The Method of Moving Averages

The method of moving averages are of two types:

1. Simple Moving Averages
2. Weighted Moving Averages

Simple Moving Averages

If the observed values of a variable $Y$ are $y_1, y_2,\cdots, y_n$ corresponding to the time periods $t_1, t_2,\cdots, t_n$, respectively, the $k$-period simple moving averages are defined as

\begin{align*}
a_1 &= \frac{1}{k} \sum_{i=1}^{k} y_i\\
a_2 &= \frac{1}{k} \sum_{i=2}^{k+1} y_i,\\
a_3 &= \frac{1}{k} \sum_{i=3}^{k+2} y_i \\
\vdots &= \quad \vdots\\
a_m &= \frac{1}{k} \sum_{i=m}^{n} y_i
\end{align*}

where $a_1, a_2, \cdots, a_m$ is the sequence of $k$-period simple moving averages. That is, the $k$-period simple moving averages are calculated by averaging first $k$ observations and then repeating this process of averaging the $k$ observations by dropping each time the first observation and including the next one. This process is continued till the last $k$ observations have been averaged. For example, the 3-period simple moving averages are given as:

\begin{align*}
a_1 &= \frac{1}{3} (y_1+y_2+y_3) = \frac{1}{3} \sum_{i=1}^{3} y_i\\
a_2 &= \frac{1}{3} (y_2+y_3+y_4) = \frac{1}{3} \sum_{i=2}^{4} y_i\\
a_3 &= \frac{1}{3} (y_3+y_4+y_5) = \frac{1}{3} \sum_{i=3}^{5} y_i\\
\vdots &= \quad \vdots\\
\text{and so on}
\end{align*}

Each of these simple moving average of the sequence $a_1, a_2, a_3,\cdots$ is placed against the middle of each successive group. The $k$-period moving successive totals $S_1, S_2, S_3, \cdots$ are obtained by the following relations

\begin{align*}
S_1 = \sum_{i=1}^{k} y_i\\
S_2 &= S1+ y_{k+1}-y_1\\
S_3 &= S_2 + y_{k+2} – y_2\\
\vdots &= \quad \vdots\\
\text{so on}
\end{align*}

The $k$-period simple moving averages are obtained by dividing these $k$-period moving successive totals ($S_1, S_2, S_3, \cdots$) by $k$, as given in the following relations

\begin{align*}
a_1 &= \frac{S_1}{k}\\
a_2 &= a_1 + \frac{y_{k_1}0y_1} {k}\\
a_3 &= a_2 + \frac{y_{k+2} -y_2}{k}\\
\vdots &= \quad \vdots\\
\text{so on}
\end{align*}

• When $k$ is odd, the sequence $a_1, a_2, a_3, \cdots$ will be placed against the middle of its time period.
• When $k$ is even, the sequence $a_1, a_2, a_3, \cdots$ of simple moving averages will be placed in the middle of two time periods. It is necessary to centralize these averages. For centralization, further 2-period moving averages of the former $k$-period moving averages are computed which are called $k$-period centered moving averages.

Weighted Moving Averages

For observed values ($y_1, y_2, \cdots, y_n$) of a variable $Y$ corresponding to the time periods $t_1, t_2, \cdots, t_n$, respectively, the $k$-period weighted moving averages with weights $w_1, w_2, \cdots, w_k$ are defined as

\begin{align*}
a_1 &= \frac{1}{\sum w} \sum_{i=1}^{k} y_i w\\
a_2 &= \frac{1}{\sum w} \sum_{i=2}^{k+1} y_i w\\
a_3 &= \frac{1}{\sum w} \sum_{i=3}^{k+2} y_i w\\
\vdots &= \vdots\\
a_m &= \frac{1}{\sum w} \sum_{i=m}^{n} y_i w\\
\end{align*}

where $a_1, a_2, \cdots, a_m$ is a sequence of $k$-period weighted moving averages with weights $w_1, w_2, \cdots, w_k$, respectively. The $k$-period weighted moving averages are calculated by taking the weighted average of first $k$ observed values with weights $w_1, w_2, \cdots, w_k$ and then repeating this process of averaging the $k$ observations by dropping each time the first observation and including the next one. This process is continued until the last $k$ observations have been averaged.

Merits

• The method of moving averages is simple and easy.
• This method is appropriate to remove, seasonal variations, cyclical fluctuations, and irregular variations.

Demerits

• Some values at the beginning and the end of the series are lost.
• Moving averages are greatly affected by extreme values.
• The method does not provide a mathematical formula for the trend.

Example: Calculate 3-years simple moving averages for the following time series. Also, plot actual data and moving averages on a graph. Also, find the 3-years weighted moving averages with weights 2, 2, 1, respectively.

Year	1970	1971	1972	1973	1974	1975	1975	1977
Production	170.0	154.8	156.6	158.9	140.3	154.2	160.7	178.3

Solution:

Year	Production	3-year Simple MT	3-year simple MA	3-year WMT	3-year WMA
1970	170.0
1971	154.8	481.3	160.43	806.1	161.22
1972	156.5	470.2	156.73	781.5	156.30
1973	158.9	455.7	151.90	771.1	154.22
1974	140.3	453.4	151.13	752.6	150.52
1975	154.2	455.2	151.73	749.7	149.94
1976	160.7	493.2	164.40	808.1	161.62
1977	178.3

*MT=moving total, MA=moving averages, WMT=weighted MT, WMA=Weighted MA