Solved – Why does using residuals as predictor not alter coefficient

interpretationmultiple regressionresiduals

I am working with a data set describing a set of US Senators, their votes on a climate bill, and several characteristics of the state they represent. One of the variables, 'imptot', describes the combined importance for their state's economy of four industries. These industries cause greenhouse gas emissions and would be strongly affected by climate legislation. I also have an indicator of the general greenhouse gas intensity ('GHGintensity') of each state's economy, meaning its greenhouse gas emissions divided by its GDP. These two variables are positively correlated.

I would like to know what's the influence of greenhouse gas intensity on a Senator's vote, after controlling for the total importance of these four industries. In other words, I would like to find out how much variance in voting behavior can be explained exclusively by greenhouse gas intensity, not taking into account the variance that can be explained by the presence of these four specific industries.

I reasoned that if I regressed GHGintensity (dep. var.) on imptot (indep. var.), the residuals of this regression (a new variable called 'GHGresiduals') would represent the variance in GHGintensity that could not be explained by imptot.
However, when I then ran regressions of these variables on voting behavior ('vote'), the following outcomes resulted for the fixed effects portion of the (multilevel logit) model:

               Estimate Std. Error z value Pr(>|z|)
(Intercept)      -0.045      0.607  -0.074    0.941
imptot            0.000      0.022   0.003    0.997
impbentot         0.085      0.530   0.160    0.873
GHGintensity     -0.176      0.076  -2.303    0.021
member            5.399      1.152   4.689    0.000
pubop             1.991      0.233   8.535    0.000

               Estimate Std. Error z value Pr(>|z|)
(Intercept)      -0.839      0.387  -2.169    0.030
imptot           -0.034      0.021  -1.622    0.105
impbentot         0.085      0.530   0.160    0.873
GHGresiduals     -0.176      0.076  -2.303    0.021
member            5.399      1.152   4.689    0.000
pubop             1.991      0.233   8.535    0.000

As you can see, GHGintensity in the first specification had the same coefficient as GHGresiduals in the latter. The coefficients of imptot, however, were different. Because GHGintensity and GHGresiduals are different variables, I did not expect this result. I had expected the effect of greenhouse gas intensity to change, because my intuition was that I had now removed the component of this variable that was correlated with imptot.

Was this intuition incorrect? Should I do a different analysis to answer the questions I am asking? How should I interpret the results I obtained, and why did coefficients not change the way I expected?

EDIT: I am unfortunately not in a position to share the data, but here's the associated R code:

dataSenate$GHGresiduals <- residuals(lm(GHGintensity~imptot,data=dataSenate))

Smodelgeo <- formula(vote ~
  imptot +
  impbentot + 
  GHGresiduals  +
  member +
  pubop +
  (1 | state) + (1 | yearbill))

Sfitgeo <- glmer(as.formula(Smodelgeo), dataSenate, family=binomial,na.action = na.omit)

Best Answer

In the meantime, I have come across an excellent, accessible article on this subject (in the context of linear regression, but its conceptual explanations were helpful nonetheless):

"What residualizing predictors in regression analyses does (and what it does not do)" by Lee H. Wurm and Sebastiano A. Fisicaro (Journal of Memory and Language, Volume 72, April 2014, p 37–48).

It explains why residualizing one predictor does not change its effect size, and how the technique is often employed in a context where in fact, regular multiple regression would have been suitable.

This text has led me to conclude that in my case, residualizing GHGintensity was not necessary. Simply put, the question of "what amount of the variance in voting behavior can be uniquely explained by GHGintensity" can be answered adequately with a typical multiple regression model without residualized predictors. In fact, this is exactly what multiple regression is for.

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