Analysis of Time Series
Mohamed M. Shoukri, Mohammad A. Chaudhary · 2007
CONTENTS 5.1 Introduction ............................................................................................... 159 5.2 Simple Descriptive Methods ................................................................... 162 5.2.1 Multiplicative Seasonal Variation Model .................................. 163 5.2.2 Additive Seasonal Variation Model ........................................... 169 5.2.3 Detection of Seasonality: Nonparametric Test ......................... 172 5.2.4 Autoregressive Errors: Detection and Estimation ................... 175 5.2.5 Modeling Seasonality and Trend Using Polynomial and Trigonometric Functions ...................................................... 177 5.3 Fundamental Concepts in the Analysis of Time Series ....................... 179 5.3.1 Stochastic Processes ..................................................................... 180 5.3.2 Stationary Series ........................................................................... 180 5.3.3 The Autocovariance and Autocorrelation Functions .............. 181 5.4 Models for Stationary Time Series ......................................................... 183 5.4.1 Autoregressive Processes ............................................................ 184 5.4.1.1 AR(1) Model ................................................................... 184 5.4.1.2 AR(2) Model (Yule’s Process) ....................................... 185 5.4.2 Moving Average Processes .......................................................... 188 5.4.2.1 First-Order Moving Average Process MA(1) ............. 188 5.4.2.2 Second-Order Moving Average Process MA(2) ................................................................ 189 5.4.3 The Mixed Autoregressive Moving Average Processes .......... 190 5.5 ARIMA Models ......................................................................................... 191 5.6 Forecasting ................................................................................................. 195 5.6.1 AR(1) Model .................................................................................. 195 5.6.2 AR(2) Model .................................................................................. 197 5.6.3 MA(1) Model ................................................................................. 198 5.7 Modeling Seasonality with ARIMA: The Condemnation Rates Series Revisited .............................................................................. 199 A time series is an ordered sequence of observations. Although the ordering is usually through time, particularly in terms of some equally spaced intervals, the ordering may also be taken through other dimensions such as space.