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).

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).