# Why does the variance of the OLS estimator goes to zero as n increases while it is a random variable?

I have a question regarding the OLS estimator. In my class, the teacher asked us to critic this comment:

“We write the unconditional variance of the OLS as $$V(\hat{b}^{ols}) = 1/N 𝜎^2 E(X_i'X_i )^{−1}$$. But this goes to 0 when N increases, it’s impossible because the OLS is a random variable !”

It is true that the unconditional variance of the estimator of the OLS goes to 0 as N increases, but I don't see to which extent is this incompatible with the fact that the estimator is a random variable. Could you help me with this? Thank you!

• Haven't you just written your critique? What more are you looking for? – whuber Jan 9 at 21:01
• Well we have to prove why this statement is false. Intuitively, it seems to me that this is the case because I don't see why a random variable could not have a variance that tends to zero, but I'm not sure if what I'm saying is true? – Raphaël Huleux Jan 9 at 21:25
• There's an imprecision in the language "this goes to 0 when N increases". What actually happens? – Glen_b Jan 9 at 23:12
• "Critique a comment" and "prove it false" are different things. In this case, the meaning of the statement is problematic because (a) it does not stipulate what actually happens "when N increases" and (b) it adduces a nonsensical reason, namely that a sequence of variances cannot converge to zero simply because the variances are associated with random variables. Being "false," therefore, isn't even a quality one can attribute to this statement! – whuber Jan 9 at 23:37