# Including correlated variables in multiple regression model.

While doing multiple regression, some of my predictors are correlated. But it's not collinear. The results are given below.

y - dependent variable a,b,c - independent variables

Correlation

          y           a          b             c
y         1.0000000   0.5418774  0.5047409     0.4394508
b         0.5418774   1.0000000  0.8532455     0.7017283
c         0.5047409   0.8532455  1.0000000     0.6983398
d         0.4394508   0.7017283  0.6983398     1.0000000


vif

 a             b             c
3.995067      3.965175      2.127463


Can I include both the variables in the model? After including both the variables in the model I am getting decent results and p value for coefficients for both variables are also significant.

Residuals:
Min       1Q   Median       3Q      Max
-13.3627  -1.3381   0.1509   1.4823   9.1753

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)    7.62649    0.02669  285.79   <2e-16 ***
a              0.78397    0.01473   53.23   <2e-16 ***
b              0.31077    0.01899   16.36   <2e-16 ***
c              0.21349    0.01214   17.59   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.133 on 54635 degrees of freedom
Multiple R-squared:  0.3048,    Adjusted R-squared:  0.3048
F-statistic:  7985 on 3 and 54635 DF,  p-value: < 2.2e-16


I read Collinearity implies correlation but not vice- versa, so is it okay to include correlated variables in a linear model if they are not collinear?

• What is it you want to learn from, or do with, this model? Aug 4, 2017 at 18:52
• I want to identify the coefficients for each variable from this model. Aug 4, 2017 at 18:55
• You have the fitted coefficients, so you're fine. The CIs for the coefficients will be approximately 2X wider than they would have been ($\sqrt{\approx 4} \approx 2$) if the VIFs had been 1. So you have less precision in your estimates. If you compute the CIs and incorporate them into your interpretations, you are fine. Aug 4, 2017 at 19:03
• Yeah. I just wanted to confirm that the correlation doesn't influence my the model. Thanks for the reply. Aug 4, 2017 at 19:31