Why Should We Standardize Regression Variables? Many textbooks and articles (such as this one) advise to standardize variables before entering them into our regression models; i.e., (variable - mean) / standard deviation
However, I just came across a counter example. How do you rationalize this?
Let's say:
$$Y = 2X$$
$$X = \{-3, -2, -1, 0, 1, 2, 3 \}$$
$$Y = \{-6, -4, -2, 0, 2, 4, 6 \}$$
After standardizing,
$$X_{sd} = \{-1.3887301, -0.9258201, -0.46291,  0.0,  0.46291,  0.9258201,  1.3887301 \}$$
$$Y_{sd} = \{-1.3887301, -0.9258201, -0.46291,  0.0,  0.46291,  0.9258201,  1.3887301 \}$$
$$\Rightarrow Y_{sd} = X_{sd}$$
So, in this example, if we standardize the variables, the desired slope (i.e., the effect size) vanishes!!!

The article you are referring (correctly) advises to center covariates involving higher order polynomal terms or interactions. The reason is simple: it reduces collinearity across related terms and thus improves stability of coefficient estimates and their interpretation.
Your example does not make much sense because $Y$ seems to be the response variable. Any transformation on the response changes the result, including standardization.
