# 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 ${t_1}$ ${Y_1}$ ------ ------ ${t_2}$ ${Y_2}$ ------ ${t_3}$ ${Y_3}$ $\frac{{{a_1} + {a_2}}}{2} = {A_1}$ ${t_4}$ ${Y_4}$ $\frac{{{a_2} + {a_3}}}{2} = {A_2}$ ${t_5}$ ${Y_5}$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$

The first average is ${a_1}$ which is calculated as

. 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 --- --- --- ---