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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).

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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.

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  • $\begingroup$ thank you, that was it already! $\endgroup$ – pf11 Sep 9 '20 at 13:19

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