OLS Estimator for ARMA(1,1)

Question

We consider the ARMA(1,1) model:

$$ y_t = \beta y_{t−1} + \varepsilon_t + \theta \varepsilon_{t−1}. $$

I would like to demonstrate that $y_{t-2}$ provides an instrumental variable estimator for $\beta$. How can I do that ?

Thank you!

Answer 1

First of all, ARMA(1,1) process can be rewritten in this way:

$y_t$ = $\epsilon_t$ + ($\beta$+$\theta$)$\sum_{j=1}^{\infty}$$\beta^{j-1}$$\epsilon_{t-j}$

Therefore,
A = E[$y_{t-2}$$\epsilon_t$] = E[$\epsilon_{t-2}$$\epsilon_t$ + ($\beta$+$\theta$)$\sum_{j=3}^{\infty}$$\beta^{j-3}$$\epsilon_{t-j}$$\epsilon_t$] ---------- eq.(1)

Because $\epsilon_t$ is white noise, E[$\epsilon_{t-2}$$\epsilon_t$]=0, and similarly for other terms in eq.(1). Overall A=0.

Likewise, if you calculate
B = E[$y_{t-2}$$\epsilon_{t-1}$], you will get zero,

which means that $y_{t-2}$ is not correlated with error term $u_t$ = $\epsilon_t$ + $\theta$$\epsilon_{t-1}$. We also know that $y_{t-2}$ is correlated with $y_{t}$ through $y_{t-1}$. This all points to $y_{t-2}$ as a reliable instrument for $y_{t-1}$.