Lecture 15

Introduction to Time Series

Business executives and economists constantly pore over time series. We should know how to analyze time series and use them for forecasting purposes.

A time series consists of data pertaining to a given unit or entity at a number of points in time. For example, the accounting department of Sears generates time series on sales, costs, taxes, profits, assets, debts, dividends, and many other variables. Often our aim is to forecast the value of a variable (say, sales) for a particular month in the future. In order to do this statisticians have developed certain techniques for describing and analyzing time series data. They have developed procedures to break down a time series into such elements as its trend, seasonal variation, and cyclical variation. Depending on the purpose, an analyst may be interested only in one or more of these components. For example, when the Social Security Administration studies the long-term viability of its trust fund, it will only be interested in the long term trend of the trust fund, and not in its within-year variations. On the other hand, a roadside hotdog vendor will be interested in its sales figures for each day of the week.

The traditional time-series model

The classical approach to the analysis of time series, devised primarily by economic statisticians, expressed the value of a time series for a particular month as comprised of trend, seasonal variation, cyclical variation, and irregular movements. For example, the value of a company’s sales in January 1994 is viewed as equal to

T x S x C x I

where T is the trend value of the firm’s sales during that month, S is the seasonal variation attributable to January, C is the cyclical variation occurring that month, and I is the irregular variation that occurred then.

Trend: A trend is relatively smooth long-term movement of a time series. Trends can be upward, downward or horizontal.

Seasonal Variation: In a particular month the value of an economic variable is likely to differ from what would be expected on the basis of its trend, due to seasonal factors. For example, ice cream sales are stronger during the summer months than in winter months – this is a very regular predictable phenomenon.

Cyclical Variation: Another reason why an economic variable may differ from its trend value is that it may be influenced by the so-called business cycles.

Irregular Variation: After having been multiplied by both S and C, the trend value has been altered to reflect seasonal and cyclical forces. However, besides these forces, a variety of short-term, erratic forces are also at work. These irregular forces are too unpredictable to be useful for forecasting purposes.

Given a real life time series (for instance, the per capita real GNP for U.S. during 1953:I – 1998:III), there are different procedures to extract T, S, and C. We will briefly go through them.

Trend Extraction:

    1. Linear or non-linear least squares regression to estimate the trend in a series.
    2. Moving averages
    3. Exponential smoothing

Extraction of Seasonality-Deseasonalization of a series:

    1. Ratio-to-Moving Average Method
    2. Dummy Variable technique

Identification of Cyclical Component:
Business cycles have four phases: trough, expansion, peak, and recession.
Other often-used terms are depression, growth cycles, and prosperity.