Answer: The roots of the characteristic equation lie outside the unit circle

## Explanation For an autoregressive (AR) process to be stationary, the roots of the characteristic equation must lie **outside** the unit circle. This condition ensures that the process does not exhibit explosive behavior and that its statistical properties (mean, variance, autocovariance) remain constant over time. ### Key Points: 1. **Stationarity Condition**: In time series analysis, an AR(p) process is stationary if all roots of the characteristic equation have absolute values greater than 1 (i.e., lie outside the unit circle in the complex plane). 2. **Mathematical Representation**: For an AR(p) process: $$X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + ... + \phi_p X_{t-p} + \epsilon_t$$ The characteristic equation is: $$1 - \phi_1 z - \phi_2 z^2 - ... - \phi_p z^p = 0$$ Stationarity requires that all roots $z_i$ satisfy $|z_i| > 1$. 3. **Why Outside the Unit Circle?**: When roots lie outside the unit circle, the process has finite variance and mean, and shocks to the system die out over time rather than accumulating. 4. **Common Misconceptions**: - Roots **on** the unit circle indicate a unit root process (non-stationary) - Roots **inside** the unit circle would imply explosive behavior (also non-stationary) This is a fundamental concept in time series analysis and ARIMA modeling.

Author: Nikitesh Somanthe