How to calculate the linear regression model of function $y=\alpha + \beta k + \beta x$?

Question

I have a linear function $y=\alpha + \beta k + \beta x$ and observation data that consist of pairs of $x$ and $y$. $\alpha$, $\beta$, and $k$ are unknown parameters. I want to estimate the value of $\alpha + \beta k$ and $\beta$. I don't need the values of $\alpha$ and $k$. Is it possible to solve this problem by using the Ordinary Least Squares estimator? Or if there are easier ways to solve this problem?

Answer 1

Is k another variable or is it constant?. If it is a constant, lump the first two terms into , one big variable called K and use ordinary least squares method to estimate K and beta with your labeled data (as you do not care about alpha and k). If it is not, then you need to know the value of k's in order to use OLS to figure out the coefficients. and keep the model $y=\alpha + \beta(k+x)$ and your new pairwise labeled data would be $(x_i+k_i,y_i)$