Given a random walk $x_t$,
$$x_t=x_{t-1}+\varepsilon_t,$$
consider estimating the slope coefficient $\beta$ in
$$x_t=\beta x_{t-1}+\varepsilon_t$$
by OLS. This question and the following answer noted that $\hat \beta^{OLS}$ has a skewed distribution. My question is,
Are any of OLS assumptions violated in the model $x_t=\beta x_{t-1}+\varepsilon_t$ given that $x_t$ is a random walk? If so, what are the violations?
It is generally assumed that the explanatory variables have finite moments at least up to second order. In this case, as the explanatory variable is a random walk, its variance is not finite. This makes the matrix $Q=\hbox{plim } X′X/n$ not finite, with the consequences discussed below.
The explanatory variable $x_{t-1}$ is not fixed (it is stochastic as it depends on $\epsilon$) and is not independent of the error term $\epsilon_t$. This makes OLS in general biased and inference is not valid in small samples.
The explanatory variable and $\epsilon_t$ are not independent of each other but they are contemporaneously uncorrelated, $E(x_t, u_t) = 0 \forall t$. In the classical regression model this will open the possibility for the the OLS estimator to be consistent in large samples.
If the matrix $Q = \hbox{plim } X′X/n$ were finite and positive definite matrix, then the F-test statistic will follow asymptotically follows the $\chi^2$ distribution. As pointed by @ChristophHanck this matrix is not finite in this context. Hence, the Mann and Wald theorem is not applicable and inference based on OLS will not be reliable even in large samples.
You may be interested in this answer, which discusses similar issues in the context of a stationary AR(q) process.
One of the key assumptions I would list among standard OLS assumptions is that there is no weak LLN for the "average of the $X'X$-matrix" $1/T\sum_tx_{t-1}^2$. Instead, we have weak convergence to a functional of Brownian motion provided we scale by $T^2$, viz.
$$ T^{-2}\sum_{t=1}^Tx^2_{t-1}\Rightarrow\sigma^2\int_0^1W(r)^2d r $$
I would, btw, not quite agree with @Alecos statement in the link you posted that there is no analytical solution to the distribution of the OLSE - we know the asymptotic distribution of the OLSE, when scaled with the suitable superconsistent rate $T$, to be
\begin{eqnarray*} T\left(\hat{\beta}^{OLS}-1\right)&=&T\frac{\sum_{t=1}^Tx_{t-1}\epsilon_{t}}{\sum_{t=1}^Tx_{t-1}^2}\\ &=&\frac{T^{-1}\sum_{t=1}^Tx_{t-1}\epsilon_{t}}{T^{-2}\sum_{t=1}^Tx_{t-1}^2}\\ &\Rightarrow&\frac{\sigma^2/2\{W(1)^2-1\}}{\sigma^2\int_0^1W(r)^2d r}\\ &=&\frac{W(1)^2-1}{2\int_0^1W(r)^2d r}, \end{eqnarray*} the "Dickey-Fuller-distribution" (JASA 1979).
