ACF and PACF of an ARMA(1,1) Process

Consider the ARMA(1,1) process $X_t - \phi X_{t-1} = \varepsilon_t + \theta \varepsilon_{t-1},$ where $|\phi| < 1$, $|\theta| < 1$, and $\varepsilon_t \overset{\text{i.i.d.}}{\sim} N(0, \sigma^2)$. 1. Derive $\gamma(h) = \text{Cov}(X_t, X_{t+h})$ for $h \ge 0$ and the autocorrelation function $\rho(h) = \gamma(h)/\gamma(0)$. 2. Show that $\rho(1)$ can be positive or negative depending on $\phi$ and $\theta$, and describe when each case occurs. 3. Sketch how the PACF behaves for an ARMA(1,1) and explain how the signs of $\phi$ and $\theta$ affect short-lag correlations.