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I am trying to run a weighted least squares model that looks something like this (but could be different):

$y = beta_0 + beta_1 x + beta_2 log(x) + epsilon$

with weights $w_1, w_2, ..$

However, I know, from external knowledge, that whatever the model the outcome must asymptotically converge to a constant for large values of $x$. How can I get an OLS estimate with this constraint.

As an example, let's say if I knew the asymptote $c$, then I can add two fake data points to my model, with very high values of $x$ and very high weights $w$ and $y=c$, and run the normal WLS model and it would give me what I need - except I don't know the value of $c$. Is there a way to impose this constraint - maybe through adding a custom error term to the model?

The model you are looking for is this:

$Y=fracA1+e^-(beta_0+beta_1x+beta_2log(x))$, this could not be obtained but a very similar was obtained.

This code in R might work:

R=data.frame(X=c(1,2,3,4,5,6,7,8,9),Y=c(1,2,3,3,3,3,3,3,3)) # Data in which X is a line, and Y has an still unknown limit.
model=nls(formula = Y~A/(1+exp(-(b0+b1*X))),data=R)
summary(E)

In the result you can see how $A$ says that the limit of 3 (previously unknown) is calculated.

There is a limitation to take into account, is explained in this link, is summarized in the impossibility for all possible models to exist, the "most inside" model should be linear.

The model $beta_0+beta_1x+beta_2log(x)$ could not be used, the model $beta_0+beta_1x$ could be used, take this into account.

First steps with Non-Linear Regression in R

Singular Gradient Error in nls with correct starting values