AR(1) Stationarity Conditions and Yule-Walker Equations

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Time Series Analysis.

Your Task
Consider the autoregressive process $X_t = \phi X_{t-1} + \epsilon_t$, where $\epsilon_t \sim \text{WN}(0, \sigma_\epsilon^2)$.
(1) State the necessary and sufficient condition on $\phi$ for weak (covariance) stationarity.
(2) Derive the autocovariance function $\gamma(k)$ and autocorrelation function $\rho(k)$ using the Yule-Walker equations.
(3) Prove that the Partial Autocorrelation Function (PACF) cuts off sharply after lag 1.

Constraints

• Transitions and increments satisfy the Markov or martingale property with respect to natural filtration.
• Stopping times are almost surely finite.