Centering only removes so-called "non-essential" collinearity--meaning that part of collinearity that is caused by the predictors having non-zero means. Cohen, Cohen, West, & Aiken (2003, p. 264) offer an algebraic explanation for this phenomenon. "Essential" collinearity (i.e., collinearity that is due to skewness/asymmetry in the predictors' univariate distributions) cannot be removed by centering.

The standard errors are affected by collinearity because collinearity (=redundancy) increases the uncertainty about a particular predictor's unique contribution to the regression model.

Note that in the case of moderated regression, the regression coefficient for the product term is unaffected by centering. So regardless of whether you center or not, the regression coefficient for the product term (and its standard error) will be the same. However, centering usually leads to an easier interpretation of the lower-order regression coefficients, especially when the predictors do not have a meaningful zero point. So there is good reason to center predictors in moderated regression even beyond the collinearity issue.

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Lawrence Erlbaum Associates Publishers.

Thank you for your reply. Your answer is very precise and my mind is now clear. Thanks!

Yuto Yasuda Because multicollinearity can cause the standard errors of the estimated coefficients to be inflated, high standard errors in the independent variable might be a sign of multicollinearity. This can happen when two or more independent variables in a regression model have a strong correlation.

Because it has no effect on the overall fit of the model, multicollinearity can be difficult to identify. However, it can lead the estimated coefficients to be unstable and difficult to understand. Checking the variance inflation factor (VIF) for each independent variable in the model is one technique to discover multicollinearity. If the VIF is more than 10, multicollinearity may exist.

One method for dealing with multicollinearity is to center the data. This entails removing a variable's mean value from each observation of that variable. Centering can assist to minimize variable correlation and increase the stability of computed coefficients.

Here are some sites that you may find useful:

James D. Miller's "Multicollinearity in Regression Analysis: Problems, Detection, and Solutions" (https://www.statisticssolutions.com/multicollinearity-in-regression-analysis-problems-detection-and-solutions/)

UCLA Institute for Digital Research and Education's "Multicollinearity" (https://www.ats.ucla.edu/stat/mult pkg/faq/general/multicollinearity.htm)

Stat Trek's "Multicollinearity" (https://stattrek.com/regression/multicollinearity.aspx)

Yuto, I published an article at the end of 2013 in which I developed an approach that deals with the issue you raise. This new approach, which I call Sequential Residual Centering (SRC), eliminates the portion of multicollinearity called "nonessential ill-conditioning," which is the multicollinearity among the component terms of an interaction (which Christian Geiser explained in the first reply). For instance, for the highest-order interaction xwz, nonessential ill-conditioning would be the multicollinearity among the following related terms (x, w, z, xw, xz, wz, xwz). [Of course, these terms may also be characterized by "essential ill-conditioning," which refers to their multicollinearity with other specified predictors that are not also component terms of the interaction.]

When VIF values are inflated above 10 (as Fatemeh Khozaei indicated in the above discussion), my SRC approach obtains the same regression slope values but lowers both the standard errors and the VIF (and therefore the p values) for the component terms of an interaction, which improves the sensitivity and statistical conclusion validity of moderated regression (i.e., reduces Type II error). I conducted simulations which validated the SRC under a variety of conditions. SRC is relatively easy to understand and conduct using any regression software. One can still use mean-centered data (with its interpretational advantages that Christian Geiser pointed out in the first reply) even as one uses the SRC approach.

The citation for my article is: Francoeur, R. B. (2013). Could sequential residual centering resolve low sensitivity in moderated regression? Simulations and cancer symptom clusters. Open Journal of Statistics, 3, 24-44. You can locate and download it under my profile on ResearchGate (https://www.researchgate.net/profile/Richard_Francoeur) or in the December 2013 issue of the Open Journal of Statistics (http://dx.doi.org/10.4236/ojs.2013.36A004).

Good luck with your research.

Multicollinearity in moderated regression models is a red herring (i.e., a distraction clue you do not need to worry about). There will almost always be a correlation between X and Z and their product XZ. However, whether you reduce this collinearity by centering or other means, will have absolutely no effect on the statistical test, parameter coefficient, or effect size of the product term. It will change the meaning of the coefficients on the individual terms X and Z. While centering will eliminate most of the "nonessential ill-conditioning," there ALWAYS exists a transformation (subtraction of constants from X and Z that eliminated all of the "nonessential ill-conditioning." Hence, the entire correlation between X and Z and their product is nonessential. see Article Multicollinearity is a red herring in the search for moderat...

Conducting a traditional moderated regression can often be accomplished simply as one run, as Gary Mcclelland mentioned, without involving mean-centering or any other subtraction of constants from X and Z to eliminate the "nonessential ill-conditioning." This provides simultaneous estimates for all statistical tests, parameter coefficients, or effect sizes for all predictors, including the product term.

However, in cases where the p value needs to be exact to ensure proper inferences (e.g., it may fall just above p = .05, and the researcher wants to be sure that it really is not statistically significant), it can be helpful to run a sequentially estimated model (the model is run more than once) based on what I refer to in my earlier response as "sequential residual centering", SRC. This is different from a single run of the moderated regression in which all parameters are estimated simultaneously, which may or may not involve mean-centering.

The model is estimated sequentially to derive estimates for only the three-way term (if specified) in the first model run, then the two-way terms in a second model run, and finally the one-way terms in a third model run. For each sequence of terms, this approach derives the very same slope values (and preserves their same meaning) as in the initial moderated regression, but provides a lower variance inflation factor (VIF). The lower VIF is then used to derive a lower standard error for the slope, shifting the p value to be lower (e.g. the original p value above .05 may now fall below .05; or perhaps another p value that fell below .05 may now fall below p = .01, becoming highly significant).

Of the three sequences of terms run separately in the three model runs, it is necessary to run only the first sequence to derive estimates for the highest-order interaction term itself (e.g. a three-way interaction), in order to determine whether the p value for that interaction term shifts and changes the statistical conclusions about the interaction. (The analyst might decide to run the other two sequences only if the p value for that highest-order interaction term shifts in a way that changes the statistical conclusions).

Thus, the use of residual-centering in a sequentially estimated model, which may be run more than once for piecemeal estimation of the standard errors and p-values, is NOT the same as mean-centering (or other subtraction of constants, from X and Z) to eliminate nonessential ill-conditioning within only one run of the moderated regression for simultaneous estimation of the standard errors and p-values. It may be called for in certain situations of multicollinearity.