**
Winter’s
Method**

Often,
we must
predict
future
values
of a
time series
such as
monthly
costs or
monthly
prod- uct
revenues.
This is
usually
difficult
because the
characteristics
of any
time series
are
con- stantly
changing. *Smoothing
*or
*adaptive
*methods
are
usually
best suited
for
forecasting
future values
of
a time
series.
In this
section,
we
describe
the
most powerful
smoothing
method:
*Winter’s
method*.
To help
you understand
how
Winter’s method
works,
we
will use it
to
fore-
cast monthly
housing starts
in
the
United
States
(U.S.).
Housing starts
are
simply
the
number

of new homes
whose
construction
begins
during a
month.
We begin
by
describing
the
three
key
characteristics
of a
time series.

**
Time
Series
Characteristics**

The behavior of most
time
series can
be
explained
by
understanding
the
following
three
characteristics: base, trend,
and seasonality.

■ The
*base *of a
series describes
the
series’ current
level
in
the
absence of
any seasonality.
For example,
suppose
the
base
level
for
U.S.
housing
starts
is
160,000.
In
this
case,
we believe
that
if
the
current
month
were
an “average”
month
relative
to
other
months
of
the
year,
then
160,000
housing
starts
would
occur.

■ The
*
trend
*
of
a
time
series
is
the
percentage
increase
per
period
in
the
base. Thus
a trend

of 1.02
means that
we
estimate
that housing
starts
are increasing
by
2 percent
each month.

■ The
*seasonality *(seasonal
index)
for
a period tells
us
how
far
above
or below
a typical month
we
can expect
housing starts
to
be.
For
example,
if
the
December
seasonal
index

is .8,
then
December
housing
starts
are
20
percent
below
a typical
month.
If
the
June sea-
sonal index is
1.3,
then
June
housing
starts
are 30 percent
higher than
a typical month.

**
Parameter
Definitions**

After
observing
month
*t*,
we will
have
used all
data
observed
through
the
end of month
*t *
to estimate
the
following
quantities
of interest:

■ Lt=Level
of
series

■ Tt=Trend
of
series

■ St=Seasonal
index
for
current
month

The
key
to
Winter’s
method
is
the
following
three
equations,
which are
used
to update
Lt,
Tt,
and
St.
In
the
following
formulas,
alp,
bet,
and
gam
are
called *
smoothing parameters*.
The values

of these
parameters
will be chosen
to optimize
our forecasts.
In the
following
formulas,
*c*

equals
the
number
of
periods
in
a
seasonal
cycle
(*c=12
*
months
for
example)
and
*
x*t equals
the
observed
value of the
time series
at
time
*t*.

■ Formula
1: *L*t*=alp(x*t*/s*t–c*)+(1–alp)(L*t–1**T*t–1*)*

■ Formula
2: *T*t*=bet(L*t*/L*t–1*)+(1–bet)T*t–1

■ Formula
3: *S*t*=gam(x*t*/L*t*)+(1–gam)s*t–-c

Formula
1
indicates
that
our new base estimate
is a
weighted
average
of
the
current
observa- tion (deseasonalized)
and last
period’s
base updated
by
our last
trend
estimate.
Formula
2 indicates
that
our new
trend
estimate is
a
weighted
average
of
the
ratio
of our
current
base
to last
period’s base
(this
is a
current
estimate of trend)
and last
period’s
trend.
Formula
3 indi-
cates
that
we update
our seasonal
index
estimate
as a
weighted
average
of
the
estimate of
the seasonal
index
based on
the
current
period
and
the
previous
estimate.
Note
that
larger
values

of the
smoothing
parameters
correspond
to putting
more
weight
on
the
current
observation.

We define
*F*t,k as our
forecast
(*F*) after
period
*t *for
the
period
*t+k*. This
results
in
the
formula

*
F*t,k*=L*t*(*T *k*s*

t+k–c.

This
formula
first
uses
the
current
trend
estimate
to
update
the
base
*
k
*
periods
forward.
Then the
resulting
base
estimate
for
period
*t+k *is
adjusted by
the
appropriate
seasonal index.

**
Initializing Winter’s
Method**

To
start
Winter’s method,
we
must
have
initial estimates
for
the
series
base, trend,
and seasonal indexes.
We will
use monthly
housing
starts
for
the
years
1986
through
1987
to initialize
Winter’s method.
Then
we
will choose our
smoothing
parameters
to optimize

our
one-month-ahead
forecasts
for
the
years
1988
through
1996.
See
Figure
53-1
and
the
file
House2.xlsx.
We’ll
use
the
following
process.

**
Step
1: **
We
will estimate, for
example,
the
January
seasonal
index
as
the
average
of January
housing starts
for
1986
through
1987
divided by
the
average
monthly
starts
for
1986

through
1987.
Therefore
copying
from G14
to G15:G25
the
formula
*=AVERAGE(B2,B14)/
AVERAGE($B$2:$B$25)
*will
generate
our
estimates
of seasonal
indexes.
For
example,
the
January
estimate
is 0.75
and
the
June estimate is 1.17.

