= Solution
For a causal <autoregressive model> of order $p$, the population <partial autocorrelation function> cuts off after lag $p$, whereas the <autocorrelation function> generally decays. For an invertible <moving-average model> of order $q$, the <autocorrelation function> cuts off after lag $q$, whereas the partial <correlation coefficients> generally decay. For a mixed <autoregressive moving-average model>, both generally decay. Exponential decay may alternate in sign or show damped oscillations. Sample <correlation coefficients> only approximate these patterns, so isolated crossings of the significance bands are not exact order tests.
In PDF Figure 1, the <autocorrelation function> alternates sign, with a large negative lag-one <correlation coefficient> and geometrically decreasing magnitude. The <partial autocorrelation function> has essentially one substantial spike, negative at lag one and about $-0.8$; later values mostly lie within the displayed bands. Thus \b[a stationary <AR(1)> with a negative coefficient is the natural first model], with $\widehat\phi$ initially near $-0.8$. The trace fluctuates around an approximately constant mean and shows no evident deterministic trend.
Fit that model, compare nearby low-order alternatives using <likelihood function> and an <Akaike information criterion> or <Bayesian information criterion>, and inspect the residual <autocorrelation function> and residual <variance>. A few small later <PACF> spikes are expected in a plot with many lags. The roughly $\pm1.96/\sqrt T$ <white noise> bands are a guide, rather than simultaneous guarantees for every lag.
The sign alternation also suggests concentration of spectral power near the high-frequency end: the <AR(1)> denominator $1+\phi^2-2\phi\cos\lambda$ is smallest near $\lambda=\pi$ when $\phi<0$. From estimated $\gamma(0)$ and lag-one <correlation coefficient> one can estimate innovation <variance> as $\widehat\sigma_\varepsilon^2\approx\widehat\gamma(0)(1-\widehat\phi^2)$. The persistence parameter controls the decay rate and <long-run variance of a stationary process>. These plots do not establish a <normal distribution>, <independence>, or the absence of nonlinear dependence; <correlation coefficients> of squared residuals can provide a separate <variance> diagnostic.
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