Consider the linear regression model

where X is an n × k matrix of random regressors, u is an n-vector of error terms, and β is a k-vector of parameters. Suppose X has full column rank with probability one. It is a standard textbook claim that the ordinary least squares (OLS) estimator

of β is unbiased if

, where

signifies almost sure equality. Specifically, it is claimed that unbiasedness follows from the law of iterated expectations and the relation

. As it turns out, this argument is flawed.

(a) Show by example that

does not imply existence of

.

(b) Provide stronger conditions under which

exists (and equals β).