Consistent estimator

[Jordi Geeraerts]

Dear all,

Suppose we have a lot of observations.
Suppose the following model : y = b*x + e , where e is distributed as N(0,epsilon).
Can a consistent estimator (OLS for example) for 'b' become inconsistent if 'epsilon' becomes very large? Or does it stays consistent but because relative to the huge value of epsilon, it remains close to the true value?

Thank you in advance for an answer,
Jordi

[Joro Kolev]

B = inv(x'x)*x'y = inv(x'x)*x'(b*x + e) = b + inv(x'x)*x'e, where b is the true population parameter and B is the OLS estimator of b.

What you need for consistency of OLS is that
1. inv(x'x/n) converges to some finite quantity, and in your setup this is satisfied
2. x'e/n to converge to 0 as the sample size n grows without bound.

Therefore your question reduces to the question "Does e having an infinite variance somehow imply that x'e/n will not converge to 0 as n grows?"

I think the answer is No, and having a finite variance of e is not a requirement for OLS consistency.

[Jordi Geeraerts]

Dear Mr Kolev,
Thank you very much for your answer!