FORECASTING
I. Introduction
-- What is forecasting?
 Scientific (educated) guess
 Based on past data or experience
 Rarely perfect
 Group forecast more accurate
 Shorter time horizon, more accurate
-- Why forecasting?
-- Forecasting time horizon
 Short range – usually less than 3 months
 Medium range – 3 months to 3 years
 Long range – over 3 years
-- Which forecasting method to use?
 Time horizon
 Costs vs. accuracy
 Easy to understand?
-- Pattern of Data
 Trend – gradual upward or downward movement
 Seasonality – repeated pattern
 Cycle – background pattern over long period of time
 Randomness – unexplainable, unpredictable, unknown
fluctuations
II. Qualitative Forecasting Methods
- no past data is available, for managerial decision,
long term forecast
 Jury of executive opinion
 The Delphi technique – systematic survey of experts
 Sales force composite
 Consumer market survey
III. Quantitative Forecasting Methods
1) Smoothing Techniques:
- for data with no clear pattern, short term forecast

Naive: Ft+1 = Yt

n Period Moving Average (MA-n): equal weight for n data
Forecast = (most recent n data)/n
 n Period Weighted Moving Average (WMA-n): flexible
weight assignment
Forecast = (weight * most recent n data)/(weight)

Exponential Smoothing with smoothing constant : very
little data record required, more weight on recent
data, weight decreases exponentially
Ft+1 = Ft + (Yt - Ft) = Yt + (1-)Ft
0   < 1,
-- Example: Forecast the score for test 5 by the above
smoothing techniques with  and F1 = 77.
Test
Avg.Score
1
72
2
82
3
85
4
90
5
?
-- Relationship between smoothing techniques
 average of past data, but differ in weight distribution
 all methods lag behind actual data change
 more weight on most recent data, more responsive, less
smooth
-- Thinking challenge:
1. Which method requires the least amount of data?
2. What happens if the  in the exponential smoothing
method increases?
2) Trend Projection by Linear Regression
- for data with linear trend pattern, medium term
Step 1. Find a trend line y = a + bx to fit the past data
the best (minimizing mean squared errors)
 xy  n x y
b
 best slope:
2
 x2  nx

best intercept: a  y  b x
where x and y are averages of x’s and y’s
-- Example continues:
X
y
xy
x2
y2
* Excel commands:
 calculating b:“=slope(range of y’s, range of x’s)”
 calculating a:“=intercept(range of y’s, range of x’s)”
Step 2. Trend projection
 Trend projection forecast: Fnext = a + b*xnext
-- Example continues:
Step 3. Find forecast interval, if necessary
 Forecast interval = Fnext +- Z(1+ /2 *Sy,x

standard error: Sy,x =
y
2
 a y  b xy
n2
-- Example continues:
* Excel commands:
 calculating Sy,x:”=steyx(range of y’s, range of x’s)”
3) Causal Model by Linear Regression
- same set up as trend projection, except that x is an
independent variable (other than time), y is a
dependent variable, medium term
-- Example: The sales manager of a large apartment rental
complex feels the demand for apartments may be related to
the number of newspaper ads placed during the previous
month. She has collected the data shown below.
Ads
15
9
40
20
25
25
15
35
Rental
6
4
16
6
13
9
10
16
If the number of ads placed in this month is 30, what would
be her estimate of rentals in the coming month?
-- Correlation coefficient r: measures the strength of
linear relationship between x and y
 -1  r  +1
n xy   x y
 r =
n x 2  ( x ) 2 * n y 2  ( y ) 2
* Excel command:
 calculating r:”=correl(range of y’s, range of x’s)”



