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 4-quarters at a time, and in a monthly time series, 12-monthly moving averages are calculated. Suppose the given time series is in years and we have decided to calculate 3-year 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)
3-year Moving Totals
3-year 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. 4-year moving averages are calculated as:

Years (t)
Variable (Y)
4-year Moving Averages
4-year 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 4-year moving averages. The process continues until the end of the series to get 4-years 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 5-year, 7-year and 9-year 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-Year Moving
7-Year Moving
9-Year 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-year 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
4-Year Moving Total
4-Year Moving Average
2-values Moving Total
4-year 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
—
—
—
—