The variance of OLS estimator (ß ^ ) should be greater than variance of true parameter (ß).True/False? Discuss with reasoning.

Question

The variance of OLS estimator (β ^ ) should be greater than variance of true parameter (β).True/False? Discuss with reasoning.

Answer 1

The answer is - it depends. It depends on your definitions. An estimator for a parameter can have variance. A true parameter, being true, has no variance. However, the framer of this question might be asking about the variance of the population with a given distribution that the parameter is referring to. That population can have a variance, which would however be another parameter. (the variance of ß in this case).

Generally the variance of the estimator of the parameter will be smaller than the variance of the population. In the case of the OLS estimators, the variance of the estimator is equal to the variance of the residuals (which is NOT the same as the true parameter) divided by the the sum of the squares of the x errors (for ß₁) or n times the squares of the x errors (for ß₀) , leading to a value smaller than the MSE (Variance of the Y errors or residuals). BUT this is not what the student asked for.

What the student asked for is the variance of the estimator for ß_hat , which is a value vs the actual ß which has no variance or zero. So variance of ß_hat will always be larger than the variance of ß. (student, this is probably NOT what your assignment/homework/test/book was really asking for)