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EBSCO Research Starters Copyright 2008 EBSCO Publishing Inc. All Rights Reserved

RESEARCH STARTERSACADEMIC TOPIC OVERVIEWS

Time SeriesStatistics > Time Series

Abstract

Time series data are used in business forecasting to examine pat-

terns, trends, and cycles from the past in order to predict patterns,

trends, and cycles in the future. In general, there are three objec-

tives for the use of time series data: Understand the mechanism

underlying the observed data, extrapolate the data in order to

predict their effect on future behavior or data values, and control

a process or system so that its outcome is more favorable to the

business. Time series data can take several distinctive forms andcan be inuenced both by deterministic and stochastic variables.

Although there are several approaches to modeling time series

data, they often yield different results. Model building and fore-

casting using time series data is a complicated process that is part

art and part science. Human beings will always have to deter-

mine which variables to include, where the line of best t lays,

what time periods to consider, and how to interpret the results.

This can be both strength and a weakness.

Overview

Managers and other business decision makers frequently need to

determine the best course of action and develop strategic plans to

help the organization reach its goals and objectives. To optimizethe worth of such decisions, good business strategy needs to be

based on the rigorous analysis of empirical data. Sometimes the

data are readily obtainable through activities such as an analysis

of the organizations resources and abilities. Sometimes the data

require research and analysis such as data concerning competito

capabilities and offerings. Sometimes the data require creative

guesswork such as determining market needs and trends in the

future. Fortunately, there are a wide range of descriptive and

inferential statistical tools available for analyzing and interpret

ing the data that managers rely on to make decisions. One of the

tools that is particularly helpful for the latter category of analy-

sis where one needs to forecast trends as the basis for decisionmaking is time series analysis.

Uses of Time Series Data

Time series data are data gathered on one or more specic char

acteristics of interest over a period of time at intervals of regula

length. These data series are used in business forecasting to exam

ine patterns, trends, and cycles from the past in order to predict

patterns, trends, and cycles in the future. Time series analysi

typically involves observing and analyzing the patterns in his

torical data. These patterns are extrapolated to forecast future

behavior. Most statistical analysis of time series data involves

model building, the development of a concise mathematica

description of past events. These mathematical models are used

to forecast how the pattern will continue into the future. Time

series are analyzed through several techniques including nave

methods, averaging, smoothing, regression analysis, and decom

position. These analysis techniques assume that the sequence

of observations is a set of jointly distributed random variables

Through the analysis of time series data, one can study the struc-

ture of the correlation (i.e., the degree to which two events or

variables are consistently related) over time to determine the

appropriateness of the model.

Abstract

Keywords

Overview

Applications

Terms & Concepts

Bibliography

Suggested Reading

Table of Contents

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Time Series Essay by Ruth A. Wienclaw, Ph.D.

EBSCO Research Starters Copyright 2008 EBSCO Publishing Inc. All Rights Reserved Page 2

Objectives of Time Series Data

In general, there are several objectives for the use of time series

data.

First, time series data are often analyzed in order to under-

stand the mechanism underlying the observed data and to

build a model that describes the mechanism and its inuence

on the variables of interest.

Second, while understanding the mechanisms that inuence

events and trends is of interest, time series data are fre-

quently analyzed not only to understand these mechanisms,

but more importantly to extrapolate them in order to predict

their affect on future behavior or data values.

Third, although sometimes knowing this information is

sufcient for making decisions and developing strategies,

there are also situations in which the results of the analysis

of time series data can also give organizations information

that will enable them to control a process or system so that

its outcome is more favorable to the business. For example,

one might nd that changes in industry technology are pro-

gressively making a current widget design obsolete and that

sales are dropping of. If acquired in time, this information

can point out areas where the organization needs to change

(e.g., a new widget design that incorporates more technol-

ogy might be appropriate) in order to change the trend in the

future.

Forms of Time Series Data

As shown in Figure 1, real world time series data can take sev

eral distinctive forms.Stationarity

Figure 1a shows the viscosity data on a chemical product over

time. These data remain fairly constant over time and are said to

be constant about the mean (an arithmetically derived measure

of central tendency in which the sum of the values of all the data

points is divided by the number of data points). This character

istic of the data is referred to as stationarity. Stationarity exists

when the probability distribution of a time series does not change

over time. Stationarity is of interest to analysts because it allows

one to mathematically model the process with an equation with

xed coefcients that estimate future values from past history. If

the process is assumed to be stationary, the probability of a givenuctuation in the process is assumed to be the same at any given

point in time. The time series data in Figure 1b, however, do no

exhibit the same degree of constancy or stationarity. It is difcul

to mathematically model a non-stationary process using a simple

algebraic equation. However, it can be possible to use a simple

mathematical procedure to transform non-stationary processes

into ones that are approximately stationary for purposes of anal

ysis. This allows the development of models to help the analys

or decision maker better understand the underlying mechanisms

in the data series.