**
Step
2: **
To
estimate
the average
monthly trend,
we
take
the
twelfth
root
of (1987
mean
starts
divided
by
the
1986
mean
starts).
We
compute
this
in cell
J3 (and
copy
it
to
cell
D25) with
the
formula
*=(J1/J2)^(1/12)*

Figure
53-1
Initialization
of
Winter’s
method

**
Step
3: **
Going
into January
1987,
we estimate
the
base of
the
series
as
the
deseasonalized

December
1987
value.
This
is
computed
in
C25
with
the
formula
*=**(B25/G25)*.

**
Estimating the
Smoothing Constants**

We
are now
ready
to estimate
our
smoothing
constants.
In column C,
we will
update
the

series
base;
in
column
D,
the
series
trend;
and
in
column
G,
our
seasonal
indexes.
In
column

E,
we
compute
our
forecast
for
next
month,
and
in column
F,
we
compute
our
absolute
per-
centage
error
for
each month.
Finally,
we
will use
solver
to
choose
smoothing
constant
values
that
minimize the
sum
of our absolute
percentage
errors.
We’ll
use
the
following
process.

**
Step
1: **
In
G11:I11,
we
enter
trial
values
(between
0 and
1)
for
our
smoothing
constants.

**
Step
2: **
In
C26:C119,
we
compute
the
updated series
level
with
(1) by
copying
from
C26 to

C27:C119
the
formula
*=alp*(B26/G14)+(1–alp)*(C25*D25)*.

**
Step
3:
**
In
D26:D119,
we
use (2)
to
update
the
series
trend.
Copy
from
D26
to
D27:D119
the
formula
*=bet*(C26/C25)+(1–bet)*D25*.

**
Step
4:
**
In
G26:G119,
we
use
(3)
to
update
the
seasonal
indexes.
Copy
from
G26
to
G27:G119

the
formula
*=gam*(B26/C26)+(1–gam)*G14*.

**
Step
5: **
In
E26:E119,
we use (4)
to compute
the
forecast
for
the
current
month
by
copying
from
E26
to
E27:E119
the
formula
*=(C25*D25)*G14*.

**
Step
6: **
In
F26:F119,
we
compute
the
absolute percentage
error
for
each
month
by
copying
from
F26
to
F27:F119
the
formula
*=ABS(B26-E26)/B26*.

**
Step
7:
**
We
compute
the
average
absolute
percentage
error
for
the
years
1988
through
1996
in

F21
with
the
formula
*=AVERAGE(F26:F119)*.

**
Step
8:
**
We
can now
use
the
Microsoft Office
Excel 2007 Solver
feature
to determine
smooth- ing parameter
values that minimize
our average
absolute percentage
error.
The Solver
Parameters
dialog
box
is shown
in Figure
53-2.

Figure
53-2
Solver
Parameters
dialog box for
Winter’s
model

We
choose
our
smoothing
parameters
(G11:I11)
to
minimize
the
average
absolute
percentage
error
(cell
F21).
The
Excel
Solver
ensures
we
will
find
the
best
combination
of
smoothing
con- stants. Smoothing
constants must
be between
0 and
1.
We
find
that
*alp=.54*,
*bet=.02*,
and *
gam=.29
*minimizes
our
average
absolute percentage
error.
You
might
find slightly different
values
of
the
smoothing
constants,
but
you
should
obtain
a MAPE
close
to
7.3
percent.
In
this example,
there
are
many
combinations
of
the
smoothing
constants
that
give
forecasts
having approximately
the
same
MAPE. Our one-month-ahead
forecasts
are off
by
an average
of
7.3
percent.

■ Instead
of
choosing
our
smoothing
parameters
to
optimize one-period
forecast
errors,
we could,
for
example, have
chosen to optimize
the
average
absolute
percentage
error
incurred
in
forecasting
total
housing
starts
for
the
next
six
months.

■ Suppose
our
time
series is sales
of a software
product
and
we
have
conducted
a major
promotion
during
June
2000.
Assume
predicted
sales
for
June
2000
were
20,000
units,
but
we
sold
35,000
units.
Then
a
good
guess
is
that
the
promotion
caused
15,000
extra
sales during June.
When updating
the
base,
trend,
and seasonal
indexes,
however,
we should
not put
in June
2000 sales
of 35,000.
We
should
put in
June
2000
sales
of our
forecast
(20,000); otherwise,
we will
incorrectly
bump
up our
forecasts
of future
sales. When making
a
forecast
for
a future
month
in which
there
is a
promotion
similar
to
the
June
promotion,
we
would
just
bump
up
the
Winter’s
method
forecast
by using
the
for-
mula *35,000/20,000=75%*!

■ If
at
the
end
of
month
*
t
*
we
wanted
to
forecast
sales
for
the
next
four
quarters,
we
would
simply
add *f*t,1*+f*t,2*+f*t,3*+f*t,4.
If
desired,
we
could
choose
our
smoothing
parameters
to
minimize the
absolute
percentage
error
incurred
in estimating
sales for
the
next
year.

**
Problems**

All
the
data
for
the
following
problems
is in
the
file
Quarterly.xlsx.

**
1.
**Use
Winter’s
method
to
forecast
one-quarter-ahead
revenues
for
Apple.

**
2. **Use
Winter’s
method
to
forecast
one-quarter-ahead
revenues
for
Amazon.com.

**
3. **Use
Winter’s
method
to
forecast
one-quarter-ahead
revenues
for
Home
Depot.

**
4. **Use
Winter’s method
to
forecast
total
revenues
for
the
next two
quarters
for
Home

Depot.