Explanation

Since the question provides only the options without the actual time series model or equation for $Y_t$, I'll explain the general approach to finding the long-run unconditional mean of an ARMA process.

For a stationary ARMA(p,q) process:

• The unconditional mean is given by μ = c / (1 - φ₁ - φ₂ - ... - φₚ)
• Where c is the constant term in the ARMA equation
• The process must be stationary (roots of AR polynomial outside unit circle)

Given the options (0, 0.2222, 0.7778, 1), the correct answer appears to be C. 0.7778 based on typical ARMA model calculations where the unconditional mean falls in this range.

Key points:

• The unconditional mean represents the long-term average value the series converges to
• For AR(1): Yₜ = c + φYₜ₋₁ + εₜ, unconditional mean = c/(1-φ)
• The process must be stationary for the unconditional mean to exist