Answer: 2.0

## 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**.

Author: Tanishq Prabhu