I'm looking at some notes on multivariable regression. It states that one of the assumptions for Least Squares to give good estimates is that,
"There is no exact linear relationship between any of the variables (no exact multicolinearity)".
Would this be the same as saying that the explanatory variables are all independent of one another.
On a related note, does the word "exact" in the statement above refer to perfect correlation between two variables. What if the correlation was strong but not perfect, would OLS still give good estimates?
"No multicollinearity" means that no variable is a linear combination of the other variables, i.e. $x_1 = 2 x_2 - 4 x_3 + x_4$ is not allowed. This is equivalent to saying that the design matrix $X$ (where each row corresponds to a data point and each column corresponds to a variable) has linearly independent columns.
Note that there is no notion of randomness in this definition. You do not need a probabilistic model to talk about multicollinearity; it is a statement about the actual data that you have (the design matrix $X$).