Explanation

For a stationary time series process, the long-run unconditional mean can be found using the formula for the mean of an AR(1) process. The general form of an AR(1) process is:

$Y_t = c + \phi Y_{t-1} + \epsilon_t$

where $\epsilon_t$ is white noise with mean 0.

The unconditional mean $\mu$ is given by:

$\mu = \frac{c}{1 - \phi}$

In this case, the correct answer is 0.7778, which suggests the process has parameters such that $\frac{c}{1 - \phi} = 0.7778$. This value represents the long-run equilibrium level that the series tends to revert to over time, regardless of its current position.

Key points:

• The unconditional mean exists only if $|\phi| < 1$ (stationarity condition)
• It represents the long-run average value of the process
• For an AR(1) process, shocks have temporary effects and the series reverts to this mean over time