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@@ -812,6 +812,55 @@ Hence it is possible to derive the likelihood function and simultaneously estima
 \subsubsection{Residual analysis}
 See \ref{residual-analysis}
 
+\subsection{ARMA(p,q)-models}
+An $ARMA(p,q)$ model combines $AR(p)$ and $MA(q)$:
+$$X_t = \alpha_1 X_{t-1} + \dots + \alpha_p X_{t-p} + E_t + \beta_1 E_{t-1} + \dots + \beta_q E{t-q}$$
+where $E_t$ are i.i.d. innovations (=a white noise process).\\
+\vspace{.2cm}
+It‘s easier to write $ARMA(p,q)$’s with the characteristic polynomials: \\
+\vspace{.2cm}
+$\Phi(B)X_t = \Theta(B)E_t$, where \\
+$\Phi(z) = 1 - \alpha_1 z - \dots - \alpha_p z^p$, is the cP of the $AR$-part, and \\
+$\Theta(z) = 1 + \beta_1 z + \dots + \beta_1 z^q$  is the cP of the $MA$-part
+
+\subsubsection{Properties of ARMA(p,q)-Models}
+The stationarity is determined by the $AR(p)$-part of the model:\\
+If the roots of the characteristic polynomial $\Phi(B)$ exceed one in absolute value, the process is stationary.\\
+\vspace{.2cm}
+The invertibility is determined by the $MA(q)$-part of the model:\\
+If the roots of the characteristic polynomial $\Theta(B)$ exceed one in absolute value, the process is invertible.\\
+\vspace{.2cm}
+Any stationary and invertible $ARMA(p,q)$ can either be rewritten in the form of a non-parsimonious $AR(\infty)$ or an $MA(\infty)$.\\
+In practice, we mostly consider shifted $ARMA(p,q)$: $Y_t = m + X_t$
+
+\begin{table}[H]
+    \centering
+    \begin{tabular}{l|l|l}
+        & ACF & PACF \\
+        \hline
+        $AR(p)$ & exponential decay & cut-off at lag $p$ \\
+        $MA(q)$ & cut-off at lag $q$ & exponential decay \\
+        $ARMA(p,q)$ & mix decay/cut-off & mix decay/cut-off \\   
+    \end{tabular}
+    \caption{Comparison of $AR$-,$MA$-, $ARMA$-models}
+\end{table}
+
+\begin{itemize}
+    \item In an $ARMA(p,q)$, depending on the coefficients of the model, either the $AR(p)$ or the $MA(q)$ part can dominate the ACF/PACF characteristics.
+    \item In an $ARMA(p,q)$, depending on the coefficients of the model, either the $AR(p)$ or the $MA(q)$ part can dominate the ACF/PACF characteristics.
+    
+\end{itemize}
+
+\subsubsection{Fitting ARMA-models to data}
+See $AR$- and $MA$-modelling
+
+\subsubsection{Identification of order (p,q)}
+May be more difficult in reality than in theory:
+\begin{itemize}
+    \item We only have one single realization of the time series with finite length. The ACF/PACF plots are not «facts», but are estimates with uncertainty. The superimposed cut-offs may be difficult to identify from the ACF/PACF plots.
+    \item $ARMA(p,q)$ models are parsimonius, but can usually be replaced by high-order pure $AR(p)$ or $MA(q)$ models. This is not a good idea in practice, however!
+    \item In many cases, an AIC grid search over all $ARMA(p,q)$ with $p+q < 5$ may help to identify promising models.
+\end{itemize}
 
 \scriptsize