Suppose Yi = aui + bvi + i for i = 1,..., 100. The i are IID with mean 0 and variance 1. The u’s and v’s are fixed not random; these two data variables have mean 0 and variance 1: the correlation between them is r. If r = ±1, show that the design matrix has rank 1. Otherwise,

let aˆ, bˆ be the OLS estimators. Find the variance of aˆ; of bˆ; of aˆ − bˆ.
What happens if r = 0.99? What are the implications of collinearity for applied work? For instance, what sort of inferences about a and b are made easier or harder by collinearity?
Comments. Collinearity sometimes means r = ±1; more often, it means r .
= ±1. A synonym is multicollinearity. The case r = ±1 is better called exact collinearity. Also see lab 7 at the back of the book.