-- Interpretation:
 r > 0:
 r = 0:
 r < 0:
 r2 = coefficient of determination: % of variation in
the dependent variable (y) is explained by regression
equation (linear relationship).
-- Example continued: How strong is the relationship between
the ads placed and the rentals?
4) Decomposition of Time Series
- for data with both trend and seasonality patterns,
short to medium term
-- Idea:
1. Decompose the past data (filter out the seasonal
influence from original data)
2. Forecast trend pattern and seasonality pattern
separately
3. Combine the forecasts using the multiplicative
model: Yt = Tt * St
-- Example: Data of a popular brand of sweater sale (by
quarters) over the past three years:
2000
Sale
2001
Sale
2002
Sale
(t)
(Yt)
(t)
(Yt)
(t)
(Yt)
1
12
5
16
9
18
2
25
6
32
10
45
3
76
7
80
11
84
4
52
8
62
12
60
Forecast the sale of each season in 2003.
Step 1. Draw the historical data diagram to check if
there is an obvious seasonal pattern
Step 2. Calculate the "seasonal indexes" (S) for each
season.
 Seasonal Index = Avg. Seasonal Sale / Avg Sale
Y (spring) =
===> S(spring) =
Y (summer) =
===> S(summer) =
Y (fall) =
===> S(fall) =
Y (winter) =
===> S(winter) =
Y =
Step 3. Deseasonalize the original data: Tt = Yt / S
2000
(t)
1
2
3
4
Sale
(Tt)
2001
(t)
5
6
7
8
Sale
(Tt)
2002
(t)
9
10
11
12
sale
(Tt)
Step 4. Calculate the trend based on the deseasonalized
demand by linear regression Tt = a + bt
a =
b =
Step 5. Use trend projection to forecast the demand with
trend only.
T13 =
T14 =
T15 =
T16 =
Step 6. Use seasonal indexes to modify the forecasts to
reflect the seasonality patterns: F = T * S
F13 =
F14 =
F15 =
F16 =
IV. Choosing a Forecasting Method
- all criteria are a function of the forecast error

forecast error = actual realization (Y)– forecast (F)
1. Bias – measures the direction of forecast
Bias = (forecast error)/n
2. Mean Absolute Deviation (MAD)
- same penalty for small and large errors
MAD = (|forecast error|)/n
3. Mean Squared Error (MSE)
– more penalty on large errors
MSE = (forecast error2)/n
4. Mean Absolute Percent Error (MAPE)
- relative error, scale independent
MAPE = (|forecast error|/Actual Data)/n
-- Comparing different forecasting methods
 Based on past performance
 Measure dependent
 Choose  (for exp. Smoothing) and n (for MA) to
minimize MAD, MSE, or MAPE
-- Example: Same data as earlier
Exp Smoothing with alpha =
0.1
Test Avg Score
Forecast
Err
|Err|
Err Square |Err|/Actual
1
72
77
-5
5
25
0.069444
2
82
76.5
5.5
5.5
30.25
0.067073
3
85
77.05
7.95
7.95
63.2025
0.093529
4
90
77.845
12.155
12.155
147.744
0.135056
Bias
MAD
MSE
MAPE
5.15125 7.65125
66.54913
0.091276
Exp Smoothing with alpha =
0.2
Test Avg Score
Forecast
Err
|Err|
Err Square |Err|/Actual
1
72
77
-5
5
25
0.069444
2
82
76
6
6
36
0.073171
3
85
77.2
7.8
7.8
60.84
0.091765
4
90
78.76
11.24
11.24
126.3376
0.124889
Bias
MAD
MSE
MAPE
5.01
7.51
62.0444
0.089817
Trend Projection with a =
Test Avg Score
Forecast
Err
1
72
73.7
2
82
79.4
3
85
85.1
4
90
90.8
Bias
68
|Err|
-1.7
2.6
-0.1
-0.8
MAD
0
b=
5.7
Err Square |Err|/Actual
1.7
2.89
0.023611
2.6
6.76
0.031707
0.1
0.01
0.001176
0.8
0.64
0.008889
MSE
MAPE
1.3
2.575
0.016346
V. Control of the Forecasting Process
- to check if the performance of a forecasting method
changes
-- Tracking Signal control chart:
 plotting statistic TS(t) = RSFE(t)/MAD(t)
 RSFE(t) = running sum of the forecast error through
time t
 MAD(t) = MAD through time t
 3 sigma control chart limits: UCL = 3, LCL = -3, CL = 0
-- Interpretation?
-- Example: A forecasting method provides the following
forecasts in the past 6 periods. Does its performance change
over time?
T
Y
F
Error
RFSE(t) MAD(t)
TS(t)
1
8
6
2
7
9
3
10
6
4
2
6
5
12
8
6
11
7