Explanation

For an ARMA(1,1) model: $Y_t = c + \phi Y_{t-1} + \theta \epsilon_{t-1} + \epsilon_t$

Given: $Y_t = 0.3 + 0.5Y_{t-1} - 0.6\epsilon_{t-1} + \epsilon_t$

Where:

• $c = 0.3$
• $\phi = 0.5$
• $\theta = -0.6$

The long-run mean (unconditional mean) for a covariance-stationary ARMA(1,1) process is:

$E[Y_t] = \frac{c}{1 - \phi}$

Substituting the values:

$E[Y_t] = \frac{0.3}{1 - 0.5} = \frac{0.3}{0.5} = 0.6$

Key points:

• The MA component ($\theta \epsilon_{t-1}$) does not affect the long-run mean
• The process must be covariance-stationary, which requires $|\phi| < 1$ (here $|0.5| < 1$)
• The long-run mean depends only on the constant term and the AR coefficient

Therefore, the correct answer is D: $E[Y_t] = 0.6$