1. ARMA(1,1) Model Definition

The ARMA(1,1) process is given by:

$$ y_t = a_1 y_{t-1} + \epsilon_t + \theta_1 \epsilon_{t-1} $$

where $\epsilon_t$ is a white noise process (mean zero, constant variance, uncorrelated over time).

2. Stationarity Condition: AR Part

  • The process is stationary if the mean, variance, and autocovariances do not depend on time.
  • The key restriction comes from the autoregressive (AR) part:
    • The characteristic equation is $1 - a_1 L = 0$, where $L$ is the lag operator.
    • The root is $L = 1/a_1$.
    • Stationarity requires:

$$ |a_1| < 1 $$

  • If $|a_1| \geq 1$, the process is nonstationary: the mean and variance can grow over time.

3. MA Part: Invertibility

  • The moving average (MA) part does not affect stationarity, but it does affect invertibility (whether the process can be written as an infinite AR process).
  • Invertibility restriction:

$$ |\theta_1| < 1 $$

  • This ensures the MA part can be expressed as a convergent AR(∞) process.

4. Combined ARMA(1,1) Stationarity

  • Summary:
    • The ARMA(1,1) process is stationary if and only if:
      • $|a_1| < 1$
      • The MA parameter $\theta_1$ can take any value for stationarity, but invertibility requires $|\theta_1| < 1$.
  • If these conditions hold, the mean, variance, and autocovariances of $y_t$ are finite and do not depend on time.

5. Interpretation

  • Why is $|a_1| < 1$ required?
    • If $|a_1| \geq 1$, shocks to the process persist or grow, so the process does not settle to a constant mean/variance.
  • MA part:
    • The MA term only affects the short-run dynamics, not the long-run stationarity.
  • Box-Jenkins methodology:
    • When fitting ARMA models, always check the AR roots for stationarity and the MA roots for invertibility.

In summary:

  • For an ARMA(1,1) process, stationarity requires $|a_1| < 1$.
  • Invertibility requires $|\theta_1| < 1$.
  • These restrictions ensure the process is well-behaved and suitable for time series analysis.