Interpreting results of regression of variable with residuals from previous regression I have regression output like so:
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      e   R-squared:                       0.284
Model:                            OLS   Adj. R-squared:                  0.282
Method:                 Least Squares   F-statistic:                     197.4
Date:                Sun, 25 Mar 2018   Prob (F-statistic):           5.19e-38
Time:                        06:08:18   Log-Likelihood:                -206.19
No. Observations:                 500   AIC:                             416.4
Df Residuals:                     498   BIC:                             424.8
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     -0.4526      0.036    -12.525      0.000      -0.524      -0.382
education      0.2178      0.016     14.051      0.000       0.187       0.248
==============================================================================
Omnibus:                        4.162   Durbin-Watson:                   1.929
Prob(Omnibus):                  0.125   Jarque-Bera (JB):                4.201
Skew:                           0.201   Prob(JB):                        0.122
Kurtosis:                       2.799   Cond. No.                         5.92
==============================================================================

The dependent variable is the residuals from an earlier regression from gender to ln(wages) and the independent variable is education from the same data.
What does it mean for education to be negatively correlated with the residuals as shown here? Does it mean that, in the original regression, error was in fact lower for individuals with higher education?
 A: You have calculated the semi-partial correlation for ln(wages) and education controlling for gender—though you have done so using a follow-up regression instead of just calculating the correlation.  For my follow-up below, I will use the variables $w$ for ln(wages), $s$ for education (schooling) and $g$ for gender.
In brief, if you calculate the residuals from the models where you use gender to predict the other two variables (using R syntax, happy to provide alternatives in comments):
u <- lm(w~g)$resid
v <- lm(s~g)$redid

you can then take the correlation of $u$ and $v$ to calculate the partial correlation.  If you take the correlation of $u$ with $s$, you obtain the semi-partial correlation.  The first correlation removes any effects of education ($s$) from both the dependent and other independent variable  The second correlation removes the effect education from the other independent variable.


Again, you used a regression approach (instead of correlation), but the interpretations would generally be the same.
Addendum #1
I wouldn't post an answer with an addendum usually.  However, in this case, I think what I originally typed might still be useful.  The issue is that it doesn't align with your data analysis.  In particular, you used the residuals from a model predicting $w$ from $g$.  The explanation I provided would require your analyses replacing the residuals (in your model) with $w$ and replacing education (in your model) with the residuals $v$.
