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Consider a standard ARMA(1,1) process such as

$$x_t - \beta x_{t-1} = \theta u_{t-1} + u_t$$

where $u_t$ is i.i.d. $u_t \sim N(0,\sigma^2)$. I know how to derive mean and variance with stationary condition ($|\beta| < 1$), but how can I derive mean and variance in general form for all values of $\beta$? This means without stationary or weak dependence of ARMA(1,1) process.

Thanks

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For the first, where $|\beta| < 1.0$, you can write it using the lag operator.

$x_t (1 - \beta L) = (1 + \theta L) u_t $

$X_t = \frac{(1 + \theta L) u_t}{(1- \beta L)} $

Since $|\beta| < 1.0 $, this is an infinite sum that converges:

$X_t = \sum_{i=0}^\infty \beta^{i}( 1 + \theta L) u_{t-i}$

The $u_{t}$ are independent and normal with mean zero and variance $\sigma^2$ so you have a converging infinite sum of iid random variables so you should be able to calculate the mean and the variance. I leave that as an exercise for the reader.

I'm not sure if the second part is possible because the series doesn't converge in that case because $X_t$ is not stationary. Hopefully someone else can say something about that part.

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