Forecasting

Quantitative Approaches to Forecasting

The Components of a Time Series Measures of Forecast Accuracy Using Smoothing Methods in

Forecasting Using Trend Projection in Forecasting Qualitative Approaches to Forecasting

Quantitative Approaches to Forecasting Quantitative methods are based on an analysis of

historical data concerning one or more time series. A time series is a set of observations measured at

successive points in time or over successive periods of time.

If the historical data used are restricted to past values of the series that we are trying to forecast, the procedure is called a time series method.

If the historical data used involve other time series that are believed to be related to the time series that we are trying to forecast, the procedure is called a causal method. E.g. Assess the effectiveness of magazine, newspaper, and TV advertising on sales.

Time Series Methods Time series methods are:

smoothing trend projection

Components of a Time Series The trend component accounts for the gradual shifting of the time series over a long period of time.

Any regular pattern of sequences of values above and below the trend line is attributable to the cyclical component of the series. The ups and downs in business activities

Components of a Time Series

The seasonal component of the series accounts for regular patterns of variability within certain time periods, such as over a year.

The irregular component of the series is caused by short-term, unanticipated and non-recurring factors that affect the values of the time series. One cannot attempt to predict its impact on the time series in advance.

Measures of Forecast Accuracy Mean Squared Error

The average of the squared forecast errors for the historical data is calculated. The forecasting method or parameter(s) which minimize this mean squared error is then selected.

Mean Absolute DeviationThe mean of the absolute values of all forecast

errors is calculated, and the forecasting method or parameter(s) which minimize this measure is selected. The mean absolute deviation measure is less sensitive to individual large forecast errors than the mean squared error measure.

Smoothing Methods In cases in which the time series is fairly

stable and has no significant trend, seasonal, or cyclical effects, one can use smoothing methods to average out the irregular components of the time series.

Four common smoothing methods are: Moving averages Centered moving averages Weighted moving averages Exponential smoothing

Smoothing Methods Moving Average Method

The moving average method consists of computing an average of the most recent n data values for the series and using this average for forecasting the value of the time series for the next period.

Sales of Comfort brand headache medicine forthe past ten weeks at Rosco Drugsare shown on the next slide. If Rosco Drugs uses a 3-periodmoving average to forecast sales,what is the forecast for Week 11?

Example: Rosco Drugs

Past Sales

Week Sales Week Sales

1 110 6 120

2 115 7 130

3 125 8 115

4 120 9 110

5 125 10 130

Example: Rosco Drugs

Example: Rosco Drugs Excel Spreadsheet Showing Input Data

A B C1 Robert's Drugs2

3 Week (t ) Salest Forect+1

4 1 1105 2 1156 3 1257 4 1208 5 1259 6 120

10 7 13011 8 11512 9 11013 10 130

Example: Rosco Drugs Steps to Moving Average Using Excel

Step 1: Select the Tools pull-down menu.Step 2: Select the Data Analysis option.Step 3: When the Data Analysis Tools dialog

appears, choose Moving Average.Step 4: When the Moving Average dialog box

appears:Enter B4:B13 in the Input Range box.Enter 3 in the Interval box.Enter C4 in the Output Range box.Select OK.

Example: Rosco Drugs Spreadsheet Showing Results Using n = 3

A B C1 Robert's Drugs2

3 Week (t ) Salest Forect+1

4 1 110 #N/A5 2 115 #N/A6 3 125 116.77 4 120 120.08 5 125 123.39 6 120 121.7

10 7 130 125.011 8 115 121.712 9 110 118.313 10 130 118.3

Smoothing Methods Centered Moving Average Method

The centered moving average method consists of computing an average of n periods' data and associating it with the midpoint of the periods. For example, the average for periods 5, 6, and 7 is associated with period 6. This methodology is useful in the process of computing season indexes.

Smoothing Methods Weighted Moving Average Method

In the weighted moving average method for computing the average of the most recent n periods, the more recent observations are typically given more weight than older observations. For convenience, the weights usually sum to 1.

Smoothing Methods Exponential Smoothing

Using exponential smoothing, the forecast for the next period is equal to the forecast for the current period plus a proportion () of the forecast error in the current period.

Using exponential smoothing, the forecast is calculated by:

[the actual value for the current period] + (1- )[the forecasted value for the current

period], where the smoothing constant, , is a number between 0 and 1.

Trend Projection If a time series exhibits a linear trend, the method

of least squares may be used to determine a trend line (projection) for future forecasts.

Least squares, also used in regression analysis, determines the unique trend line forecast which minimizes the mean square error between the trend line forecasts and the actual observed values for the time series.

The independent variable is the time period and the dependent variable is the actual observed value in the time series.

Trend Projection Using the method of least squares, the formula for

the trend projection is: Tt = b0 + b1t.

where: Tt = trend forecast for time period t

b1 = slope of the trend line

b0 = trend line projection for time 0

b1 = ntYt - t Yt

nt 2 - (t )2

where: Yt = observed value of the time series at

time period t = average of the observed values for Yt

= average time period for the n observations

0 1b Y b t

Yt