# OLS Estimator for ARMA(1,1)

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!

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 $$y_{t-1}$$.

• nice answer. just two quick things: 1) In your second to last sentence, you meant not correlated with the error term $u_t$. 2) what I don't get about instrumental variables is the following: I understand that, by using $y_{t-2}$, you then meet the OLS assumption of uncorrelatedness with the error term so the regression is "legal". But doesn't the IV coefficient have totally different meaning since $y_{t-1} is gone ? Why is the IV helpful to the original model ? Thanks and apologies for the possibly stupid question. Commented May 1, 2023 at 6:25 • Hi, thanks for the comment. Yup, I meant 'not correlated' and edited it accordingly. I am not sure what you mean by$y_{t-1}$is gone. In this example, the estimate for$\alpha$would be sum($y_{t-2}$$y_{t})/sum(y_{t-2}$$y_{t-1}$). There would still be$y_{t-1}\$ in the formula for IV estimation. If I remember correctly, IV estimates generally tend to be larger than OLS estimates. But I have no idea if this continues to hold in time-series although I am inclined to say yes. Commented May 1, 2023 at 15:41
• Thanks. I'm going to have review IV so don't worry about my question. Your answer to the OP was fine. I'm just vague on the IV specifics but I have a book that covers IV quite nicely so I'll dig it up. I'll also put the title of it in another comment once I dig up the book. Commented May 5, 2023 at 8:22