I think I lack somewhat deeper understanding of this topic, but I thought stationarity is required in order for OLS to have asymptotic properties.
"But stationarity is not at all critical for OLS to have its standard asymptotic properties"(Wooldridge, 2012)
I thought stationarity is needed or otherwise OLS would not be consistent, but I guess I'm wrong. Could someone tell me why stationarity is not critical for LLN?
Thanks in advance!
(Stationarity of what? What kind/level of stationarity?)
Given the standard linear regression specification (without any specific stochastic assumptions)
$$\mathbf y = \mathbf X\beta +\mathbf u $$ as a matter of mathematics we have $$\hat\beta_{OLS} = \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf y=\beta + \left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf u$$
For consistency of $\hat\beta_{OLS}$ we need (as sample size $n$ goes to infinity) $$\operatorname {plim}\left [\left(\mathbf X'\mathbf X\right)^{-1}\mathbf X'\mathbf u \right ]= \mathbf 0 \Rightarrow \left(\operatorname {plim}\frac 1n\mathbf X'\mathbf X\right)^{-1} \operatorname {plim}\left (\frac 1n\mathbf X'\mathbf u \right )= \mathbf 0$$
This requires
a) that $\left(\operatorname {plim}\frac 1n\mathbf X'\mathbf X\right)^{-1} < \infty$, and that it converges to a positive definite matrix, which is a condition usually just assumed, and it will be satisfied if the "Grenander conditions" are satisfied (in short, as $n\rightarrow \infty$, no regressor degenerates to a sequence of zeros, no single observation dominates the sum of squares of its series, and the regressor matrix always has full rank). These conditions exclude some kinds of non-stationarity of the regressors, but they do not require covariance-stationarity (which is the one usually meant under the term "stationarity").
b) that $$\operatorname {plim}\left (\frac 1n\mathbf X'\mathbf u \right )= \mathbf 0\Rightarrow \left [\begin{matrix} \operatorname {plim}\frac 1n\sum_{i=1}^nx_{1i}u_i \\ ...\\ \operatorname {plim}\frac 1n\sum_{i=1}^nx_{ki}u_i \end{matrix}\right ] =\mathbf 0 $$
Now Markov's Law of Large Numbers, in order to hold requires that $$\frac 1{n^2}\operatorname {Var}\left(\sum_{i=1}^nx_{1i}u_i\right)\rightarrow 0,\; \text {as}\; n\rightarrow \infty$$
Here too, this condition excludes some kinds of non-stationarity, but it does not require covariance stationarity (for example, both the mean and the variance of each $x_{ji}$ and each $u_i$ may be different -we only need that the variance of the sum is of smaller order than $n^2$). If this condition holds then the Law of Large Numbers applies and we have (abusing notation a bit)
$$\operatorname {plim}\left (\frac 1n\mathbf X'\mathbf u \right )= \left [\begin{matrix} \operatorname {lim}\frac 1n\sum_{i=1}^nE(x_{1i}u_i) \\ ...\\ \operatorname {lim}\frac 1n\sum_{i=1}^nE(x_{ki}u_i) \end{matrix}\right ]$$
For this to be equal to the zero-vector we need that each regressor is contemporaneously uncorrelated with the error term, $E(x_{ji}u_i)=0,\; \forall j,i$. This is a condition related to stochastic dependence/independence, and has nothing to do with stationarity.
For asymptotic normality of $\hat\beta_{OLS}$ we examine
$$\operatorname {plim}\sqrt n(\hat\beta_{OLS} -\beta)= \operatorname {plim}\left[\sqrt n\left(\frac 1n \mathbf X'\mathbf X\right)^{-1}\frac 1n\mathbf X'\mathbf u\right] = \left(\operatorname {plim}\frac 1n\mathbf X'\mathbf X\right)^{-1} \operatorname {plim}\left (\frac 1{\sqrt n}\mathbf X'\mathbf u \right )$$
The first plim was discussed previously. For the Lindeberg-Feller Central Limit Theorem to hold for the second plim, what is required is
a) that each regressor series is comprised of independent r.v.'s,
b) that the errors are independent from each other,
(both these can be relaxed)
c) that the expected values and the variances of the rv's involved are finite, but not necessarily equal
d)and finally that "no term dominates the whole", which is expressed as a condition on the relative magnitude of the variances involved.
So again, some forms of non-stationarity are excluded, but covariance-stationarity is not needed.
