1. Slides by
John
Loucks
St. Edward’s
University

2. Chapter 6 Forecasting Quantitative Approaches to Forecasting
Components of a Time Series
Measures of Forecast Accuracy
Smoothing Methods
Trend Projection
Trend and Seasonal Components
Regression Analysis
Qualitative Approaches

3. Quantitative Approaches to Forecasting
A forecast is a prediction of what will happen in the future.
A time series is a set of observations measured at successive points in, or over successive periods of, time.
Quantitative forecasting approaches are based on an analysis of historical data concerning one or more time series.
If the data used are limited to past values of the series we are trying to forecast, the procedure is called a time series method.
If the data used involve other time series that are believed to be related to the time series being forecasted, the procedure is called a causal method.

4. Time Series Methods
The objective of time series methods is to discover a pattern in the historical data and then extrapolate this pattern into the future.
The forecast is based solely on past values of the variable that we are trying to forecast and/or on past forecast errors.
Three time series methods are:
smoothing
trend projection
trend projection adjusted for seasonal influence

5. 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.

6. 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.

7. 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.
Three common smoothing methods are:
Moving averages
Weighted moving averages
Exponential smoothing

8. Smoothing Methods Moving Average Method
The moving average method consists of computing an average of the most recent n data values in the series and using this average for forecasting the value of the time series for the next period.
The term moving indicates that, as a new observation becomes available for the time series, it replaces the oldest observation in the equation, and a new average is computed.

9. Example: Gasoline Sales
The data below show the number of gallons of gasoline (in 1,000s) sold by a gasoline distributor in Bennington, Vermont, over the past 12 weeks. The time series appears to be stable over time, so smoothing methods are applicable.
Week Sales Week Sales

10. Example: Gasoline Sales
Moving Average
To forecast gasoline sales for week 13 (in 1,000s) using a 3-week moving average, we need to compute the average of sales for weeks 10, 11, and 12. The week 13 forecast (F13) is:

11. 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.

12. Example: Gasoline Sales
Three-Period Weighted Moving Average
To forecast gasoline sales (in 1,000s) for week 13 using a 3-week weighted moving average, we will compute the weighted average of sales for weeks 10, 11, and 12 using the weights of 1/6, 2/6, and 3/6. The week 13 forecast (F13) is:
F13 = 1/6(20) + 2/6(15) + 3/6(22) = =

13. 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.
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.

14. Smoothing Methods Exponential Smoothing Ft+1 = aYt + (1 – a)Ft where:
Ft+1 = forecast of the time series for period t+1
Yt = actual value of the time series in period t
Ft = forecast of the time series for period t
a = smoothing constant (0 < a < 1)
(To start the calculations, we let F1 equal the
actual value of the time series in period 1.)

15. Example: Gasoline Sales
Exponential Smoothing ( = .2, 1 -  = .8)
F = 17
F2 = .2Y1 + .8F1 = .2(17) + .8(17) = 17.00
F3 = .2Y2 + .8F2 = .2(21) + .8(17) = 17.80
F4 = .2Y3 + .8F3 = .2(19) + .8(17.80) = 18.04
F5 = .2Y4 + .8F4 = .2(23) + .8(18.04) = 19.03
F6 = .2Y5 + .8F5 = .2(18) + .8(19.03) = 18.83
F7 = .2Y6 + .8F6 = .2(16) + .8(18.83) = 18.26

16. Example: Gasoline Sales
Exponential Smoothing ( = .2, 1 -  = .8)
F8 = .2Y F7 = .2(20) + .8(18.26) = 18.61
F9 = .2Y F8 = .2(18) + .8(18.61) = 18.49
F10 = .2Y F9 = .2(22) + .8(18.49) = 19.19
F11 = .2Y F10 = .2(20) + .8(19.19) = 19.35
F12 = .2Y F11 = .2(15) + .8(19.35) = 18.48
F13 = .2Y F12 = .2(22) + .8(18.48) = 19.18

17. Mean Squared Error (MSE)
An important consideration in selecting a forecasting method is the accuracy of the forecast. Clearly, we want forecast errors to be small.
The mean squared error (MSE) is an often-used measure of the accuracy of a forecasting method.
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.

18. Example: Gasoline Sales
Mean Squared Error

19. 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.

