## About Time Series

Time Series is designed specifically to study and analyze linear time series, both univariate and multivariate, using Mathematica. It consists of this documentation, one Mathematica package file, and data files.
Mathematica package files are collections of programs written in the Mathematica language, so Time Series can only be used in conjunction with Mathematica. The Mathematica package file provided with Time Series is TimeSeries.m. It contains many of the functions and utilities necessary for time series analysis. MovingAverage, MovingMedian and ExponentialMovingAverage, commonly used for smoothing data, are included in Mathematica.
The primary purpose of the manual is to introduce and illustrate how to use the functions contained in the package. Part 1, User's Guide to Time Series, serves as a more detailed guide to the time series subject. Relevant concepts, methods, and formulas of linear time series analysis as well as more detailed examples are presented so as to make the whole Time Series as self-contained as possible. It is hoped that Time Series can serve as both an instructional resource and a practical tool so it can be used for pedagogical purposes as well as for analysis of real data. For those who want to pursue the detailed derivations and assumptions of different techniques, appropriate references to standard literature are given at the end of the manual. Part 2, Summary of Time Series Functions, summarizes the Mathematica functions provided by TimeSeries.m . It gives the definitions of the functions and examples illustrating their usage. Only those formulas that help define terms and notations are included. This concise summary is meant to be a quick and handy reference for the more advanced user or one familiar with the application package.
The organization of Part 1 is as follows. We introduce the commonly used stationary time series models and the basic theoretical quantities such as covariance and correlation functions in Section 1.2. Nonstationary and seasonal models are discussed in Section 1.3. Various elementary functions that check for stationarity and invertibility and compute correlations both in the univariate and multivariate cases are described in these two sections. A variety of transformations including linear filtering, simple exponential smoothing, and the Box-Cox transformation, which prepare data for modeling, are presented in Section 1.4. Model identification (i.e., selecting the orders of an ARMA model) is dealt with in Section 1.5. The calculation of sample correlations and applications of information criteria to both univariate and multivariate cases are described. Different algorithms for estimating ARMA parameters (the Yule-Walker method, the Levinson-Durbin algorithm, Burg's algorithm, the innovations algorithm, the long AR method, the Hannan-Rissanen procedure, the maximum likelihood method, and the conditional maximum likelihood method) are presented in Section 1.6. Other useful functions and diagnostic checking capabilities are also developed in this section. Section 1.7 is devoted to forecasting using the exact and approximate best linear predictors. Spectral analysis is the theme of Section 1.8. Functions to estimate the power spectrum and smoothing of spectra in time and frequency domains using a variety of windows are provided. In Section 1.9 we present functions to implement the Kalman filter technique. Structural models and univariate ARCH, GARCH, ARCH-in-mean, and GARCH-in-mean models are discussed in Section 1.10. The procedures and functions discussed in earlier sections are used to analyze four different data sets in Section 1.11.
Data sets used in the illustrative examples are also provided with the application package so the results of the examples can be reproduced if desired. These data sets are contained in data files; they can be found in the Data subdirectory of the TimeSeries directory.