1. We start by stating the problem: Find the formulas for the autocovariance at lags $k=0,1,2,3$ for an ARMA(1,1) model. 2. The ARMA(1,1) model is given by: $$ X_t = \phi X_{t-1} + Z_t + \theta Z_{t-1} $$ where $Z_t$ is white noise with variance $\sigma^2$. 3. The autocovariance function $\gamma(k)$ for an ARMA(1,1) model satisfies the recursive relation: $$ \gamma(k) = \phi \gamma(k-1) + \sigma^2 \theta \phi^{k-1} \quad \text{for } k \ge 1 $$ with $\gamma(0)$ given by: $$ \gamma(0) = \frac{(1 + \theta^2 + 2 \phi \theta) \sigma^2}{1 - \phi^2} $$ 4. Using the recursion for $k=1$: $$ \gamma(1) = \phi \gamma(0) + \theta \sigma^2 $$ 5. For $k=2$: $$ \gamma(2) = \phi \gamma(1) $$ 6. For $k=3$: $$ \gamma(3) = \phi \gamma(2) = \phi^2 \gamma(1) $$ 7. Summarizing: - $$\gamma(0) = \frac{(1 + \theta^2 + 2 \phi \theta) \sigma^2}{1 - \phi^2}$$ - $$\gamma(1) = \phi \gamma(0) + \theta \sigma^2$$ - $$\gamma(2) = \phi \gamma(1)$$ - $$\gamma(3) = \phi \gamma(2)$$ These formulas allow computing the autocovariances at lag 0,1,2,3 for the ARMA(1,1) process given parameters $\phi$, $\theta$, and noise variance $\sigma^2$.