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."