Financial Risk Manager Part 1

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.