OLS with I(1) regressor and correlated error variance

Question

I have two processes:

• integrated I(1) process: $x_t = x_{t-1} + u_t = \sum_{s=1}^t u_s$ with $x_0=0$ and $u_t \sim iid(0, \sigma_u^2)$.
• an error term $v_t \sim iid(0, 1)$.

My objective is to show that an OLS estimator of $\beta$ in $y_t = \beta x_{t-1} + x_{t-1}^2 v_t$ is inconsistent. (think of it as $x_{t-1}^4$ being the variance of an error rewritten as $\nu_t = v_t x_{t-1}^2$)

Obviously, the error term is correlated with $x_{t-1}$ directly via the variance. However I thought the expectation $\mathbb{E}[ x_{t-1} x_{t-1}^2 v_t]$ would still be zero (by Law of Iterated Expectations).
But in this case, how would the estimator be inconsistent? Because it diverges/needs to be "shut down" by a rate which depends on $\sigma_u$ ?
I can't find why the LIE would not work here (can develop $x_{t-1}$ in any way, it still does not matter, I can t get rid of $v_t$), even though, in simulations (with gaussian noise), even with large sample size, the corresponding sample mean is generally 'far' from zero).

Answer 1

To demonstrate consistency we must show, as always, that the sampling error $\hat\beta-\beta$ goes to zero in probability. Recall $$ \hat\beta=\frac{\sum_tx_{t-1}y_t}{\sum_tx_{t-1}^2} $$ Plugging in for $y_t$ gives, after some manipulation, $$ \hat\beta-\beta=\frac{\sum_tx_{t-1}\nu_t}{\sum_tx_{t-1}^2}=\frac{\sum_tx_{t-1}^3v_t}{\sum_tx_{t-1}^2} $$ It shall prove useful to rewrite this as $$ \hat\beta-\beta=\frac{\frac{1}{T^2}\sum_tx_{t-1}^3v_t}{\frac{1}{T^2}\sum_tx_{t-1}^2} $$ From, e.g., Theorem 4.2 in Ibragimov and Phillips (2008), we may deduce that $$ \frac{1}{T^2}\sum_tx_{t-1}^3v_t\to_d\int_0^1W_u(r)^3dW_v(r) $$ Here $W_u(r)$ is the Brownian motion associated with $u_t$ and $W_v(r)$ that associated with $v_t$. That is $1/\sqrt{T}\sum_{t=1}^{[Tr]}u_t\to_dW_u(r)$.

It is a fairly standard result that $$ \frac{1}{T^2}\sum_tx_{t-1}^2\to_d\int_0^1W_u(r)^2dr $$ Hence, $$ \hat\beta-\beta\to_d\frac{\int_0^1W_u(r)^3dW_v(r)}{\int_0^1W_u(r)^2dr} $$ which is some nonstandard random variable, and hence $O_p(1)$. Thus, $\hat\beta-\beta$ is not $o_p(1)$ and hence does not converge to zero in probability.

As for the use of the LIE, recall that with nonstationary time series, expected values often do not exist, in which case the LIE cannot be applied.