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? – gung Aug 4 '17 at 18:52
• I want to identify the coefficients for each variable from this model. – user2728024 Aug 4 '17 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. – gung Aug 4 '17 at 19:03
• Yeah. I just wanted to confirm that the correlation doesn't influence my the model. Thanks for the reply. – user2728024 Aug 4 '17 at 19:31