1. Chapter 11: Forecasting Models
© 2007 Pearson Education

2. Forecasting Forecasting is attempting to predict the future
Decision makers want to reduce uncertainty by predicting future values such as sales or investment return

3. Steps in Forecasting Determine the objective of the forecast
Identify items to be forecast
Determine time horizon
Select the forecasting model(s)
Gather data
Validate model
Make forecast and implement results

4. Types of Forecasts
Qualitative - subjective methods based on intuition and experience
Time Series – based on historical data and assume the past indicates the future
Causal Models – data based where there may be a cause and effect relation between variables

5. Qualitative Forecasting Models
Delphi Method – an iterative group process where a group of experts attempt to reach consensus
Jury of Executive Opinion – uses opinions of high level managers often combined with statistical models

6. Qualitative Forecasting Models
Sales Force Composite – each salesperson estimates sale in his/her own region and forecast are combined for an overall forecast
Consumer Market Survey – future purchase plans are solicited from customers

7. Measuring Forecast Error
Measures how accurate the forecast was
For time period t:
Forecast error = Actual value – Forecast value
= At - Ft

8. Methods of Measuring Overall Forecast Error
Mean Absolute Deviation (MAD)
MAD = ∑ |At – Ft| / T
where T = the number of time periods
Mean Squared Error (MSE)
MSE = ∑ (At – Ft)2 / T

9. Methods of Measuring Overall Forecast Error
Mean Absolute Percent Error (MAPE) Measure error as a percent of actual values
MAPE = 100 ∑ [ |At – Ft| / At ] / T

10. Time Series
A time series is where the same value is recorded at regular time intervals
Examples: daily stock price, monthly sales, annual revenue, etc.

11. Components of a Time Series
Trend – long term upward or downward movement
Seasonality – the pattern that occurs every year
Cycles – the pattern that occurs over a period of years
Random variations – caused by chance and unusual events

12. Time Series Components

13. Time Series Decomposition
A time series can be broken down into its individual components
Two approaches:
Multiplicative decomposition
Forecast = Trend x Seasonality x Cycles x Random
Additive decomposition
Forecast = Trend + Seasonality + Cycles + Random

14. Stationary and Nonstationary Time Series Data
If a time series has an upward or downward trend, it is nonstationary
If it has no trend, it is stationary

15. Moving Averages
Smooth out variations in a time series when values are fairly steady
Some number (k) of consecutive periods are averaged
k-period moving average =
∑ (actual values in previous k periods) k

16. Wallace Garden Supply Example

17. Weighted Moving Averages
A moving average where some periods are weighted more heavily than others
K-period weighted moving average =
∑ (wi Ai) / ∑ (wi)
where, wi = weight for period i
Ai = actual value for period i

18. Wallace Garden Supply With Weighted Moving Averages
Period Weights
last month 3
2 month ago 2
3 months ago 1

19. 3-Month Weighted Moving Average

20. Using Solver to Find the Optimal Weights
The weights are the decision variables (changing cells)
Minimize some measure of forecast error (MAD, MSE, or MAPE) as the Target cell
Note this is a nonlinear objective
Weights must be nonnegative
Go to file 11-3.xls

21. Exponential Smoothing
Another smoothing method
Does not require extensive past data
Ft+1 = Ft + α x (At – Ft)
Ft+1 = forecast for period (t+1)
Ft = forecast for period t
α = a weight (smoothing constant)
At = actual value for period t

22. Wallace Garden Supply With Exponential Smoothing
Assume the smoothed value for the first month is the actual value
Use α = 0.1 and also α = 0.9

24. Trend Analysis Fits a straight or curved line through a time series
We will cover only linear trends
A scatter diagram shows the trend
Excel can both create the scatter diagram and fit the linear trend line

25. Midwestern Electric Co. Example
Go to file 11-5.xls

26. The Trend Equation Ŷ = b0 + b1X where,
Ŷ = forecast average dependent value
X = independent value (time)
b0 = Y-intercept
b1 = slope of the line

27. Least Squares Method
The b0 and b1 values are found using the least squares method, which seeks to minimize the sum of squared errors
SSE = ∑ (Y – Ŷ)2
Where,
Error = Y - Ŷ

28. Least Squares Method for Best-Fitting Line

29. Least Squares Line With Excel
Can use regression in the Analysis ToolPak add-in
The time (X) values are transformed to 1, 2, 3, etc.
Go to file 11-6.xls

30. Seasonality Analysis
When a seasonal pattern repeats yearly, this can be used for future forecasts
Need monthly or quarterly data
A seasonal index is the ratio of the average value in that season, over the annual average

31. Eichler Supplies Seasonality Example
Have monthly demand data for 24 months
Calculate overall average monthly demand
Calculate ratio for each month
Go to file 11-7.xls

32. Decomposition of a Time Series
Decomposition breaks a time series down into its components (Trend, Seasonal, Cyclical, and Random)
Two types of models
Multiplicative
Additive

