Suppose that $\{X_t\}_{t\in\mathbb Z}$ is an AR($1$) process given by $$ X_t=\mu+\phi X_{t-1}+\varepsilon_t $$ for $t\in\mathbb Z$, where $\mu\in\mathbb R$, $|\phi|<1$ and $\{\varepsilon_t\}_{t\in\mathbb Z}$ are iid random variables such that $\operatorname E\varepsilon_0=0$ and $\operatorname E\varepsilon_0^2=\sigma^2<\infty$. I am trying to evaluate the conditional expectation $$ \operatorname E[\varepsilon_s\varepsilon_t\mid X_1,\ldots,X_{n-1}] $$ with $s,t=2,\ldots,n-1$.
Since $\varepsilon_s=X_s-\mu-\phi X_{s-1}$ and $\varepsilon_t=X_t-\mu-\phi X_{t-1}$ for $s,t=2,\ldots,n-1$, the random variables $\varepsilon_s$ and $\varepsilon_t$ with $s,t=2,\ldots,n-1$ should be measurable with respect to the sigma algebra generated by $X_1,\ldots,X_{n-1}$. Hence, $$ \operatorname E[\varepsilon_s\varepsilon_t\mid X_1,\ldots,X_{n-1}]=\varepsilon_s\varepsilon_t $$ for $s,t=2,\ldots,n-1$. Is this correct?
Any help is much appreciated!
Yes that should be correct. It's usually called the 'pulling out what is known' property of conditional expectation.
https://en.m.wikipedia.org/wiki/Conditional_expectation#Basic_properties