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Chapter 9: Short-Term Forecasting

PowerPoint Slides Prepared By:Alan Olinsky Bryant University

Management Science: The Art of Modeling with Spreadsheets, 2e

S.G. Powell

K.R. Baker

© John Wiley and Sons, Inc.

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Introduction

Regression analysis can sometimes be useful in short-term forecasting.

A better approach is to base the forecast of a variable on its own history, thereby avoiding the need to specify a causal relationship and to predict the values of explanatory variables.

Our focus in this chapter is on time series methods for forecasting.

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Forecasting with Time-Series Models

We make use of historical data for the phenomenon we wish to forecast.

We seek a routine calculation that may be applied to a large number of cases and that may be automated, without relying on any qualitative information about the underlying phenomena.

Short-term forecasts are often used in situations that involve forecasting many different variables at frequent intervals.

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Hypothesized Models

The major components of such a model are usually the following: a base level a trend cyclic fluctuations

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Three Components of Time Series Behavior

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The Moving Average Model

The n-period moving average builds a forecast by averaging the observations in the most recent n periods:

where xt represents the observation made in period t, and At denotes the moving average calculated after making the observation in period t.

At = (xt + xt–1 + … + xt–n+1) / n

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Convention

We adopt the following convention for the steps in forecasting: Make the observation in period t Carry out the necessary calculations Use the calculations to forecast period (t + 1)

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Worksheet for Calculating Moving Averages

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Number of Periods to Include in Moving Average

There is no definitive answer to this question, but there is a trade-off to consider.

Suppose the mean of the underlying process remains stable: If we include very few data points, then the moving average

exhibits more variability than if we include a larger number of data points. In that sense, we get more stability from including more points.

Suppose there is an unanticipated change in the mean of the underlying process:If we include very few data points, our moving average will tend to track the changed process more closely than if we include a larger number of data points. In that case, we get more responsiveness from including fewer points.

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Moving Average Calculations in a Stylized Example

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Comparison of 4-week and 6-week Moving Averages

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Measures of Forecast Accuracy

MSE: the Mean Squared Error between forecast and actual

MAD: the Mean Absolute Deviation between forecast and actual

MAPE: the Mean Absolute Percent Error between forecast and actual

MSE = 2)()1(

1t

v

utt xF

vu

MAD =

v

uttt xF

vu )1(

1

MAPE =

v

ut t

tt

x

xF

vu )1(

1

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Comparison of Measures of Forecast Accuracy

The MAD calculation and the MAPE calculation are similar: one is absolute, the other is relative.

We usually reserve the MAPE for comparisons in which the magnitudes of two cases are different.

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Excel Tip: Moving Average Calculations

Excel’s Data Analysis tool (Tools►Data Analysis►Moving Average) contains an option for calculating moving averages.

Excel assumes that the data appear in a single column, and the tool provides an option of recognizing a title for this column, if it is included in the data range.

Other options include a graphical display of the actual and forecast data and a calculation of the standard error after each forecast.

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The Exponential Smoothing Model

Exponential smoothing weighs recent observations more than older ones.

The parameter a is some number between zero and one, called the smoothing constant.

We refer to St as the smoothed value of the observations, and we can think of it as our “best guess” as to the value of the mean.

Our forecasting procedure sets the forecast Ft+1 = St.

St = xt + (1 – )St–1

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Comparison of Weights Placed on k-year-old Data

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Worksheet for Exponential Smoothing Calculations

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Comparison of Smoothed and Averaged Forecasts

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Exponential Smoothing Calculations in a Stylized Example

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Excel Tip: Implementing Exponential Smoothing

Excel’s Data Analysis tool contains an option for calculating forecasts using exponential smoothing.

The Exponential Smoothing module resembles the Moving Average module, but instead of asking for the number of periods, it asks for the damping factor, which is the complement of the smoothing factor, or (1 – a).

Again, there is an option for chart output and an option for a calculation of the standard error.

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Exponential Smoothing with a Trend

St = xt + (1 – )St–1

Tt = (St – St–1) + (1 – )Tt–1

where St is the smoothed value after the observation has been made in period t, and Tt is the estimated trend.

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Trend Model Calculations with a Trend in the Data

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Holt’s Method

This more flexible procedure uses two smoothing constants, as shown in the following formulas:

St = xt + (1 – )(St–1 + Tt–1)

Tt = (St – St–1) + (1 – )Tt–1

Ft+1 = St + Tt

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Holt's Method with a Trend in the Data

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Exponential Smoothing with Trend and Cyclic Factors

We can take the exponential smoothing model further and include a cyclical (or seasonal) factor.

For a cyclical effect, there are two types of models: an additive model and a multiplicative model.

See text for formulas.

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*Using CB Predictor

CB Predictor is an alternative to constructing spreadsheet formulas for the specific purposes of short-term forecasting.

CB Predictor also provides a number of analyses and reports automatically, whereas we might view the time required to add such reports to our own spreadsheet models counterproductive.

Thus, for certain kinds of applications, it may make sense to rely on CB Predictor rather than our own custom spreadsheet models.

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*CB Predictor:Alternative Forecasting Procedures

Nonseasonal data with no trend Single Moving Average Single Exponential Smoothing

Nonseasonal data with a trend Double Moving Average Double Exponential Smoothing

Seasonal data with no trend Seasonal Additive Seasonal Multiplicative

Seasonal data with a trend Holt-Winters’ Additive Holt-Winters’ Multiplicative

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*CB Preditor:Single Moving Average

Data for the Curtis Distributors example

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CB Predictor Window: Input Data Tab

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CB Predictor Window: Data Attributes Tab

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CB Predictor Window: Method Gallery Tab

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Single Moving Average Window in CB Predictor

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CB Predictor: Results Tab

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CB Predictor:Single Exponential Smoothing

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Advanced Options Window in CB Predictor

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Preferences Window in CB Predictor

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A Portion of the Report from CB Predictor

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Summary

Moving averages and exponential smoothing are widely used for routine short-term forecasting.

By making projections from past data, these methods assume that the future will resemble the past.

However, the exponential smoothing procedure is sophisticated enough to permit representations of a linear trend and a cyclical factor in its calculations.

Exponential smoothing procedures are adaptive.

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Summary

Implementing an exponential smoothing procedure requires that initial values be specified and a smoothing factor be chosen.

The smoothing factor should be chosen to trade off stability and responsiveness in an appropriate manner.

Although Excel contains a Data Analysis tool for calculating moving-average forecasts and exponentially-smoothed forecasts, the tool does not accommodate the most powerful version of exponential smoothing, which includes trend and cyclical components.

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