Explanation

For an AR(p) model of the form:

$Y_t = \alpha + \beta_1 Y_{t-1} + \beta_2 Y_{t-2} + \ldots + \beta_p Y_{t-p} + e_t$

The long-term mean (unconditional mean) is given by:

$E(Y_t) = \frac{\alpha}{1 - \beta_1 - \beta_2 - \ldots - \beta_p}$

Given parameters:

• $\alpha = 0.4$
• $\beta_1 = 1.5$
• $\beta_2 = -0.7$

Calculation:

$E(Y_t) = \frac{0.4}{1 - (1.5 - 0.7)} = \frac{0.4}{1 - 0.8} = \frac{0.4}{0.2} = 2$

Verification of stationarity: The sum of AR coefficients is $1.5 + (-0.7) = 0.8 < 1$, which confirms the process is stationary and the long-term mean exists.

Therefore, the long-term mean of the time series is 2.0.