Weak dependence of MA

Question

So I have this MA : $x_t$ = $e_t$ + A$e_{t-1}$ How do I compute covariance between $x_t$ and $x_{t+1}$?

I'm completely missing the intuition here as my book just states the result. Thanks.

Answer 1

$\renewcommand\cov{\operatorname{Cov}}$ First, by defintion of covariance, $$\cov(X_t,X_{t+1}) = E[X_tX_{t+1}] - E[X_t]E[X_{t+1}].$$ For an $\operatorname{MA}$ process, process $\{e_t\}$ is a white noise process so $E[e_t] = 0$ for all $t$.

Then use that $X_{t+1} = e_{t+1} + Ae_{t}$ so \begin{align*} E[X_tX_{t+1}] &= E[(e_t + Ae_{t-1})(e_{t+1} + Ae_t)] \\&= E[e_te_{t+1}] + AE[e_t^2] + AE[e_{t-1}e_{t+1}] + A^2E[e_{t-1}e_t]. \end{align*}

From the definition of a white noise process, $E[e_te_s] = 0 $ for $t\neq s$. Thus, $$ E[X_tX_{t+1}] = 0+AE[e_t^2] + 0+0 = AE[e_t^2]. $$