Forecasting
• Forecasting is attempting to predict the
future
• Decision makers want to reduce
uncertainty by predicting future values
such as sales or investment return
Steps in Forecasting
1. Determine the objective of the forecast
2. Identify items to be forecast
3. Determine time horizon
4. Select the forecasting model(s)
5. Gather data
6. Validate model
7. Make forecast and implement results
Types of Forecasts
1. Qualitative - subjective methods
based on intuition and experience
2. Time Series – based on historical
data and assume the past indicates
the future
3. Causal Models – data based where
there may be a cause and effect
relation between variables
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Qualitative Forecasting Models
1. Delphi Method – an iterative group
process where a group of experts
attempt to reach consensus
2. Jury of Executive Opinion – uses
opinions of high level managers often
combined with statistical models
Qualitative Forecasting Models
3. Sales Force Composite – each
salesperson estimates sale in his/her
own region and forecast are combined
for an overall forecast
4. Consumer Market Survey – future
purchase plans are solicited from
customers
Measuring Forecast Error
Measures how accurate the forecast was
For time period t:
Forecast error = Actual value – Forecast value
= At - Ft
2
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
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
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.
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Components of a Time Series
1. Trend – long term upward or
downward movement
2. Seasonality – the pattern that occurs
every year
3. Cycles – the pattern that occurs over
a period of years
4. Random variations – caused by
chance and unusual events
Time Series Components
Time Series Decomposition
• A time series can be broken down into its
individual components
• Two approaches:
1. Multiplicative decomposition
Forecast = Trend x Seasonality x Cycles x Random
2. Additive decomposition
Forecast = Trend + Seasonality + Cycles + Random
4
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
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
Wallace Garden Supply Example
5
Using ExcelModules
for Forecasting
• Install ExcelModules from the CD-ROM
• Excel files on CD-ROM are color coded
– Input cells for data are yellow
– Output cells for forecasts and error
measurement are green
• ExcelModules will appear in the main
menu bar of Excel
Go to file 11-2.xls
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
Wallace Garden Supply With
Weighted Moving Averages
Period Weights
last month 3
2 month ago 2
3 months ago 1
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3-Month Weighted Moving Average
Using ExcelModules for
Weighted Moving Averages
• Select the Weighted Moving Averages
option within the ExcelModules menu
Go to file 11.3-xls
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
7
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
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
Go to file 11-4.xls and use ExcelModules
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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
Midwestern Electric Co. Example
Go to file 11-5.xls
The Trend Equation
Ŷ = b0 + b1X
where,
Ŷ = forecast average dependent value
X = independent value (time)
b0 = Y-intercept
b1 = slope of the line
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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
Least Squares Method
for Best-Fitting Line
Least Squares Line With Excel
• Can use ExcelModules, or
• 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
10
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
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
Decomposition of a Time Series
• Decomposition breaks a time series
down into its components (Trend,
Seasonal, Cyclical, and Random)
• Two types of models
1. Multiplicative
2. Additive
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Multiplicative Decomposition
Sawyer Piano House Example
• Want to forecast sales of grand pianos
• Have quarterly data for the past 5 years
• Steps:
1. 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
Steps Continued
2. Find the trend equation using the
unseasonalized values
3. Calculate forecasts
• Use the trend equation to make an
unseasonalized forecast
• Multiply the unseasonalized forecast by the
seasonal index
4. Calculate forecast error
Go to file 11-8.xls
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
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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
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
The Regression Equation
Forecast average selling price =
-8.125 + 97.789(Home size)
Slope interpretation: On average the price
of a home will increase by $97.789
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)
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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
Correlation Coefficient Examples
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
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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 = -8.125 + 97.789(3.1) = 295
which is $295,000
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
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:
295 + 1.96 (42.602)
Which is $211,500 to $378,500
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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
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
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
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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)
Causal Multiple Regression
Example Using ExcelModules
Go to file 11-10.xls
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 11-10.xls
17
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
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)
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
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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
• Analysis ToolPak can produce the
correlation matrix
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