Simple Regression (Trend line Analysis)
Ft = a + b t
Ft = forecast for time t
a = y intercept of the line
b = the slope of the line
n = # of observations
a
X
n
t

b t
n
b
Single Moving Averages
Long Form:
Ft 1 
( xt  xt 1  ........  xt  N 1 )
N
N = # periods in moving average
t  N 1

(  Xi)
i t
N
Short Form:
 X  X tN 
Ft 1  Ft   t

N

n (tX t )   t  X t
n t 2  ( t ) 2
Weighted Moving Average
Ft+1
=
w1Xt
+
w2Xt-1
N
w
Where
i 1
1
+
wNXt-N+1
1
Average Percentage Change
PCt  ( X t  X t 1 ) / X t 1
= percentage change
n
APC  ( PCt ) /( n  1)
t 2
n = number of observations
 n X t  X t 1 
 
 (n  1)
X
t

2
t 1


 X  X1 X 3  X 2
X  X n1 
 2

 ....  n
 (n  1)
X
X
X
1
2
n 1


Ft m  [( APC )(m)( X t )]  X t
 X t [ APC (m)  1]
Where
m
= number of periods ahead to be forecast
Ft+m = forecast m periods ahead
Moving Average Percentage Change
PCt  ( X t  X t 1 ) / X t 1
 t X i  X i 1 
MPCt   PCi / N   
 N
X
i t  N 1
i 1
 i t  N 1

= percentage change
t
N = # of Periods in moving
average
.
 X t  X t 1 X t 1  X t  2
X
 X tN 

 ....  t  N 1


X t 1
X t 2
X tN



N
Ft  m  [(MPCt )(m)( X t )]  X t
 X t [(MPCt  m)  1]
m = # of periods ahead to
be forecast.
Moving Average Percentage Change
Long Form:
Short Form:
 t X i  X i 1 
MPCt   PCi / N   
 N
X
i t  N 1
i 1
 i t  N 1

t
Can use Short Form to get MPCt
MPCt  MPCt 1 
PCt  PCt  N
N
Single Exponential Smoothing
Ft+1 = Ft +  (Xt - Ft)
Assume:
0≤≤1
F1 = X1
Smoothed Percentage Change
SPCt = SPCt-1 +  (PCt – SPCt-1)
Ft+m = (SPCt x (m) x (Xt) ) + Xt
= ( (SPCt x m) + 1 ) Xt
Adaptive Response Rate
Single Exponential Smoothing (ARRSES)
Ft 1  t X t  (1  t ) Ft
 Ft   t ( X t  Ft )
 t 1 
Et
Smoothed mean (ave) error

Mt
Smoothed mean absolute deviation
Et   et  (1   ) Et 1
 Et 1   (et  Et 1 )
M t   et  (1   )M t 1
 M t 1   ( et  M t 1 )
 t 1 
Et 1    ( X t  Ft )  Et 1 
Et

Mt
M t 1    X t  Ft  M t 1 
et  X t  Ft
0 ≤ ≤ 1
Linear Moving Average
(Double Moving Average)
St'  ( X t  X t 1  ....... X t  N 1 ) / N
St"  ( St'  St'1  .......  St' N 1 ) / N
at  2 St'  St"
bt  2( St'  St" ) /( N  1)
Ft  m  at  bt m
Where,
= Single Moving Average
= Double Moving Average
= basic level
= trend adjustment
= # of periods forecast ahead
= # of periods in the moving avg.
Brown’s One Parameter Linear
Exponential Smoothing (Double ES)
St'   X t  (1   ) St'1
 St'1   ( X t  St'1 )
St"   St'  (1   ) St"1
 St"1   ( St'  St"1 )
at  2 St'  St"
bt 

( St'  St" )
1
Ft m  at  bt m
Holt’s Two – Parameter Linear
Exponential Smoothing
(Formula for initial conditions- copy here)
St   X t  (1   )(St 1  bt 1 )
 ( St 1  bt 1 )   [ X t  ( St 1  bt 1 )]
bt   (St  St 1 )  (1   )(bt 1 )
 bt 1   [(St  St 1 )  bt 1 ]
Ft m  St  bt m
Brown’s Quadratic Exponential
Smoothing (Triple Smoothing)
St'   X t  (1   )St'1  St'1   ( X t  St'1)
St''   St'  (1  )St''1  S ''   (St'  S '' )
t 1
t 1
St'''   St''  (1   )St'''1  St'''1   (St''  St'''1)
at  3St'  3St''  St'''
bt 

(6  5 )St'  (10  8 )St"  (4  3 ) St''' 
2 

2(1  )
2
ct 
(St'  2St"  St''' )
2
(1  )
Ft m  at  bt m  12 ct m2
Winter’s Exponential Smoothing
L = Length of seasonality
I = Seasonal adjustment factor
X
St   t  (1   )( St 1  bt 1 )
It L
 Xt
 ( St 1  bt 1 )   
 It L

 ( St 1  bt 1 ) 

bt   (St  St 1 )  (1   )bt 1
X
I t   t  (1   ) I t  L
St
 bt 1    (St  St 1 )  bt 1 
 Xt
 It  L   
 St

 It  L 

Ft m  (St  bt m) I t  Lm
Relative Parameter Values for Moving
Average and Exponential Smoothing
Sales = pattern [trend, seasonal, cyclical] + noise [irregular]
NOISE
Low
High
Low
Medium # of MA
terms
Medium α value
High # of MA
terms
Low α value
High
Low # of MA
terms
High α value
Medium # of MA
terms
Medium α value
PATTERN
CHANGE
Classical Decomposition Method
Ratio–to–Moving Averages
X t  I t  Tt  Ct  Et
Seasonal
Component
1.
Trend
Component
 M t  Tt  Ct
Divide Xt by Mt i.e., divide the original data series by the
moving average  isolates seasonality and randomness
actual
moving average
3.
Random
Component
Calculate moving average with length of seasonal cycle 
eliminates seasonality and randomness
X t  I t  Tt  Ct  Et
I t  Et
2.
Cyclical
Component

Xt
Mt

I t  Tt  Ct  Et
Tt  Ct
Compute the “medial average” of actual/ma ratio.
The medial average is the mean (average) value for each seasonal
period after the largest and smallest values have been excluded.
 eliminates randomness
 isolates seasonality


actual
ave 

 moving average 
 It 
Et
 It
4.
Calculate a trend value
 X b t
a
b
5.
n
n  (tX)   t  X
n t 2    t 
2
Divide the moving average by the trend value
 eliminates trend
 isolates cycle

6.
n
Mt
Tt

Tt  Ct
Tt
 Ct
To Prepare a forecast:
1) A trend value is calculated for period to be forecast.
Tt = a + b t
2) Multiply this by the appropriate seasonal factor It
3) Multiply this by the appropriate cyclical factor (may not be
easy because length of cycle may vary) Ct
Ft = Tt x It x Ct