TIME-SERIES ANALYSIS, MODELLING AND FORECASTING USING SAS SOFTWARE Ramasubramanian V. IA.S.R.I., Library Avenue, Pusa, New Delhi – 110 012 ramsub@iasri.res.in 1. Introduction Time series (TS) data refers to observations on a variable that occurs in a time sequence. Mostly these observations are collected at equally spaced, discrete time intervals. The TS movements of such chronological data can be resolved or decomposed into discernible components as trend, periodic (say, seasonal), cyclical and irregular variations. A basic assumption in any TS analysis/modeling is that some aspects of the past pattern will continue to remain in the future. Here it is tacitly assumed that information about the past is available in the form of numerical data. Ideally, at least 50 observations are necessary for performing TS analysis/ modeling, as propounded by Box and Jenkins who were pioneers in TS modeling. Decomposition models are among the oldest approaches to TS analysis albeit a number of theoretical weaknesses from a statistical point of view. These were followed by the crudest form of forecasting methods called the moving averages method. As an improvement over this method which had equal weights, exponential smoothing methods came into being which gave more weights to recent data. Exponential smoothing methods have been proposed initially as just recursive methods without any distributional assumptions about the error structure in them, and later, they were found to be particular cases of the statistically sound AutoRegressive Integrated Moving Average (ARIMA) models. A detailed discussion regarding various TS components has been done by Croxton et al. (1979). A good account on exponential smoothing methods is given in Makridakis et al. (1998). A practical treatment on ARIMA modeling along with several case studies can be found in Pankratz (1983). A reference book on ARIMA and related topics with a more rigorous theoretical flavour is by Box et al. (1994). 2. Time Series Components An important step in analysing TS data is to consider the types of data patterns, so that the models most appropriate to those patterns can be utilized. Four types of TS components can be distinguished. They are (i) Horizontal  when data values fluctuate around a constant value (ii) Trend  when there is long term increase or decrease in the data (iii)Seasonal  when a series is influenced by seasonal factor and recurs on a regular periodic basis (iv) Cyclical  when the data exhibit rises and falls that are not of a fixed period Note that many data series include combinations of the preceding patterns. After separating out the existing patterns in any TS data, the pattern that remains unidentifiable form the ‘random’ or ‘error’ component. Time plot (data plotted over time) and seasonal plot (data plotted against individual seasons in which the data were observed) help in visualizing these patterns while exploring the data.