Symbolic ARMA Model Analysis

This chapter extends the APPL language to include the analysis of ARMA (autoregressive moving average) time series models. ARMA models provide a parsimonious and flexible mechanism for modeling the evolution of a time series. Some useful measures of these models (e.g., the autocorrelation function or the spectral density function) are oftentimes tedious to compute by hand, and APPL can help ease the computational burden. Many problems in time series analysis rely on approximate values from Monte Carlo simulations or the central limit theorem rather than exact results. This chapter describes our time series extension to APPL which can compute autocorrelation and partial autocorrelation functions of ARMA models which would otherwise require extremely tedious pencil and paper work or simulation to find, along with several other tools for working with ARMA models. Simulation is not used in any of the results. The time series extension to APPL provides procedures for ARMA models to: calculate an autocorrelation (TSCorrelation), including autocorrelations of models with heteroskedastic error terms, calculate a partial autocorrelation (TSPartialCorrelation), calculate a mean (TSMean), calculate a variance or covariance (TSVariance), plot an autocorrelation or partial correlation function (TSPlot), calculate a spectral density function (SDF), perform a unit roots analysis to determine whether or not an AR(p) model is stationary or an MA(q) model is invertible (UnitRoots), forecast an AR(p) model and calculate a confidence interval (TSForecast), generate a realization of a process (TSRealization), perform an exploratory time series analysis (ETSA) which displays several of the previous commands at once. The remainder of this chapter is organized as follows. Section 11.1 describes the ARMA model. Section 11.2 describes the data structure for storing ARMA models and explains the ways that the mean, variance, autocorrelation, partial autocorrelation and forecasts are computed. Section 11.3 illustrates how to use the time series extension through a series of examples.