Causality MA(1) process

Question

Given is the MA(1) process:

$X_t = Z_t + \theta Z_{t-1}$

Where,

$Z_t \sim WN(0,1)$

For what values of $\theta$ is $X_t$ a causal function?

I know how to show causality for a AR(p) process with use of the backshift operator, but in this case I don't see how to solve it. Help is very much appreciated.

Answer 1

For what values of θ is Xt a causal function?

Always, by definition. Note that $X_t$ is measurable with respect to $(W_t,W_{t-1})$ hence also measurable with respect to the past sigma-algebra $\sigma(W_s;s\leqslant t)$.