# 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

$$||\hat{\beta}||^2 := \sum_{i = 1}^{d} \hat\beta_i ^2 = c$$

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:

1. Define matrix $$N$$ which is a diagonal matrix with the diagonal elements being $$diag(\Sigma_{XX}) )$$ where $$\Sigma_{XX}$$ is the covariance matrix of $$X$$.
2. 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).

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.