Should we control for the sub-item of a composite variable

[Qingfeng Liu]

Dear Statalist:

I want to study the effect of X on Y.
X is a composite variable, eg. X=A*exp(B)-C
Should I control for A,B,C when do regression?
On one hand, according to "A Crash Course in Good and Bad Controls", A,B,C are "Common Causes" for X and Y.
On the other hand, controlling for A,B,C may generate multicollinearity problems.

So in the regression, should we also control the effects of A,B,C separately?

[Carlo Lazzaro]

Qingfeng:
the risk of your approach is regression overfitting, like in the following toy-example (please see -foreign- variables, when considered continuous or categorical):

. use "C:\Program Files\Stata18\ado\base\a\auto.dta"
(1978 automobile data)

. gen wanted=( rep78*foreign )


. regress wanted i.rep78 i.foreign

      Source |       SS           df       MS      Number of obs   =        69
-------------+----------------------------------   F(5, 63)        =    821.57
       Model |  274.400362         5  54.8800725   Prob > F        =    0.0000
    Residual |  4.20833333        63  .066798942   R-squared       =    0.9849
-------------+----------------------------------   Adj R-squared   =    0.9837
       Total |  278.608696        68   4.0971867   Root MSE        =    .25845

------------------------------------------------------------------------------
      wanted | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       rep78 |
          2  |  -1.03e-14   .2043265    -0.00   1.000     -.408314     .408314
          3  |   -.087963   .1889489    -0.47   0.643    -.4655473    .2896213
          4  |   .0601852   .1974852     0.30   0.762    -.3344575    .4548279
          5  |   .9166667   .2110276     4.34   0.000     .4949618    1.338372
             |
     foreign |
    Foreign  |    3.87963   .0869457    44.62   0.000     3.705883    4.053377
       _cons |   1.02e-14   .1827552     0.00   1.000    -.3652072    .3652072
------------------------------------------------------------------------------

. estat vif

    Variable |       VIF       1/VIF  
-------------+----------------------
       rep78 |
          2  |      4.42    0.226230
          3  |      9.06    0.110343
          4  |      7.77    0.128738
          5  |      6.16    0.162226
   1.foreign |      1.65    0.604870
-------------+----------------------
    Mean VIF |      5.81

. regress wanted c.rep78 c.foreign

      Source |       SS           df       MS      Number of obs   =        69
-------------+----------------------------------   F(2, 66)        =   1139.65
       Model |  270.768288         2  135.384144   Prob > F        =    0.0000
    Residual |  7.84040724        66  .118794049   R-squared       =    0.9719
-------------+----------------------------------   Adj R-squared   =    0.9710
       Total |  278.608696        68   4.0971867   Root MSE        =    .34467

------------------------------------------------------------------------------
      wanted | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       rep78 |   .2377382   .0523998     4.54   0.000     .1331186    .3423578
     foreign |   3.985004   .1119138    35.61   0.000     3.761561    4.208447
       _cons |  -.7181674   .1659245    -4.33   0.000    -1.049446   -.3868885
------------------------------------------------------------------------------

. estat vif

    Variable |       VIF       1/VIF  
-------------+----------------------
     foreign |      1.54    0.649255
       rep78 |      1.54    0.649255
-------------+----------------------
    Mean VIF |      1.54

.

It is noteworthy that, in both cases, the VIF is not alarming.

[Maarten Buis]

It is by no means given that A, B, and C are common causes for X and Y. This would require that A, B, and C are related to Y independent of X. This is mainly a theoretical question, as it is hard (in practice, impossible) to figure that out empirically:

The only reason that a model \(y=\beta_0 + \beta_1 A + \beta_2 B + \beta_3 C + \beta_4 X + \varepsilon \) is identified is the non-linear relationship between A, B, C and X. Relying on only the functional form to identify a model is way to fragile for my taste. I am not alone in that: the common strategy is to leave A, B, and C out and thus assume that all the effect of A, B, and C on Y is through X. If you are really worried about that, then you should find a different way to identify the model: maybe a direct measurement of X independent of A, B, and C or maybe an instrumental variable.

[daniel klein]

I agree with Maarten that this is mainly a theoretical question.

The model \(y= b\left(Ae^{B}-C\right)\) imposes a linear relationship between the composite and the outcome. The model \(y=b_{1}A+b_{2}B+b_{3}C+b_{4}\left(Ae^{B}-C\right)\) implies, among other things, that the association between \(A\) and \(y\) depend on the values of \(B\) (and vice versa). But the (conditional) main effects seem misspecified. Anyway, you need to figure out the data-generating process that you wish to approximate with your model.

[daniel klein]

It is by no means given that A, B, and C are common causes for X and Y. This would require that A, B, and C are related to Y independent of X.

Perhaps more fundamentally, from the original post, it is not even clear that X exists independently from A, B, and C. Think of the body mass index. It's probably quite controversial, I believe non-sensical, to think of weight and height as "causes" of BMI.

Edit: For those who are more interested, my statement relates closely to the discussion of reflective vs. formative models; I leave it at these keywords.