The prob. limit of the OLS estimator of AR(1) process with AR(1) errors

Given the model: \begin{aligned} Y_t &= \delta Y_{t-1}+u_t, \\ u_t &= \rho u_{t-1}+\epsilon_t, \end{aligned} where $$\epsilon_t\sim i.i.d. (0,\sigma^2)$$, $$|\delta|,|\rho|<1$$. Then how to find the prob. limit of the OLS estimator $$\hat\delta$$?

I have tried to use WLLN, since the $$Y_t$$ is not i.i.d., however the variance of $$Y_t$$ doesn't converge to 0. WLLN doesn't applies. So how to find the prob. limit of the OLS estimator $$\hat\delta$$?

Actually i calculated $$\hat\delta$$ directly using OLS estimator formula, and get $$$$\hat\delta=\delta+\sum_{t=2}^T y_tu_{t-1}/\sum_{t=2}^Ty_{t-1}^2$$$$. I wonder if i can expand $$Y_t$$ to infinite horizon to conclude $$Y_t$$ as covariance-stationary process or i can just expand $$Y_t$$ to finite horizon as $$Y_t=\sum_{j=0}^{t-1}\delta^ju_{t-j}$$. In order to use WLLN, I also used the above finite horizon expansion to calculate the $$E(\sum_{t=2}^Ty_{t-1}^2/T-2)$$ and find the probability limit is $$\frac{(1+\rho \delta)\sigma^2}{(1-\rho \delta)(1-\delta^2)(1-\rho^2)}$$. However, if i just consider the finite horizon expansion,then the fourth moment expection $$E[(\sum_{t=2}^Ty_{t-1}^2/T-2)^2]$$ is evidently doesn't converge to zero. So now i am quite confused. Is the OLS estimator $$\hat\delta$$ converge to some distribution in probability?

update: uploaded the full question Thanks for everyone, i'd appreciate it if you can add your comment to this question.

• I guess you mean $\vert\delta\vert,\vert\rho\vert<1$ and not 0. Dec 22, 2021 at 11:59
• @orsos yes,sry it is a typo Dec 22, 2021 at 12:02

In this case, the OLS estimator for $$\delta$$ is inconsistent, i.e., $$plim(\hat{\delta}) \neq \delta$$. I think the easiest way to see this, is to subtract $$\rho y_{t-1}$$ in the first equation. You get: \begin{align} y_{t}-\rho y_{t-1}&=\delta y_{t-1}+u_t-\rho(\delta y_{t-2}+u_{t-1}) \end{align} Rearranging the terms yields: $$y_t=(\delta+\rho)y_{t-1}-\delta \rho y_{t-2}+(u_t-\rho u_{t-1})$$ Basically, $$y_{t-2}$$ is an omitted variable in your original model and you can apply the usual omitted variable bias methology. I haven't calculated it yet, but this prob limit should look like: $$plim(\hat{\delta})=\delta-\delta\rho\frac{cov(y_{t-2},y_{t-1})}{var(y_{t-1})}$$ Whatever the second term will look like when it is simplified.