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So, at present, I am studying for a Time Series midterm.

After looking around for a while, I had difficulty finding a unified table describing the behaviour of different time-series models based on their behaviour as a function of their ACF, PACF, Periodogram or CCF plots. As such, I was wondering if we might be able to together put together such a table (e.g. below), plausibly with examples? Here's what I have so far:

$$ \begin{align} \text{Model} \ & || \quad \text{ACF} &|\quad &\text{PACF} &|\quad \text{Periodogram}\\ AR(p) \ & || \quad \text{Tails off} &|\quad &\text{Cuts off after lag } p &|\quad \text{?} \\ MA(q)\ & || \quad \text{Cuts off after lag } q &|\quad &\text{Tails off} &|\quad \text{?} \\ ARMA(p,q) \ & || \quad \text{Tails off} &|\quad &\text{Tails off} &|\quad \text{?} \\ others \ & || \quad \text{cyclic at lag } \frac{1}{\omega} &|\quad &\text{?} &|\quad \text{spike at }\omega \\ \vdots \ & | \ \quad \cdots \end{align}$$

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  • $\begingroup$ Need to use the self study tag $\endgroup$ – Michael R. Chernick Mar 6 '17 at 5:31
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For AR, MA, and ARMA, the peaks in the periodogram will depend on the parameters $\phi$s.

For instance, in the case of AR(2), the spectral shapes are determined by: $|\phi_1(1-\phi_2)|<|4\phi_2|$. When $\phi_1^2+4\phi_2\leq0$, the periodogram presents a peak spectrum, otherwise it will displays a long shape. You can find a quite informative diagram for the special case AR(2) in the following book, page 337:

Cryer, J. D., & Chan, K.-S. (2008). Time Series Analysis: With Applications in R. Springer Science & Business Media.

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