How do I find the variance of this ARMA(2,1) model? $$ X_t=0.5X_{t-2} + e_t + e_{t-1} $$ I know the formula for ARMA(1,1), but when trying to solve I just keep getting an endless path of higher auto-covariances to find.
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2 Answers
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Assuming the series are stationary, you know that $Var[X_t]=Var[X_{t-1}]=Var[X_{t-2}]$. You can figure out the rest yourself.
Also, your process is a restricted ARMA(2,1), because you set coefficient of the first lag to 0.
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First note that a stationary process solution exists for your equation(Assume {$e_{t}$} is White Noise). And you can refer to any time series book to know that this process is causal, which basically means $X_t$ only depends on the previous innovations $e_i$ where $i \leqslant t$. So we know $Cov(e_t,X_{t-2})=Cov(e_{t-1},X_{t-2})=0$, then taking variance on both sides we have the variance $\gamma_0$ satisfies $\gamma_0=0.25*\gamma_0+2\sigma^2_e$, here $\sigma^2_e$ is the variance of the white noise series. Thus we know $\gamma_0=(8/3)*\sigma^2_e$.
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