= Solution
Take the driving signs $a_t=\alpha(t)$ to be independent across times, with $\mathbb Ea_t=0$ and $a_t^2=1$. Temporal independence of the drivers is an assumption needed in addition to their stated equal one-time <probabilities>. For $X_t=\delta y(t)=a_{t-2}a_{t-1}+a_t$,
$$
\boxed{\mathbb EX_t=\mathbb Ea_{t-2}\,\mathbb Ea_{t-1}+\mathbb Ea_t=0.}
$$
As a useful scale check, $X_t$ takes values $-2,0,2$ with <probabilities> $1/4,1/2,1/4$. Its <variance> is therefore $\boxed{\mathbb EX_t^2=2}$, rather than the unit <variance> of a single driving sign.
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