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\begin{center}
\textsclarge{Analysis Checklist}
\end{center}

When faced with an analysis of a real data set, the following steps should be followed:

\begin{enumerate}[1.]
  \item \textbf{Initial graphical checks:} Assess
  \begin{enumerate}[(i)]
  
  \item \textit{mean and variance stationarity}: identify any systematic changes in mean and or variance;
  
  \item \textit{correlation patterns}: examine evidence of strong positive or negative dependence;
  
  \item \textit{periodic or seasonal components}: identify any evident seasonalities (eg annual, quarterly, monthly cycles) that influence the mean level in a deterministic fashion;
  
  \item \textit{autocorrelation}: plot the acf and examine for indications of non-stationarity (ie a slowly decaying acf).
      
  \end{enumerate}
  \item \textbf{Pre-processing:} use pre-processing tools to attempt to form the decomposition
  \[
X_t = m_t + s_t + Y_t \qquad t=1,2,\ldots,n
\]
where $m_t$ is the deterministic mean at time $t$, $s_t$ is the deterministic seasonal component, and $Y_t$ is the stationary component of the variation.   This may require
  \begin{enumerate}[(i)]
  
  \item log or other transformations of the data;
  
  \item smoothing to obtain $m_t$;
  
  \item parametric or non-parametric modelling of $s_t$.
  
  \end{enumerate}
  
  The \texttt{R} functions \texttt{decompose} \texttt{stl}, \texttt{HoltWinters} can be used.
  
  \medskip
  \textbf{Note:} Pre-processing needs to be carried out with care if forecasting is a goal, as non-parametric procedures may not be suitable for forecasting out of sample (ie for future values).
  

  \item \textbf{Differencing:} If there is no evident mean variation, but the acf decays slowly, then there may be \textit{stochastic} non-stationarity, that is, a random walk like structure in the data.  This can be removed by \textit{differencing}, that is for observed series $\{x_t\}$ we might difference $d=1,2,3,\ldots$ times
      \[
      y_t = (1-B)^d x_t
      \]
      to try to obtain stationarity.   In addition, there may be \textit{stochastic seasonality} with season $s$, so \textit{ seasonal differencing} of order $D=1,2,3,\ldots$ at lag $s$
      \[
      y_t = (1-B^s)^D x_t
      \]
      might be used.
      
  \item \textbf{Model fitting and selection:}  Having processed observed series $\{x_t\}$ into possibly stationary process $\{y_t\}$, we may then
    \begin{enumerate}[(i)]

  \item fit $\ARMA(p,q)$ models for a range of values of $p$ and $q$ using \texttt{arima};

  \item select the preferred model using AIC or BIC;
  
  \item summarize the preferred model in terms of parameter estimates, fitted acf and possibly fitted values or forecasts.

  \end{enumerate}
    
  
  \item \textbf{Residual checks:}  Having selected an $\ARMA(p,q)$ model, obtain the residuals from the fit, and plot them and their acf.  The ideally, the residuals should resemble a white noise process, that is, 
       \begin{enumerate}[(i)]

  \item have zero mean and constant variance: this can be checked in a plot.
  
  \item be uncorrelated; this can be checked by computing the residual acf.
  
  \end{enumerate}  
  

\end{enumerate}
  \textbf{Note:}  In a real data analysis, different valid approaches may lead to different models being selected, but no model might fit the data perfectly well.

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