Method of Moving Averages

Suppose that there are n times periods denoted by {t_1},{t_2},{t_3}, \ldots ,{t_n}and the corresponding values of 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 short time series, we use period of 3 or 4 values. For long time series, the period may be 7, 10 or more. For quarterly time series, we always calculate averages taking 4-quarters at a time. In monthly time series, 12-monthly moving averages are calculated. Suppose the given time series is in years and we have decided to calculate 3-years moving average. The moving averages denoted by {a_1},{a_2},{a_3},  \ldots ,{a_{n - 2}} are calculated as below:

Years (t)

Variable (Y)

3-Years moving totals

3-Years 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 calculate 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. 4-years moving averages are calculated as under:

Years (t)

Variable (Y)

3-Years moving averages

3-Years 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 \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 of the 4-years moving averages. The process is continued till the end of the series to get 4-years moving average 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 5-years, 7-years and 9-years 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:

 

 

5-Years Moving

7-Years Moving

9-Years 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 4-years moving average 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

4-Years Moving Total

4-Years Moving Average

2-values Moving Total

4-years 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

---

---

---

---

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