20. Trend Projection
Using the method of least squares, the formula for the trend projection is: Tt = b0 + b1t.
where: Tt = trend value in period t
b0 = intercept of the trend line
b1 = slope of the trend line
where: Yt = actual value of the time series in period t
= average value of the time series
= average value of t
n = number of periods

21. Example: Bicycle Sales
Consider the time series for bicycle sales (in 1,000s) of a particular manufacturer over the past 10 years, as shown below. Although the data show some up-and-down movement over the past 10 years, the time series for the number of bicycles sold seems to have an overall increasing or upward trend.
Year Sales Year Sales

22. Example: Bicycle Sales
Trend Projection
t Yt tYt t 2

23. Example: Bicycle Sales
Trend Projection (continued)
= 55/10 = = 264.5/10 = 26.45
ntYt - t Yt (10)(1545.5) - (55)(264.5)
b1 = = = 1.1
nt 2 - (t) (10)(385) - (55)2
= (5.5) = 20.4
Tt = (t)
T11 = (11) = (or 32,500 bicycles)

24. Forecasting with Trend and Seasonal Components
Steps of Multiplicative Time Series Model
1. Calculate the centered moving averages (CMAs).
2. Center the CMAs on integer-valued periods.
3. Determine the seasonal and irregular factors (StIt ).
4. Determine the average seasonal factors.
5. Scale the seasonal factors (St ).
6. Determine the deseasonalized data.
7. Determine a trend line of the deseasonalized data.
8. Determine the deseasonalized predictions.
9. Take into account the seasonality.

25. Example: Television Sales
The data below show quarterly television sales (in thousands of units) for a particular manufacturer over the past four years.
Quarter
Year

26. Example: Television Sales
1. Calculate the centered moving averages.
There are four distinct seasons in each year. Hence, take a four-period centered moving average to eliminate seasonal and irregular factors. For example:
1st CMA = ( )/4 = 5.35
2nd CMA = ( )/4 = 5.60
etc.

27. Example: Television Sales
2. Center the CMAs on integer-valued periods.
The first centered moving average computed in step 1 (5.35) will be centered on quarter 2.5 of year 1. The second centered moving average computed (5.60) will be centered on quarter 3.5 of year 1.
Note that the moving averages from step 1 do not center themselves on integer-valued periods because n (the number of seasons) is an even number.
In this case, 2-period CMAs are computed for the original 4-period CMAs. These 2-CMAs will center themselves on integer-valued periods.

28. Example: Television Sales
2. Center the CMAs on integer-valued periods.

29. Example: Television Sales
3. Determine the seasonal & irregular factors (St It ). Isolate the trend and cyclical components. For each period t, this is given by:
St It = Yt /(Moving Average for period t )

30. Example: Television Sales
3. Determine the seasonal & irregular factors (St It ).

31. Example: Television Sales
4. Determine the average seasonal factors.
Averaging all St It values corresponding to that season:
Season 1: ( ) /3 =
Season 2: ( ) /3 =
Season 3: ( ) /3 =
Season 3: ( ) /3 =

32. Example: Television Sales
5. Scale the seasonal factors (St ).
Average the seasonal factors = (
)/4 = Then, divide each seasonal factor by the average of the seasonal factors.
Season 1: / =
Season 2: / =
Season 3: / =
Season 4: / =
Total =

33. Example: Television Sales
6. Determine the deseasonalized data.
Divide the data point values, Yt , by St .

34. Example: Television Sales
6. Determine the deseasonalized data.

35. Example: Television Sales
7. Determine a trend line of the deseasonalized data.
Using the least squares method for t = 1, 2, ..., 16, gives:
Tt = (t)

36. Example: Television Sales
8. Determine the deseasonalized (trend) predictions.
Substitute t = 17, 18, 19, and 20 into the least squares equation:
Year 5, Quarter 1: T17 = (17) = 7.617
Year 5, Quarter 2: T18 = (18) = 7.765
Year 5, Quarter 3: T19 = (19) = 7.913
Year 5, Quarter 4: T20 = (20) = 8.061

37. Example: Television Sales
9. Take into account the seasonality.
Multiply each deseasonalized (trend) prediction by its seasonal factor to give the following forecasts for year 5:
Year 5, Quarter 1: F17 = 7.617(0.9308) = 7.090
Year 5, Quarter 2: F18 = 7.765(0.8363) = 6.494
Year 5, Quarter 3: F19 = 7.913(1.0915) = 8.637
Year 5, Quarter 4: F20 = 8.061(1.1414) = 9.201

38. Qualitative Approaches to Forecasting
Delphi Approach
A panel of experts, each of whom is physically separated from the others and is anonymous, is asked to respond to a sequential series of questionnaires.
After each questionnaire, the responses are tabulated and the information and opinions of the entire group are made known to each of the other panel members so that they may revise their previous forecast response.
The process continues until some degree of consensus is achieved.

39. Qualitative Approaches to Forecasting
Scenario Writing
Scenario writing consists of developing a conceptual scenario of the future based on a well defined set of assumptions.
After several different scenarios have been developed, the decision maker determines which is most likely to occur in the future and makes decisions accordingly.

40. Qualitative Approaches to Forecasting
Subjective or Interactive Approaches
These techniques are often used by committees or panels seeking to develop new ideas or solve complex problems.
They often involve "brainstorming sessions".
It is important in such sessions that any ideas or opinions be permitted to be presented without regard to its relevancy and without fear of criticism.

41. End of Chapter 6