Method of Moving Averages
Suppose that there are $$n$$ time periods denoted by $${t_1},{t_2},{t_3}, \ldots ,{t_n}$$ and the corresponding values of the $$Y$$ variable are $${Y_1},{Y_2},{Y_3}, \ldots ,{Y_n}$$. First of all we have to decide the period of the moving averages. For a short time series we use a period of 3 or 4 values, and for a long time series the period may be 7, 10 or more. For a quarterly time series we always calculate averages taking 4quarters at a time, and in a monthly time series, 12monthly moving averages are calculated. Suppose the given time series is in years and we have decided to calculate 3year moving averages. The moving averages denoted by $${a_1},{a_2},{a_3}, \ldots ,{a_{n – 2}}$$ are calculated as below:
Years (t)

Variable (Y)

3year Moving Totals

3year Moving Averages

$${t_1}$$

$${Y_1}$$

——

——

$${t_2}$$

$${Y_2}$$

$${Y_1} + {Y_2} + {Y_3}$$

$$\frac{{{Y_1} + {Y_2} + {Y_3}}}{3} = {a_1}$$

$${t_3}$$

$${Y_3}$$

$${Y_2} + {Y_3} + {Y_4}$$

$$\frac{{{Y_2} + {Y_3} + {Y_4}}}{3} = {a_2}$$

$${t_4}$$

$${Y_4}$$

$$ \vdots $$

$$ \vdots $$

$$ \vdots $$

$$ \vdots $$

$$ \vdots $$

$$ \vdots $$

$${t_{n – 2}}$$

$${Y_{n – 2}}$$

$$ \vdots $$

$$ \vdots $$

$${t_{n – 1}}$$

$${Y_{n – 1}}$$

$${Y_{n – 2}} + {Y_{n – 1}} + {Y_n}$$

$$\frac{{{Y_{n – 2}} + {Y_{n – 1}} + {Y_n}}}{3} = {a_{n – 2}}$$

$${t_n}$$

$${Y_n}$$

——

——

The average of the first 3 values is $$\frac{{{Y_1} + {Y_2} + {Y_3}}}{3}$$ and is denoted by$${a_1}$$. It is written against the middle year $${t_2}$$. We leave the first value $${Y_1}$$ and calculates the average for the next three values. The average is $$\frac{{{Y_2} + {Y_3} + {Y_4}}}{3} = {a_2}$$ and is written against the middle years$${t_3}$$. The process is carried out to calculate the remaining moving averages. 4year moving averages are calculated as:
Years (t)

Variable (Y)

4year Moving Averages

4year Moving Averages Centered

$${t_1}$$

$${Y_1}$$

——

——

$${t_2}$$

$${Y_2}$$

\[\frac{{{Y_1} + {Y_2} + {Y_3} + {Y_4}}}{4} = {a_1}\]

——

$${t_3}$$

$${Y_3}$$

\[\frac{{{Y_2} + {Y_3} + {Y_4} + {Y_5}}}{4} = {a_2}\]

$$\frac{{{a_1} + {a_2}}}{2} = {A_1}$$

$${t_4}$$

$${Y_4}$$

\[\frac{{{Y_3} + {Y_4} + {Y_5} + {Y_6}}}{4} = {a_3}\]

$$\frac{{{a_2} + {a_3}}}{2} = {A_2}$$

$${t_5}$$

$${Y_5}$$

$$ \vdots $$

$$ \vdots $$

$$ \vdots $$

$$ \vdots $$

$$ \vdots $$

$$ \vdots $$

The first average is $${a_1}$$ which is calculated as \[\frac{{{Y_1} + {Y_2} + {Y_3} + {Y_4}}}{4} = {a_1}\]. It is written against the middle of $${t_3}$$ and $${t_4}$$. The two averages $${a_1}$$ and $${a_2}$$ are further averaged to get an average of $$\frac{{{a_1} + {a_2}}}{2} = {A_1}$$, which refers to the center of $${t_3}$$ and is written against $${t_3}$$. This is called centering the 4year moving averages. The process continues until the end of the series to get 4years moving averages centered. The moving averages of some proper period smooth out the short term fluctuations and the trend is measured by the moving averages.
Example: Compute 5year, 7year and 9year moving averages for the following data.
Years 
1990

1991

1992

1993

1994

1995

1996

19997

1998

1999

2000

Values 
2

4

6

8

10

12

14

16

18

20

22

Solution:
The necessary calculations are given below:


5Year Moving

7Year Moving

9Year Moving


Years

Values

Total

Average

Total

Average

Total

Average

1990

2

—

—

—

—

—

—

1991

4

—

—

—

—

—

—

1992

6

30

6

—

—

—

—

1993

8

40

8

56

8

—

—

1994

10

50

10

70

10

90

10

1995

12

60

12

84

12

108

12

1996

14

70

14

98

14

126

14

1997

16

80

16

112

16

—


1998

18

90

18

—

—

—

—

1999

20

—

—

—

—

—

—

2000

22

—

—

—

—

—

—

Example: Compute 4year moving averages centered for the following time series:
Years 
1995

1996

1997

1998

1999

2000

2001

2002

Production 
80

90

92

83

87

96

100

110

Solution:
The necessary calculations are given below:
Year

Production

4Year Moving Total

4Year Moving Average

2values Moving Total

4year Moving
Average Centered 
1995

80

—

—

—

—

1996

90

345

86.25

—

—

1997

92

352

88.00

174.25

87.125

1998

83

358

89.50

177.50

88.750

1999

87

366

91.50

181.00

90.500

2000

96

393

98.25

189.75

94.875

2001

100

—

—

—

—

2002

110

—

—

—

—

Cosvan
June 23 @ 3:11 pm
I appreciate the work, but how I can get notes when I need it through my email?
zunaira
October 30 @ 7:36 pm
How to find 2 year moving average
Roger
November 13 @ 8:39 pm
This has been very helpful. But what happens if I don’t have data for certain years?