# ARMA moments proof

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

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.