https://cran.r-project.org/web/views/Optimization.html
for more info on what is available.

Here is an example using a nonlinear least squares solver and non-negative bound constraints. The constraint that the coefficients sum to 1 is implied by dividing them by their sum and then dividing the coefficients found by their sum at the end:

# test data
set.seed(123)
n <- 1000
X1 <- rnorm(n)
X2 <- rnorm(n)
X3 <- rnorm(n)
Y <- .2 * X1 + .3 * X2 + .5 * X3 + rnorm(n)

# fit
library(nlmrt)
fm <- nlxb(Y ~ (b1 * X1 + b2 * X2 + b3 * X3)/(b1 + b2 + b3),
     data = list(Y = Y, X1 = X1, X2 = X2, X3 = X3),
     lower = numeric(3),
     start = list(b1 = 1, b2 = 2, b3 = 3))

giving the following non-negative coefficients which sum to 1 that are reasonably close to the true values of 0.2, 0.3 and 0.5:

Linear Regressions with constraint coefficients

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