Figure 1: Three Examples of Time Series Data

(From Mastrangelo, Simpson, & Montgomery, p. 828)

COMPDOC

Deterministic Variables

A third general form that can be taken by time series data is illus

trated in Figure 1c. Time series data that can be inuenced by

various deterministic variables are those for which there are spe

cic causes or determiners. This type of variable includes trends

business cycles, and seasonal uctuations.

Trends are persistent, underlying directions in which a factor

or characteristic is moving in either the short, intermediate,

or long term. Trends tend to be linear rather than cyclic, and

grow or shrink steadily over a period of years. For example,

a trend might be an increasing tendency for business to out-

source and offshore technical support and customer service

in many high tech companies. Trends are not necessarily lin-

Autocorrelation

Autoregression

Autoregressive Integrated Moving Average

Business Cycle

Data

Decomposition

Deterministic Variables

Forecasting

Moving Average

Seasonal Fluctuation

Stationarity

Stochastic

Time Series Data

Trend

Variable

Keywords

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ear, however. For example, trends in new industries tend to

be curvilinear as the demand for the new product or service

grows after its introduction then declines after the product or

service becomes integrated into the economy.

Another type of deterministic factor is business cycles.

These factors are continually recurring variation in total

economic activity. Business cycles tend to occur across

most sectors of the economy at the same time. For example,

several years of a boom economy with expansion of eco-

nomic activity (e.g., more jobs, higher sales) are frequently

followed by slower growth or even contraction of economic

activity.

Another category of deterministic factor is seasonal uctua-

tions. These are changes in activity that occur in a fairly reg-

ular annual pattern and which are related to seasons of the

year, the calendar, or holidays. Figure 1c shows a seasonal

uctuation in soft drink sales, with sales going up during the

warmer months and down during the colder months over a

period of four years.

Stochastic Variables

In addition to deterministic variables, time series data can be

inuenced by stochastic variables. These variables are caused

by randomness or include an element of chance or probability

and are caused by unpredictable factors. Stochastic variables can

be either irregular or random. Examples of irregular stochastic

variables include natural disasters such as earthquakes or oods,

political disturbances such as war or changes in the political party

in charge, strikes, and other external factors. Other unpredictable

or random factors include situations such as high absenteeism

due to an epidemic that could affect a businesss protability.

The real world includes many variables both deterministic and

stochastic that can affect time series data, and most real world

situations and models include both types of variables.

Applications

Approaches to Time Series Modeling

Regression Methods

There are several approaches to modeling time series data. One of

the primary tools for analyzing time series data and developing a

mathematical model of real world situations is regression meth-ods. These are a family of statistical techniques that are used to

develop a mathematical model for use in predicting one variable

from the knowledge of another variable. Regression methods

include both linear and nonlinear techniques. In linear regres-

sion, a line of best t is tted to a data set to estimate the effect

of a single independent variable. The slope of the line shows the

impact of the independent variable on the dependent variable.

Other regression methods are available for use in multivariate

and nonlinear situations.

Smoothing Techniques

Nave Forecasting

Another family of methods frequently used in building models

from time series data is smoothing techniques. One approach to

smoothing time series data is through nave forecasting models

These are simple models that assume that future outcomes

are best predicted by the more recent data in the time series.

Under this philosophy, for example, one would assume that las

months sales were a better predictor of next months sales than

were the sales from six months ago. It should be borne in mind

however, that nave forecasting models do not consider the pos-

sibility of trends, business cycles, or seasonal uctuations. For

example, if ten gross of widgets were sold last month, a nave

model would conclude that ten gross of widgets would also be

sold next month. Because of this assumption, nave forecasting

models work better on data that are reported more frequently

(e.g., daily or weekly) or in situations without trends or season

ality. One danger in nave model forecasts lies in the fact tha

they are often based on the observations of one time period. As a

result, they can easily become a function of irregular uctuationin data (e.g., Acme corporation needed a one-time purchase of

widgets to set up their new operations facility which accounts for

the one-time high demand for widgets in the previous month).

