Multicollinearity and Correlation I am a little bit confused with Collinearity and Correlation. As far as I know Collinearity (which is linear dependency between variables) implies a Correlation between the variables but the reverse is not true. Why so? If we have a strong correlation does it not imply we have a linear relationship between the variables?
 A: Nonlinear relationship can have linear correlation. That is why linear correlation doesn't imply linear relationship, the one that is called collinearity. 
Look at two cases:


*

*$X_1=a+bX_2$

*$X_1=e^{X_2}$


Where $X\sim Uniform(0,1)$
In both cases, you'll have correlation $Cov(X_1,X_2)\ne 0$, but only the first case leads to multicollinearity, a technical issue of identification due to rank deficiency of $X'X$ matrix. 
In OLS you have the following estimate of the parameters: $$\hat\beta=(X'X)^{-1}yX$$
In case of perfect collinearity you get a singular matrix $(X'X)^{-1}$. You can think of a first case as where $X_1$ is a temperature in Celsius while $X_2$ is the temperature in Fahrenheit. Obviously, it doesn't make a sense to have both variables in the model. 
A: Let us say you have a dependent variable $Y$ built as a linear combination of independent variables $Y$ = $\Sigma_i^N \alpha_i X_i$. You are in a situation of multicollinearity if there is an ind. variable $X_j$ that can be written as a linear combination of $X_j = \Sigma_{i\neq j}^N \beta_i X_i$; in plain english, one of the supposedly independent variables can be written as a linear combination of (some of) the others. 
In this nomenclature, if you only have one index $\beta_i$ (let us say, $h$) that is significantly different from 0, while the others are very similar to 0, then you will be able to measure a very strong correlation between $X_j$ and $X_h$. However, this might not be as easily measurable when you have several $\beta_i \neq 0$ (and yet you are in a situation of multicollinearity). 
As a side remark, if you have a correlation between two variables X and Y, you do not have multicollinearity.
