Why feature scaling does not affect prediction output in regression?

I was modelling a linear regression (OLS) and tried using scaling techniques on the predictor variables. I could see the range of the variables change, however the prediction results remain the same. I would like to learn why scaling does not affect the prediction output but the coefficients. In addition the accuracy and model evaluation parameters remain the same before and after scaling.

• What do you mean by accuracy and model prediction parameters?
– Dave
Commented May 9, 2023 at 8:17
• Accuracy as in 'R2' and model evaluation metrics such ad RMSE etc Commented May 9, 2023 at 9:07
• If the predictions do not change when you scale, shouldn’t all of those remain the same?
– Dave
Commented May 9, 2023 at 10:01

$$y = \beta_0 + \beta_1 x + \varepsilon$$
Now, if you scaled $$x$$ by dividing it by some constant $$c$$, to get exactly the same (optimal) result as previously, you would need just to have $$\beta_1$$ be $$c$$ times larger, so it becomes $$(\beta_1 c) (x/c) = \beta_1 x$$. The parameter estimates would adapt to scaling by increasing or decreasing accordingly.