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Given the equation $e_t = pe_{t-1} + v_t$, and given the values of p and $Var(v_t)$, how would you calculate $Var(e_t)$?

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The following is true if your process is weakly stationary, which implies that your first and second moments are time-invariant:

$Var(e_t) = Var(p*e_{t-1} + v_t)$

If $v_t$ and $e_t$ are independent it follows that:

$Var(e_t) = p²*Var(e_{t-1})+ Var(v_t)$

If weak stationarity holds, you finally get:

$Var(e_t)= \frac{1}{1-p²} Var(v_t)$

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