Answer: It's impossible to determine values of autocovariances without knowing disturbing variances.

## Explanation For an MA(1) process with zero mean: **MA(1) Model:** \[ X_t = \epsilon_t + \theta \epsilon_{t-1} \] where: - \(\epsilon_t\) is white noise with mean 0 and variance \(\sigma^2\) - \(\theta\) is the moving average coefficient (given as 0.5) **Autocovariance at lag 1:** \[ \gamma(1) = \text{Cov}(X_t, X_{t-1}) = \theta \sigma^2 \] **Given:** - \(\theta = 0.5\) - \(\sigma^2\) (variance of disturbances) is **unknown** **Calculation:** \[ \gamma(1) = 0.5 \times \sigma^2 \] Since \(\sigma^2\) is not provided, we cannot determine the numerical value of \(\gamma(1)\). The autocovariance depends on both the MA coefficient \(\theta\) **and** the variance of the disturbances \(\sigma^2\). **Key Points:** 1. For MA(1) processes, the autocovariance at lag 1 is \(\theta \sigma^2\) 2. Without knowing \(\sigma^2\), we cannot compute a numerical value 3. The autocovariance structure depends on both the model parameters and the disturbance variance Therefore, option D is correct: "It's impossible to determine values of autocovariances without knowing disturbing variances."

Author: Nikitesh Somanthe