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Demand Forecasting:
Time Series Models
Professor Stephen R. Lawrence
College of Business and Administration
University of Colorado
Boulder, CO 80309-0419

Forecasting Horizons
Long Term
• 5+ years into the future
• R&D, plant location, product planning
• Principally judgement-based
Medium Term
• 1 season to 2 years
• Aggregate planning, capacity planning, sales forecasts
• Mixture of quantitative methods and judgement
Short Term
• 1 day to 1 year, less than 1 season
• Demand forecasting, staffing levels, purchasing, inventory levels
• Quantitative methods

Short Term Forecasting:
Needs and Uses
Scheduling existing resources
• How many employees do we need and when?
• How much product should we make in anticipation of demand?
Acquiring additional resources
• When are we going to run out of capacity?
• How many more people will we need?
• How large will our back-orders be?
Determining what resources are needed
• What kind of machines will we require?
• Which services are growing in demand? declining?
• What kind of people should we be hiring?

Types of Forecasting Models
Types of Forecasts
• Qualitative --- based on experience, judgement, knowledge;
• Quantitative --- based on data, statistics;
Methods of Forecasting
• Naive Methods --- eye-balling the numbers;
• Formal Methods --- systematically reduce forecasting errors;
– time series models (e.g. exponential smoothing);
– causal models (e.g. regression).
• Focus here on Time Series Models
Assumptions of Time Series Models
• There is information about the past;
• This information can be quantified in the form of data;
• The pattern of the past will continue into the future.

Forecasting Examples
Examples from student projects:
• Demand for tellers in a bank;
• Traffic on major communication switch;
• Demand for liquor in bar;
• Demand for frozen foods in local grocery warehouse.
Example from Industry: American Hospital Supply Corp.
• 70,000 items;
• 25 stocking locations;
• Store 3 years of data (63 million data points);
• Update forecasts monthly;
• 21 million forecast updates per year.

Simple Moving Average
Forecast Ft is average of n previous observations or
actuals Dt:
Note that the n past observations are equally weighted.
Issues with moving average forecasts:
• All n past observations treated equally;
• Observations older than n are not included at all;
• Requires that n past observations be retained;
• Problem when 1000's of items are being forecast.
∑−+=
+
−+−+
=
+++=
t
nti
it
ntttt
D
n
F
DDD
n
F
1
1
111
1
)(
1


Simple Moving Average
Include n most recent observations
Weight equally
Ignore older observations
weight
today
123...n
1/n

Moving Average
Internet Unicycle Sales
0
50
100
150
200
250
300
350
400
450
Apr-01 Sep-02 Jan-04 May-05 Oct-06 Feb-08 Jul-09 Nov-10 Apr-12 Aug-13
Month
Units
n = 3

Example:
Moving Average
Forecasting

Exponential Smoothing I
Include all past observations
Weight recent observations much more heavily
than very old observations:
weight
today
Decreasing weight given
to older observations

weight
today
0 1< <α
α

weight
today
0 1< <α
α
α α( )1−

weight
today
0 1< <α
α
α α
α α
( )
( )
1
1 2
−
−

Exponential Smoothing: Concept
weight
today
0 1< <α
α
α α
α α
α α
( )
( )
( )
1
1
1
2
3
−
−
−


Exponential Smoothing: Math
[ ]

+−+−+=
+−+−+=
−−
−−
21
2
2
1
)1()1(
)1()1(
tttt
tttt
DaDDF
DDDF
αααα
ααααα

1)1( −−+= ttt FaaDF
[ ]

+−+−+=
+−+−+=
−−
−−
21
2
2
1
)1()1(
)1()1(
tttt
tttt
DaDDF
DDDF
αααα
ααααα

Thus, new forecast is weighted sum of old forecast and actual
demand
Notes:
• Only 2 values (Dt and Ft-1 ) are required, compared with n for moving average
• Parameter a determined empirically (whatever works best)
• Rule of thumb: α < 0.5
• Typically, α = 0.2 or α = 0.3 work well
Forecast for k periods into future is:
1
2
2
1
)1(
)1()1(
−
−−
−+=
+−+−+=
ttt
tttt
FaaDF
DaaDaaaDF 
tkt FF =+

Exponential Smoothing
Internet Unicycle Sales (1000's)
0
50
100
150
200
250
300
350
400
450
Jan-03 May-04 Sep-05 Feb-07 Jun-08 Nov-09 Mar-11 Aug-12
Month
Units
α = 0.2

