1. The problem is to find the autocovariance \( \gamma_k \) at lags \( k=1,2,3 \) for an ARMA(1,1) process. 2. Recall the general ARMA(1,1) model: $$ X_t = \phi X_{t-1} + Z_t + \theta Z_{t-1} $$ where \( Z_t \) is white noise with variance \( \sigma^2_Z \). 3. The autocovariance function \( \gamma_k = Cov(X_t, X_{t-k}) \) for ARMA(1,1) satisfies: $$ \gamma_0 = \frac{(1+\theta^2 + 2 \phi \theta) \sigma_Z^2}{1-\phi^2} $$ 4. For lag \( k \geq 1 \), the autocovariance is: $$ \gamma_k = \phi^{k-1} \big[\phi + \theta \big] \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2} $$ 5. Specifically, calculate for each \( k \): - \( k=1 \): $$ \gamma_1 = \phi^0 (\phi + \theta) \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2} = (\phi + \theta) \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2} $$ - \( k=2 \): $$ \gamma_2 = \phi^{1} (\phi + \theta) \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2} = \phi (\phi + \theta) \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2} $$ - \( k=3 \): $$ \gamma_3 = \phi^{2} (\phi + \theta) \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2} = \phi^2 (\phi + \theta) \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2} $$ 6. In summary, the autocovariance function for an ARMA(1,1) at lags 1, 2, and 3 are: $$ \gamma_1 = (\phi + \theta) \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2}, \quad \gamma_2 = \phi (\phi + \theta) \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2}, \quad \gamma_3 = \phi^2 (\phi + \theta) \sigma_Z^2 \frac{1+\phi \theta}{1-\phi^2} $$ This formula shows how the AR and MA parameters \( \phi \) and \( \theta \) influence the autocovariance at different lags.