33. Multiplicative Decomposition Sawyer Piano House Example
Want to forecast sales of grand pianos
Have quarterly data for the past 5 years
Steps:
Find the seasonal indices
First smooth data with moving averages
Seasonal ratio = actual value / smoothed value
Average the seasonal ratios for each quarter
Unseasonalized value = actual value / seasonal index

34. Steps Continued
Find the trend equation using the unseasonalized values
Calculate forecasts
Use the trend equation to make an unseasonalized forecast
Multiply the unseasonalized forecast by the seasonal index
Calculate forecast error
Go to file 11-8.xls

35. Causal Forecasting Models
Forecasting a dependent variable based on other (independent) variables
Uses simple or multiple regression
Example:
Dependent variable: Swimwear sales
Independent variables: selling price, competitors prices, temperature, whether schools are in session, advertising

36. Causal Simple Regression Model
Want to predict selling price of homes (Y) based on the square footage (X)
Have data on 12 homes recently sold in a specific neighborhood
Use scatter diagram to check for linear relation
Find least squares equation
Ŷ = b0 + b1X

37. Causal Simple Regression With ExcelModules
Given the X and Y data, it will automatically:
Calculate the regression equation
Calculate forecast error
Produce a scatter plot with the regression line
Go to file 11-9.xls

38. The Regression Equation
Forecast average selling price =
(Home size)
Slope interpretation: On average the price of a home will increase by $ thousand per additional thousand sq. ft.
Intercept interpretation: When X=0 the average selling price is -$8.125 thousand (has no practical meaning since no houses have 0 square feet)

39. Standard Error and Correlation
Standard Error (Sy,x) – the standard deviation of the regression equation (useful for confidence intervals on forecasts)
Correlation Coefficient (r) – measures the strength of the linear relation
-1 < r < 1

40. Correlation Coefficient Examples

41. Coefficient of Determination (R2)
Measures the proportion of variation in the dependent variable (Y) that can be explained with the independent variable (X)
0 < R2 < 1
It is the correlation squared

42. Using the Causal Simple Regression Model
To forecast the average selling price of a 3100 sq. foot home, use X = 3.10
Can use ExcelModules to produce forecast
Forecast = (3.1) = 295
which is $295,000

43. Potential Weaknesses of Causal Forecasting With Regression
We need to provide the value(s) of the independent variable(s)
Individual values of Y may be much higher or lower than the forecast average
Model is generally valid only for X values within the range of the data set

44. Approximate Confidence Interval
Helpful for showing how high or low an individual value might be
Approximate confidence interval formula:
Ŷ + Zα/2 (Sy,x)
Approximate 95% interval example:
(42.602)
Which is $211,500 to $378,500

45. Causal Simple Regression Using Excel’s Analysis ToolPak
An add-in that includes regression
Appears as “Data Analysis” at the bottom of the “Tools” menu
Go to file 11-9.xls

46. Statistical Significance Tests
If the true value of the slope (β1) does not differ significantly from 0, then Y does not change as X changes
Hypotheses:
H0: β1=0 (X is not significantly related to Y)
H1: β1≠0 (X is significantly related to Y)
Tested by both F-test and t-test
Reject H0 if p-value < α

47. Hypothesis Test Results for Home Selling Prices
From either F-test or t-test (they are equivalent for simple regression):
Reject H0 because p=0.011 is < 0.05 (alpha)
Home size is statistically significant in having ability to predict home selling price

48. Causal Multiple Regression Model
More than one independent variable
Ŷ = b0 + b1X1 + b2X2 + … + bpXp
Where,
b0 = Y-axis intercept (all X’s =0)
bi = slope for Xi
p = number of independent variables (X’s)

49. Causal Multiple Regression Using Excel’s Analysis ToolPak
All columns of independent variables must be adjacent to one another (no gaps)
The Analysis ToolPak add-in (Data Analysis) must be “turned on”
Go to file xls

50. Statistical Significance Test Of the Overall Model (F-test)
If all true slope values (βi) equal 0, then the model has no ability to predict Y
Hypotheses:
H0: β1=β2=0 (model has no ability to predict Y)
H1: at least one βi ≠ 0 (at least one variable has ability to predict Y)
F-test is used

51. Statistical Significance Test of Individual Variables (t-test)
Tests whether an individual X is helping to predict Y (in the presence of the other X’s)
Hypotheses for each Xi:
H0: βi=0 (Xi adds no ability to predict Y, given the other X’s in the model)
H1: βi ≠ 0 (Xi adds ability to predict Y, given the other X’s in the model)

52. Hypothesis Test Results for Home Selling Prices
F-test: Overall model has significant ability to predict home prices
T-tests:
Land area – is significant, given the presence of home size
Home size – is not significant, given the presence of land area

53. Multicollinearity
Why did home size become nonsignificant when land area was added?
Multicollinearity exists when 2 or more independent variables are highly correlated
Correlations among X’s can be used to detect multicollinearity