Converting standardized betas back to original variables without intercept

Question

I am working on a multiple linear regression problem with system defined as :

$Y=\sum_{j=1}^k \beta_jx_j$

I'm using the RidgeCV and LassoCV for the regression, and as mentioned in many places, the standardization of the predictor variables should be done before the Ridge or Lasso regression. Because of the physical definition of my system, I don't want to have the intercept term in the equation, that's why there is no $\beta_0$. But if I standardize the predictors with the normal way :

$z_j=\frac{x_j-\overline{x_j}}{S_j}$

when I convert the standardized betas back to origianl Bs that I want, there will be a term of intercept $\hat{\beta_0}-\sum_{j=1}^k \hat\beta_j\frac{\overline{x_j}}{S_j}$ appear as mentioned in the answer of this question Converting standardized betas back to original variables. But in my case as I said before, I don't want to have the term of intercept, how can I get my original Bs?

P.S. I'm thinking about standardize the predictors by: $z_j=\frac{x_j}{S_j}$ but I don't know if it makes any sense... I'm new to statistics so please help!

Answer 1

First, please reconsider: long experience makes us humbly suspect that even when theory indicates an intercept is unnecessary, nature may intervene in the form of errors or variation from our theory that make an intercept statistically significant and important. Leaving out the intercept can (greatly) bias all estimates, as well as providing misleading information about the uncertainties in the estimates.

But in the rare situations where an intercept definitely must be omitted, a geometric interpretation of standardization shows us what to do. Dropping the subscript $j,$ which is not needed for this discussion, your vector $\mathbf{z}$ is the result of orthogonally projecting $\mathbf{x} = (x_1,x_2, \ldots, x_n)$ onto the hyperplane orthogonal to $\mathbf{1}=(1,1,\ldots,1)$ and then rescaling that result to have a unit $L^2$ norm. Omitting the intercept means not doing the projection. Thus, the relevant analog of standardization is to rescale the unprojected vector to unit length. Accordingly, compute

$$S^2 = S(\mathbf{x})^2 = ||\mathbf{x}||^2 = \sum_{i=1}^n x_i^2$$

and then

$$\mathbf{z} = \frac{1}{S}\,\mathbf{x} = \left(\frac{x_1}{S}, \frac{x_2}{S}, \ldots, \frac{x_n}{S}\right).$$

---

In this context the standardized betas $\tilde\beta_i$ are, by definition, the estimates $\hat\gamma_i$ in the model

$$E[Y/S(\mathbf{y})] = \sum_i\gamma_i\, \mathbf{z}_i = \sum_i \gamma_i\,\frac{\mathbf{x}_i}{S(\mathbf{x}_i)}$$

where $\mathbf{y}$ is the vector of observed responses. Because (by virtue of the linearity of the expectation operator $E$) this is algebraically identical to the model

$$E[Y] = \sum_i \frac{S(\mathbf{y})\,\gamma_i}{S(\mathbf{x}_i)} \mathbf{x}_i = \sum_i \beta_i\, \mathbf{x}_i,$$

to convert the standardized betas into the estimates $\hat\beta_i$ for this, the original model, equate the coefficients and deduce that for each $i,$

$$\hat\beta_i = \frac{S(\mathbf{y})}{S(\mathbf{x}_i)}\,\tilde\beta_i.$$

This is recognizable as a simple change of units of measurement for $\mathbf{y}$ and $\mathbf{x}_i.$