How to normalize data such that an estimated OLS regression vector has pre-specified length ($ L_2$ norm)? I have the following data: $n$ observations on $d$ variables $X$ and one outcome variable $Y$; i.e. $X$ is a $n \times d$ matrix and $Y$ an $n \times 1$ vector.
I consider the following Ordinary Least Squares (OLS) model:
$$Y = X  \beta + \varepsilon$$
where $\beta$ is a $d \times 1$ vector of coefficients.
I want to scale the data such that the estimated parameter vector has a specific length, i.e. the condition I want to achieve is
$$\Vert\hat{\beta}\Vert^2 := \sum_{i = 1}^{d} \hat\beta_i ^2 = c $$
where $c$ is a constant, say $c = 1$.
More specifically, I normalize the data such that all $d$ covariates have mean = 0 and variance = 1 in the following way:

*

*Define matrix $N$ which is a diagonal matrix with the diagonal elements being $\textrm{diag}(\Sigma_{XX}) )$ where $\Sigma_{XX}$ is the covariance matrix of $X$.

*Normalize $X:=X * N^{-1}$
Is it possible to "adjust" this normalizing procedure such that the least-squares parameter vector $\hat\beta := (X^\top X)^{-1} (X^\top Y) $ has a specific norm, namely such that $\sum_{i = 1}^{d} \hat\beta_i ^2 = c$?
By "adjusting" I mean adding factor, which is itself a function of the data $\{X,Y\}$ to matrix $N$ above (or any other way to achieve the desired norm of the estimated parameter vector; I am not wedded to the normalizing approach described in steps 1 and 2 above).
 A: I’m not sure what exactly you are looking for(it would be nice to have some more context), but it’s not to hard to scale the data so that $\beta$ has norm 1. The easiest way is to notice that $X\beta=(aX)(\beta/a)$ and then just solve the linear regression once, then rescale all the columns with some scalar $a$. In order to give you a better process for your problem, I’ll need more context as to the reason for having a fixed norm.
A: Remember that the least squares process is an optimization problem: find the parameters such that the sum of the squares residuals is minimized.
In multivariable calculus, there is a notion of constrained optimization where some function is minimized (or maximized) subject to one or several conditions being satisfied.
With an equality constraint like you have, this is just a matter of using the Lagrange multiplier method. The calculus and algebra of this might lead you somewhere, and it should be possible to program a computer to solve the constrained optimization, too.
