Answer: $ E[Y_t] = 0.6 $

## 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 $**

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