diff --git a/main.tex b/main.tex index a268864..e1b3ba4 100644 --- a/main.tex +++ b/main.tex @@ -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