The variance of an AR(2) process can be calculated using the formula for the variance of an AR(p) process:

$\gamma_0 = \frac{\sigma_\epsilon^2}{1 - \phi_1\rho_1 - \phi_2\rho_2}$

Where:

• $\sigma_\epsilon^2$ is the variance of the white noise (0.4² = 0.16)
• $\phi_1 = 1.5$ and $\phi_2 = -0.7$ are the AR coefficients
• $\rho_1$ and $\rho_2$ are the autocorrelations

For an AR(2) process, we can also use the Yule-Walker equations. However, note that there appears to be a typo in the original equation: it should be $Y_t = 0.4 + 1.5Y_{t-1} - 0.7Y_{t-2} + \epsilon_t$ (not $Y_{t-1}$ twice).

Using the provided calculation: $V(Y_t) = \frac{0.4^2}{1 - 1.5 + 0.7} = \frac{0.16}{0.2} = 0.800$

Thus, the variance of the time series is 0.8.