Example:

Complicating Factors
Simple Exponential Smoothing works well
with data that is “moving sideways”
(stationary)
Must be adapted for data series which
exhibit a definite trend
Must be further adapted for data series
which exhibit seasonal patterns

Holt’s Method:
Double Exponential Smoothing
What happens when there is a definite trend?
A trendy clothing boutique has had the following sales
over the past 6 months:
1 2 3 4 5 6
510 512 528 530 542 552
480
490
500
510
520
530
540
550
560
1 2 3 4 5 6 7 8 9 10
Month
Demand
Actual
Forecast

Holt’s Method:
Double Exponential Smoothing
Ideas behind smoothing with trend:
• ``De-trend'' time-series by separating base from trend effects
• Smooth base in usual manner using α
• Smooth trend forecasts in usual manner using β
Smooth the base forecast Bt
Smooth the trend forecast Tt
Forecast k periods into future Ft+k with base and trend
))(1( 11 −− +−+= tttt TBDB αα
11 )1()( −− −+−= tttt TBBT ββ
ttkt kTBF +=+

ES with Trend
0
50
100
150
200
250
300
350
400
450
Month
Units
α = 0.2, β = 0.4

Example:
with Trend

Winter’s Method:
w/ Trend and Seasonality
Ideas behind smoothing with trend and seasonality:
• “De-trend’: and “de-seasonalize”time-series by separating base from
trend and seasonality effects
• Smooth base in usual manner using α
• Smooth trend forecasts in usual manner using β
• Smooth seasonality forecasts using γ
Assume m seasons in a cycle
• 12 months in a year
• 4 quarters in a month
• 3 months in a quarter
• et cetera

Winter’s Method:
Smooth the base forecast Bt
Smooth the seasonality forecast St
))(1( 11 −−
−
+−+= tt
mt
t
t TB
S
D
B αα
11 )1()( −− −+−= tttt TBBT ββ
mt
t
t
t S
B
D
S −−+= )1( γγ

Winter’s Method:
Forecast Ft with trend and seasonality
Smooth the seasonality forecast St
mktttkt SkTBF −+−−+ += )( 11
11 )1()( −− −+−= tttt TBBT ββ
mt
t
t
t S
B
D
S −−+= )1( γγ

ES with Trend and Seasonality
0
50
100
150
200
250
300
350
400
450
500
Month
Units
α = 0.2, β = 0.4, γ = 0.6

Example:
with
Trend and Seasonality

Forecasting Performance
Mean Forecast Error (MFE or Bias): Measures
average deviation of forecast from actuals.
Mean Absolute Deviation (MAD): Measures
average absolute deviation of forecast from
actuals.
Mean Absolute Percentage Error (MAPE):
Measures absolute error as a percentage of the
forecast.
Standard Squared Error (MSE): Measures
variance of forecast error
How good is the forecast?

Forecasting Performance Measures
)(
1
1
t
n
t
t FD
n
MFE −= ∑=
∑=
−=
n
t
tt FD
n
MAD
1
1
∑=
−
=
n
t t
tt
D
FD
n
MAPE
1
100
2
1
)(
1
t
n
t
t FD
n
MSE −= ∑=

Want MFE to be as close to zero as possible --
minimum bias
A large positive (negative) MFE means that the
forecast is undershooting (overshooting) the actual
observations
Note that zero MFE does not imply that forecasts are
perfect (no error) -- only that mean is “on target”
Also called forecast BIAS
Mean Forecast Error (MFE or Bias)
)(
1
1
t
n
t
t FD
n
MFE −= ∑=

Mean Absolute Deviation (MAD)
Measures absolute error
Positive and negative errors thus do not cancel out (as
with MFE)
Want MAD to be as small as possible
No way to know if MAD error is large or small in
relation to the actual data
∑=
−=
n
t
tt FD
n
MAD
1
1

Mean Absolute Percentage Error
(MAPE)
Same as MAD, except ...
Measures deviation as a percentage of actual data
∑=
−
=
n
t t
tt
D
FD
n
MAPE
1
100

Mean Squared Error (MSE)
Measures squared forecast error -- error variance
Recognizes that large errors are disproportionately more
“expensive” than small errors
But is not as easily interpreted as MAD, MAPE -- not as
intuitive
2
1
)(
1
t
n
t
t FD
n
MSE −= ∑=

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