Question

Suppose that normal error regression model (2.1) is applicable except that the error variance is not constant; rather the variance is larger, the larger is X. Does β₁ = 0 still imply that there is no linear association between X and Y? That there is no association between X and Y? Explain.

Answer 1

Suppose you simulate Xi∼N(0,1) and then you set Yi=1+εi, with εi∼N(0,X²ⁱ) for i=1,…,n. A scatterplot of a random sample from the pair (X, Y) will be something like:

so it is clear that X and Yare not independent: although the conditional mean of Y remains the same regardless of the value of X, the effect of X on the conditional variance of Y is evident. The linear regression model Y=β0+β1X+ε holds (with β0=1 β0=1 ) and the red line represents the fitted linear model to the sample.

So, if you understand "association" as "dependence", here you have a counterexample where X and Y are associated but linearly independent (or "no linearly associated"). Roughly speaking, the reason is that linear association looks only to relationships between Y and X that could be modeled by a straight line and it may produce this kind of effect when the dependence between them is more complex.