Averaging Models

Smoothing of time series data can also be done using averaging

models. These techniques take into account data from severa

time periods, thereby neutralizing the problem of nave models in

which the forecast is overly sensitive to irregular uctuations as

illustrated in the previous example. In the simple average model

forecasts for an upcoming time period are made using the aver

age of the values for a specied number of previous time periods

(e.g., the forecast of widgets sales for next month might be theaverage number of widgets sold per month over the past six

months). In the moving average approach, however, the average

value from previous time periods is used to forecast future time

periods and is updated in each ensuing time period by including

the new values not available in the previous average and dropping

out the date from the earliest time periods. Although the moving

average approach has the advantage of taking into account the

most recent data available, it can be difcult to choose the opti

mal length of time over which to compute the moving average

Another drawback of the moving average approach is that it does

not take in to account the effects of trends, business cycles, and

seasonal uctuations. Because of this fact, analysts often use a

weighted moving average which gives more weight to some timeperiods in the series than to others. For example, if three months

ago the company introduced a newly redesigned product into the

marketplace, the analyst might believe that the past three months

reect the markets reaction to the new design and be better

able to forecast the continuing reaction than if s/he did not have

this information. Similarly, prediction of sales from last years

holiday season may be better predictors of next years holiday

sales than sales during the summer months. A third approach to

smoothing time series data comprises exponential smoothing

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techniques. Exponential techniques use weight data from previ-

ous time periods with exponentially decreasing importance so

that the new forecast is a product of the current forecast and the

current actual value.

Autoregression

Another approach to modeling time series data is autoregres-

sion, a multiple regression technique in which future values ofthe dependent variable are predicted from past values of the vari-

able. This technique takes advantage of the relationship of values

to the values of previous time periods. The independent vari-

ables are time-lagged versions of the dependent variable so that

one tries to forecast a future value of a variable from knowledge

of that variables value in previous time periods. This approach

can be particularly useful for locating both seasonal and cyclic

effects.

Mixed/Integrated Techniques

Times series data can be modeled using mixed or integrated

techniques that utilize both moving average and autoregres-

sive techniques. One of these approaches is the autoregressiveintegrated moving average (ARIMA) model (also called the

Box-Jenkins model). The ARIMA model is an integrated tool

for understanding and forecasting using time series data that

has both an autoregressive and a moving average component.

ARIMA modeling techniques are powerful and frequently result

in a better model than either the use of moving averages or

autoregressive techniques alone. For example, ARIMA can be

used to determine the length of the weights (i.e., how much of

the past should be used to predict the next observation) and the

values of these weights.

Autocorrelation

Models based on time series data can produce spurious results

when the error terms of the model are correlated with each other

in a situation referred to as autocorrelation or serial correlation.

Autocorrelation can be problematic in the use of regression anal-

ysis because regression analysis assumes that error terms are not

correlated because they are either independent or random. When

autocorrelation occurs, the estimates of the regression coefcients

may be inefcient and both the variance of the error terms and

the true standard deviation may be signicantly underestimated.

In addition, the autocorrelation effect means that the condence

intervals and t and F tests are no longer strictly applicable. A

number of methods are available to determine whether or not

autocorrelation is present in time series data (e.g., the Durbin-Watson test). Autocorrelated data can sometimes be corrected by

the addition of independent variables or transforming variables.

Determining Which Forecast to Use

As discussed above, there are a number of techniques available

to forecast stationary time series data (i.e., those that show no

signicant trend, cyclic, or seasonal effects). However, these

approaches often yield different results. To help determine which

forecast better models a given set of data, it is necessary to deter-

mine the amount of error produced by each technique. This is

the difference between the forecasted value of a variable and the

actual value of a variable. A number of techniques are available

for doing this including mean error, mean absolute deviation

mean square error, mean percentage error, and mean absolute

percentage error. Data that are inuenced by trends can be ana

lyzed by a number of approaches, including linear regression

and regression using quadratic models. However, if these data

are also inuenced by seasonal uctuations, other data analy

sis techniques are more appropriate. One of the most frequently

used of these techniques is decomposition, in which the time

series data are broken down into the four component factors

of trend, business cycle, seasonal uctuation, and irregular or

random uctuation.

Modeling Technologies

Model building and forecasting using time series data is a com

plicated process that is part art and part science. However, there

are now a number of software packages available that can help

in this task. The techniques and concomitant software for devel-

oping forecasting models continue to improve. However, these

techniques and tools are still not highly accurate for erraticpatterns with short life cycles. No matter how rened or power

ful the tools, some randomness or noise will always remain in

time series data. It is this noise that limits the upper limit of the

forecast. In addition, forecasting cannot be done wholly based

on mathematics. Human beings will always have to determine

which variables to include, where the line of best t lies, what

periods to consider, and how to interpret the results. This can be

both a strength and a weakness.

Terms & Concepts

Autocorrelation:A problem occurring over time in regression

analysis when the error terms of the forecasting model are cor-related. Also called serial correlation.

Autoregression:A multiple regression technique used in fore-

casting in which future values of the variable are predicted from

past values of the variable.

Autoregressive Integrated Moving Average (ARIMA): An

integrated tool for understanding and forecasting using time

series data. The ARIMA model has both an autoregressive and a

moving average component. The ARIMA model is also referred

to as the Box-Jenkins model.

