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
---
---
---
---