Does centering variables for regression always result in unchanged coefficients?

[monsmatglad]

I am studying mean-centering for multiple linear regression (ols).
Specifically I'm talking about the situation when there is interaction.
When centering variables for a regression analysis, my literature tells me that the coefficients do not change? But when there is some sort of interaction between the variables, the coefficients of the non-interaction terms (the variables that take part in the interaction, but are also represented individually) of the variables do in fact change.

When it is said that when centering the variables, "the coefficients do not change", does that only apply to the non-integrated variables?

 

[andrewkirk]

When it is said that when centering the variables, "the coefficients do not change", does that only apply to the non-integrated variables?

What do you mean by 'non-integrated variables'?

 

[monsmatglad]

oops.. Was supposed to be "non-interaction"

 

[andrewkirk]

In that case, yes. Consider the model
$$y_j = a_0 + a_1x_1 + a_2x_2 +a_12x_1x_2 + a_3 x_3+\epsilon_j$$
in which there is an interaction of $x_1,x_2$ but no interactions for $x_3$.
Now centring each variable we get
$$y_j = a'_0 + a'_1(x_1-\bar x_1) + a'_2(x_2-\bar x_2) +a'_{12}(x_1-\bar x_1)(x_2-\bar x_2) + a'_3 (x_3-\bar x_3)+\epsilon_j$$
Rearranging this and matching coefficients to the first equation, we get:

• $a_0=a'_0-a'_1\bar x_1-a'_2\bar x_2-a'_3\bar x_3 +a'_12\bar x_1\bar x_2$
• $a_1=a'_1 - a'_{12}\bar x_2$
• $a_2=a'_2 - a'_{12}\bar x_1$
• $a_3=a'_3$ [no change]
  
• $a_{12}=a'_{12}$ [no change]

So the only coefficients that remain unchanged are those of any variables with no interactions, plus those of any interaction terms.

 

[WWGD]

I think this is just a property of linearity, which I believe is equivalent with a lack of interaction between variables, i.e., linearity "preserves translation" , but non-linear interactions do not.