Business Cycle:A continually recurring variation in total eco-

nomic activity. Such expansions or contractions of economic

activity tend to occur across most sectors of the economy at the

same time.

Data:(sing. datum) In statistics, data are quantiable observa

tions or measurements that are used as the basis of scientic

research.

Deterministic Variables:Variables for which there are specic

causes or determiners. These include trends, business cycles, and

seasonal uctuations.

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Decomposition:The process of breaking down time series data

into the component factors of trends, business cycles, seasonal

uctuations, and irregular or random uctuations.

Forecasting:In business, forecasting is the science of estimat-

ing or predicting future trends. Forecasts are used to support

managers in making decisions about many aspects of the busi-

ness including buying, selling, production, and hiring.

Moving Average: A method used in forecasting in which the

average value from previous time periods is used to forecast

future time periods. The average is updated in each ensuing time

period by including the new values not available in the previous

average and dropping out the date from the earliest time peri-

ods.

Seasonal Fluctuation:Changes in economic activity that occur

in a fairly regular annual pattern. Seasonal uctuations may be

related to seasons of the year, the calendar, or holidays.

Stationarity:The condition of a random process where its sta-

tistical properties do not vary with time.

Stochastic:Involving chance or probability. Stochastic variables

are random or have an element of chance or probability associ-

ated with their occurrence.

Time Series Data: Data gathered on a specic characteris-

tic over a period of time. Time series data are used in business

forecasting. To be useful, time series data must be collected at

intervals of regular length.

Trend: The persistent, underlying direction in which something

is moving in either the short, intermediate, or long term. Identi-

cation of a trend allows one to better plan to meet future needs.

Variable: An object in a research study that can have more than

one value. Independent variables are stimuli that are manipu-

lated in order to determine their effect on the dependent variables

(response). Extraneous variables are variables that affect the

response but that are not related to the question under investiga-

tion in the study.

Bibliography

Black, K. (2006).Business statistics for contemporary decision

making(4th ed.). New York: John Wiley & Sons.

Gilliland, M. & Leonard, M. (2006). Forecasting software the

past and the future. Country Commerce, 33-36. Retrieved

September 11, 2007, from EBSCO Online Database

Business Source Complete.http://search.ebscohost.com/

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Mastrangelo, C. M., Simpson, J. R., & Montgomery, D. C.

(2001). Time series analysis. In Saul I. Gass, S. I. &

Harris, C. M. (eds),Encyclopedia of Operations Research

and Management Science(pp. 828-833). New York: Wiley

Retrieved September 11, 2007, from EBSCO Online

Database Business Source Complete. http://search.ebsco-

host.com/login.aspx?direct=true&db=bth&AN=21891965

&site=ehost-live

Nazem, S. M. (1988).Applied time series analysis for business

and economic forecasting. New York: Marcel Dekker.

Pindyck, R. S. & Rubinfeld, D. L. (1998). Econometric models

and economic forecasts.Boston: Irwin/McGraw-Hill.

Suggested Reading

Craighead, C. W. (2004). Right on target for time-series

forecasting.Decision Sciences Journal of Innovative

Education,2(2), p207-212. Retrieved September 12,

2007, from EBSCO Online Database Business Source

Complete. http://search.ebscohost.com/login.aspx?direct=t

rue&db=bth&AN=13279969&site=ehost-live

De Luna, X. (2001). Guaranteed-content prediction intervals

for non-linear autoregressions.Journal of Forecasting,

20(4), 265-72. Retrieved September 12, 2007, from

EBSCO Online Database Business Source Complete.

http://search.ebscohost.com/login.aspx?direct=true&db=bt

h&AN=13638439&site=ehost-live

Jarvis, C. H. & Stuart, N. (2001). Accounting for error when

modelling with time series data: Estimating the develop-

ment of crop pests throughout the year. Transactions in

GIS,5(4), 327-343. Retrieved September 12, 2007, from

EBSCO Online Database Business Source Complete.

http://search.ebscohost.com/login.aspx?direct=true&db=bt

h&AN=5439610&site=ehost-live

Ng, S. & Vogelsang, T. J. (2002). Forecasting autoregres-

sive time series in the presence of deterministic compo-

nents.Econometrics Journal, 5(1), 196-224. Retrieved

September 12, 2007, from EBSCO Online Database

Business Source Complete. http://search.ebscohost.com/

login.aspx?direct=true&db=bth&AN=6719925&site=eho

st-live

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Essay by Ruth A. Wienclaw, Ph.D.

Ruth A. Wienclaw holds a Doctorate in industrial /organizational psychology with a specialization in organization development from

the University of Memphis. She is the owner of a small business that works with organizations in both the public and private sectors,

consulting on matters of strategic planning, training, and human /systems integration.

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