Financial Risk Manager Part 1

Explanation

For an autoregressive (AR) process of order q to be stationary, the roots of the characteristic equation must lie inside the unit circle.

Key Points:

• Stationarity Condition: In time series analysis, an AR(q) process is stationary if all roots of its characteristic equation have absolute values less than 1 (i.e., lie inside the unit circle in the complex plane).
• Mathematical Formulation: For an AR(q) process: $X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + \cdots + \phi_q X_{t-q} + \epsilon_t$ The characteristic equation is: $`$1` - \phi_1 z - \phi_2 z^2 - \cdots - \phi_q z^q = 0$$
• Root Location:
  • Inside unit circle (|z| < 1): Stationary process
  • On unit circle (|z| = 1): Unit root, non-stationary
  • Outside unit circle (|z| > 1): Explosive, non-stationary

Why Option C is Correct:

When roots lie inside the unit circle, the process exhibits mean reversion and finite variance, which are essential properties of stationarity. This ensures that shocks to the system decay over time rather than persisting indefinitely.