Where is the mistake in this proof of the consistency of the OLS estimator $\hat\beta$?

Question

Consider the linear regression model $y=X\beta+u$, where $y\in\mathbb R^T$ and $X\in\mathbb R^{T\times K}$. We make the following assumptions only:

1. $\mathrm E[u]=0$;

2. $\mathrm{Var}[u]=\sigma^2I_T$ for some $\sigma^2<\infty$.
3. $X$ is either non-stochastic OR $X$ is independent of $u$ (my understanding is that this means that all columns of $X$ are independent of $u$ so that $X_{tk}$ is independent of $u_s$ for all $t,s=1,\dots,T$ and $k=1,\dots K$).

No further assumptions are made, i.e. the components of $u$ are not independent or identically distributed. Using Assumptions 1-3, we can show that the OLS estimator $\hat\beta=(X'X)^{-1}y=\beta+(X'X)^{-1}u$ is unbiased and has variance covariance $\sigma^2(X'X)^{-1}$. In addition to these two statistical properties, I want to show that $\hat\beta$ is consistent.

We add assumption

4. $\mathrm{plim}(X'X/T)=Q$ if $X$ is stochastic and $\lim_{T\to\infty} (X'X)/T=Q$ if non-stochastic, for some positive-definite $Q$.

There are two ways to proceed. On the one hand, I can show $$\mathrm{plim\,}\hat\beta=\beta+Q^{-1}\mathrm{plim}\,(X'u/T).$$But I think here we cannot proceed because in order to use some form of the LLN to conclude that the second plim is $E(X'u)=0$ by assumption 3, we also need to make some additional assumption, like that $(X_{tk},u_t)$ is an i.i.d. sequence for all $k$. But this is not the case here.

Alternatively, I tried to use this result saying that an unbiased estimator whose variance tends to 0 is consistent. In this case the variance is $\sigma^2(X'X)^{-1}$ and with non-stochastic $X$ $$\lim_{T\to\infty}\sigma^2(X'X)^{-1}=\lim_{T\to\infty}\sigma^2(X'X)^{-1}T/T=\lim_{T\to\infty} \sigma^2Q^{-1}/T=0,$$ while with stochastic $X$ we have the same result with limits replaced by plims. This should "prove" that $\hat\beta$ is consistent according to the linked result. However, given the above I suspect that there is something wrong here, since $\hat\beta$ shouldn't be consistent with Assumptions 1-4 only. Could you help me find out what I'm doing wrong?

Answer 1

The fundamental sticking point you seem to be at is proving that $\mathrm{plim}(X'u/T) = 0$. We need to look at both the expected value and variance of $X'u/T$:

1. $\mathbb{E} {X'u \over T} = 0$.

2. $\mathbb{E} \left({X'u \over T}\right)\left({X'u \over T}\right)' = \mathbb{E} \left({X'uu' X \over T^2}\right) = {X'(\sigma^2\text{I})X \over T^2} = {\sigma^2 \over T}{X'X \over T}$

This follows straightforwardly from the assumptions.

Therefore,

3. $\mathrm{lim}_{T \to \infty}\mathbb{E}\left({X'u \over T}\right)\left({X'u \over T}\right)' = \mathrm{lim}_{T \to \infty}{\sigma^2 \over T} {X'X \over T}= 0$

But statements 1 and 3 together imply that $\mathrm{plim}(X'u/T) = 0$.

It is essential that we have the limit of ${X'X \over T} = Q$ for this to work, otherwise, you could have a regressor that $\to \infty$ as $T \to \infty$ fast enough so that $X'u/T$ would be $O(T)$ or greater and the $\mathrm{plim}$ wouldn't go to zero.

This is essentially the same as the proof in Econometrics, by Peter Schmidt, which in my edition is ~ 250 pages of proofs.