# How do I find the variance of a linear combination of terms in a time series?

I'm working through an econometrics textbook and came upon this supposedly simple problem early on.

Suppose you win $\$1$if a fair coin shows heads and lose$\$1$ if it shows tails. Denote the outcome on toss $t$ by $ε_t$. Assume you want to calculate your average winnings on the last four tosses. For each coin toss $t$, your average payoff on the last four tosses is $w_t= 0.25ε_t + 0.25ε_{t-1} + 0.25ε_{t-2} + 0.25ε_{t-3}$

Find var($w_t$), and find var($w_t$) conditional on $ε_{t-3} = ε_{t-2} = 1$

For the first part of the question (finding the variance without any conditions on $ε$), I know that:

Var($w_t$) = Var(0.25ε_t + 0.25ε_{t-1} + 0.25ε_{t-2} + 0.25ε_{t-3}

= Var(0.25ε_t) + Var(0.25ε_{t-1}) + Var(0.25ε_{t-2}) + Var(0.25ε_{t-3})

=0.0625 Var($ε_t$) + 0.0625 Var($ε_{t-1}$) + 0.0625 Var($ε_{t-2}$) + 0.0625 Var($ε_{t-3}$)

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