Simple Average
MS4102 Business Forecasting Methods
Moving Average Methods
n
Suitable for no trend/horizontal series
yt = β0 + εt,
εt ~ N(0, σ2)
where β0 may change slowly with time
Lecturer :
Room No :
Tel No :
Dr Iris Yeung
P7509
2788 8566
n
The first n data points are averaged and used to
forecast the next period
F
t +1
=
n
∑
y
t
n
t =1
1
Simple Average
n
2
Simple Moving Average
§ Suitable for no trend/ horizontal series
Forecast sales of ovens for 2000 with simple
average
yt = β0 + εt, εt ~ N(0, σ2 )
24
n
where β0 may change slowly with time
t
9800
=
= 408.33
24
24
9800 + 850
=
= 426
25
yˆ 25 =
yˆ 26
∑y
t =1
Disadvantage
uses the mean of all data to forecast and has not
placed more weight to recent observations
§ The n-period moving average calculated at time
period t-1 is the average of the n most recent
observations
M t −1 =
y t −1 + yt −2 + L + yt −n +1 + yt −n
n
3
Simple Moving Average
Simple Moving Average
§ As each new observation becomes available, a new
moving average can be computed by dropping the
oldest value and including the newest one.
yt + yt −1 + L + yt − n+ 1
n
§ Mt can also be calculated by
Mt =
M t = M t −1 +
4
n
We use the moving average calculated at time t to
forecast the y value at time t + 1
Ft +1 = M t =
y t + y t −1 + L + y t −n +1
n
y t − y t−n
n
5
6
1
Example on Simple Moving Average
(2)
(3)
(4)
(5)
Time
Period
Observed
Values
Forecast using
Three-Month
Moving
Average
Forecast using
Five -Month
Moving
Average
Jan
Feb
Mar
Apr
1
2
3
4
200.0
135.0
195.0
197.5
------176.7
---------
May
June
July
Aug
Sept
Oct
5
6
7
8
9
10
310.0
175.0
155.0
130.0
220.0
277.0
175.8
234.2
227.5
213.3
153.3
168.3
--207.5
202.5
206.5
193.5
198.0
Nov
Dec
11
12
235.0
---
209.2
244.2
191.4
203.5
350
Shipments of electric openers
(1)
Month
n
250
Y
200
n=3 ma
150
n=5 ma
100
50
0
0 1 2 3 4 5 6 7 8 9 1011 12 13
Time (months)
Choice of n for Simple Moving Average
n
300
If n = 1, the simple moving average is simply the
naïve model.
Double Moving Average
n
Suitable for linear trend series
n
0 + β1t + ε(t ),
yt be
= βshown
It can
that
If data have large randomness, use large n.
Otherwise use small n.
E (M
t
In general, we choose n such that MSE, MAPE, …
is minimum
9
2
where M t′′ =

t
2
)−

 n −1

β1
 2 
M t + M t −1 + M t − 2 + L + M t −n +1
n
is the double moving average.
Double Moving Average
)
 n − 1 1
β
) = E ( yt ) − 
E ( M t′′) = E ( M
n
ε(t ) ~ N (0,6
10
Double Moving Average
25
Time
1
2
3
4
5
6
7
8
9
10
11
Actual SMA(3) Act-SMA DMA(3) SMA-DMA
2
4
6
4
2
8
6
2
10
8
2
6
2
12
10
2
8
2
14
12
2
10
2
16
14
2
12
2
18
16
2
14
2
20
18
2
16
2
A
B Forecast eForecast
20
10
12
14
16
18
20
2
2
2
2
2
2
12
14
16
18
20
22
0
0
0
0
0
15
Actual
10
SMA(3)
DMA(3)
5
0
0
5
10
15
Time
11
2
Example on Double Moving Average
Double Moving Average
n
The current level of the data can be estimated by
n
The slope β1 of the series can be estimated by
a t = M t + (M t − M t′′) = 2 M t − M ′t ′
bt =
n
2
(M t − M ′t′)
n−1
The forecast for period t + τ is obtained by
extrapolating the trend τ periods into the future
Ft +m =at +bt m
Period
(1)
Inventory
Balance of
Product E12
(2)
Four-Month
Moving Average
of (1)
(3)
Four-Month
Moving Average
of (2)
(4)
One-Month
Ahead
Forecast
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
140.00
159.00
136.00
157.00
173.00
131.00
177.00
188.00
154.00
179.00
180.00
160.00
182.00
192.00
224.00
188.00
198.00
206.00
203.00
238.00
228.00
231.00
221.00
259.00
273.00
---
----148.00
156.25
149.25
159.50
167.25
162.50
174.50
175.25
168.25
175.25
178.50
189.50
196.50
200.50
204.00
198.75
211.25
218.75
225.00
229.50
234.75
246.00
---
------------153.25
158.06
159.62
165.93
169.87
170.12
173.31
174.31
177.87
184.93
191.25
197.62
199.93
203.62
208.18
213.43
221.12
227.00
233.81
---
--------------169.91
182.56
167.29
188.77
184.20
165.12
178.47
185.47
208.87
215.77
215.91
214.62
196.77
223.95
236.35
244.27
243.45
247.66
266.31
13
Summary
Inventory
300
250
200
150
100
50
0
Y
SMA
DMA
FORECAST
0
10
20
30
n
Mean
n
Simple moving average
n
n
n
Time period
Equal weight to all past observations
Equal weight to the past n observations
only
Double moving average
n
Unequal weight
16
Determination of buy/sell
signals
n
Use a moving average
n
n
n
Use two moving averages
n
n
http://www.pcn.com.hk/
Buy if P > MA
Sell if P < MA
Buy if MA(short) > MA (long)
Sell if MA(short) < MA (long)
18
3
19
4