correlation and linear relationship

  1. can i say that “perfect correlation” is equal to “linear relationship”? 2) Is it correct that independent variables can be “correlated”, but if they are “highly correlated” with each other, multicollinearity will be a problem?

1/ Linear relationship implies correlation coefficient of +1 or -1 (depending on the nature of the linear relationship) but the inverse is not necessarily true - the correlation coefficient as discussed in the CFA curriculum is a linear construct and it makes no sense to apply it to non-linear data. 2/ Yes.

sorry but i don’t understand your 1st explanation. Why the inverse is not necessarily true? Thanks.

!) I think it’s a semantic issue. Perfect correlation implies a linear relationship in the variables as stated. For example, if Y= A + X^2, Y would be perfectly correlated with X^2 (i.e. if you regress Y on X^2 you’d get an R^2 of 100%, but Y would be nonlinear in X. 2) Yes, but it goes further - if you had five independent variables, the univariate correlations between each pair of variables might not be that high, but there could still be multicollinearity with some of the variables: As an example, assume that X1 is only slightly correlated (assume 20%) with X2 through X5, and that X2, X3, X4, and X5 are orthogonal (i.e. noncorrelated) to each other. It’s possible that X1 might still be highly collinear with a linear combination of combination of X2, 3, 4, and 5. As a simple way of looking at it, assume that if you regressed X1 on X2, X3, X4, and X5, and got an R^2 of 0.90. This would means that almost ALL of X1 is explained by a linear combination of X2–X5. Hence, in this case, even tho there aren’t very high univariate correlations, there still is multicollinearity. In fact, this is one common way of measuring the degree of multicollinearity is to calculate for each variable the calculate 1/(1=R^2), where R^2 is the R^2 of a regression of that independent variable on all the other independent variables. This measure is known as the Variance Inflation Factor, or VIF for that IV, and is increasing in the degreee of multicollinearity.

Gosh, you guys dont need me anymore…

Just glad to have a question I can handle.