# st: Obtaining partial effects for mean centred interaction term

 From Phil1899@gmx.de To statalist@hsphsun2.harvard.edu Subject st: Obtaining partial effects for mean centred interaction term Date Wed, 01 Jul 2009 14:50:46 +0200

```Dear Statalist,

I am currently working on a regression model which contains an interaction term. My goal is to find a model which allows me to make useful interpretations. I found several threads regarding this topic and read some articles / books about it; however, I am still unsure about some aspects.

My model looks as follows:

Y = b0 + b1X + b2Z + b3XZ

I mean centred the variable as suggested by Aiken and West (1991), leading to

Y = b0 + b1cX + b2cZ + b3cXcZ,

where c in front of a variable implies that it is centred.

Wooldridge (2008) states that centreing the variables (he includes  cXcZ and X and Z in the regression; however, the results are similar to cX cZ cxcZ) implies that e.g. the coefficient b1 represents the partial effect of X on Y at the mean of Z. Further, as I understand, the t-statistic for the individual coefficient b1 is relevant for the statistical significance; thus, a F-test for joint significance of b1 and b3 is not needed.

As mentioned, Aiken and West (1991) also suggest to mean centre the variables. Moreover, they suggest the following steps to analyse e.g. the effect of X on Y for interesting values of Z:
1.	Create a new variable Zs, which is Z minus the value of Z for which we want the simple slope of Y on X. For simple slopes at the mean of Z, this transformation is the same as centreing Z.
2.	Form a new variable that is X times Zs.
3.	Regress Y on X, the Zs, and the product term, and the t-test for the X coefficient will be relevant for statistical significance.
My question is now whether one should expect to obtain similar results for the partial effects of X and Z for given values of the other variable independent of the chosen approach (Wooldridge 2008 vs. Aiken and West 1991)?
I tried both approaches (always subtracting the mean of the variables) and strangely find that the effect of X is similar in both cases, while the coefficient and t-statistic of Z change. Obviously, I thought that I made a mistake in calculations; however, I rechecked that several times. Is there another explanation for that? And if so, which results are preferable?
I would appreciate any help in this matter!
Below I listed three regression output tables, the first refers to the approach suggested by Wooldridge (2008), the second and third refer to the approach suggested by Aiken and West (1991).

Best regards
Phil

|               Robust
Y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
cX |   .8456389   .2197025     3.85   0.000     .4143864    1.276891
cZ |   .2674772   .2051609     1.30   0.193    -.1352317    .6701861
cXcZ |    .345705   .1313459     2.63   0.009     .0878871     .603523

------------------------------------------------------------------------------
|               Robust
Y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
X |   .8456389   .2197025     3.85   0.000     .4143864    1.276891
cZ |   -.055858   .2458966    -0.23   0.820    -.5385269    .4268109
XcZ |    .345705   .1313459     2.63   0.009      .087887     .603523

------------------------------------------------------------------------------
|               Robust
lnexp |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
cX |  -.3351341    .241403    -1.39   0.165    -.8089824    .1387142
Z |   .7790538   .1445994     5.39   0.000     .4952205    1.062887
cXZ |    .345705   .1313459     2.63   0.009     .0878871     .603523

References:

Aiken, L. S., & West, S. G. (1991): Multiple Regression: Testing and interpreting interactions. Thousand Oaks: Sage.

Wooldrige, J.M. : Introductory Econometrics: A Modern Approach. Mason, OH : Thomson South-Western.

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