Why high correlation coefficient doesn't guarantee high VIF? I know that there are similar questions about this topic, but my question is not really which metric should we depend on but why they are not equivalent?
There is a saying in the book Basic Econometrics

high zero-order correlations are a sufficient but not a necessary condition for the existence of multicollinearity

However, in the case of my project, I do detect the evidence of a high correlation coefficient and low VIF. May someone explains to me?
Thanks!
 A: What do you consider to be a high correlation coefficient? What do you consider to be a low VIF?
The VIF is calculated by regressing predictor $i$ on all the other predictors, and then calculating $VIF = \frac{1}{1 - R_i^2}$.  If you consider a VIF of 5 to be high, you'd only get a high VIF if $R_i^2$ was greater than or equal to 0.8.      
Now imagine that you have two predictors $i$, and $j$ that have a correlation coefficient of 0.8, which is fairly high, but no other predictors are correlated with predictor $i$ or $j$.  When you then regress $i$ on all the other variables, $R_i^2$ will be slightly greater than the square of the correlation coefficient between $i$ and $j$, which is $0.8^2 = 0.64$ (smaller than the 0.8 needed to achieve a VIF of 5). In other words, the VIF's for $i$ and $j$ will be small, even though the correlation between them is high.
Basically the VIF for predictor $i$ captures how well all other predictors can explain predictor $i$.  But to get a VIF that's considered high (greater than or equal to 5), there has to be a very strong fit when regressing predictor $i$ on the other variables.  As demonstrated, it's definitely possible to have a "high" correlation between two variables, but still have "low" VIF's.
As far as your quote, the opposite can happen as well.  You could have low pairwise correlations, but have high VIF's.  This is because it's possible that there's a strong relationship between predictor $i$ and all the other variables together, even though there's not a high correlation between predictor $i$ any other predictor independently.
