Chapter 8, Problem 1P

Consider the following time series data:

Using the naïve method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy:

1. a. Mean absolute error
2. b. Mean squared error
3. c. Mean absolute percentage error
4. d. What is the forecast for week 7?

To determine

Find the value of mean absolute error.

Answer to Problem 1P

The mean absolute error is 4.4.

Explanation of Solution

Week	Time Series Value	Forecast	Forecast Error	Absolute Value of Forecast Error
1	18
2	13	18	−5	5
3	16	13	3	3
4	11	16	−5	5
5	17	11	6	6
6	14	17	−3	3
			Total	22

$\begin{array}{c}MAE=\frac{{\displaystyle {\sum }_{}^{}\left|e_r\right|}}{n-k}\\ =\frac{22}{5}\\ =4.4\end{array}$

Thus, the mean absolute error is 4.4.

To determine

Obtain the mean squared error.

Answer to Problem 1P

The mean squared error is 20.8.

Explanation of Solution

The mean squared error is obtained as given below:

Absolute Value of Forecast Error	Squared Forecast Error
5	25
3	9
5	25
6	36
3	9
22	104

$\begin{array}{c}MSE=\frac{{\displaystyle {\sum }_{}^{}{\left|e_r\right|}^2}}{n-k}\\ =\frac{104}{5}\\ =20.8\end{array}$

Thus, the mean squared error is 20.8.

To determine

Find the mean absolute percentage error.

Answer to Problem 1P

The mean absolute percentage error is 31.88.

Explanation of Solution

The mean absolute percentage error is obtained is given below:

Week	Time Series Value $\left(y_i\right)$	Forecast	Forecast Error $e_i$	Percentage Error $\frac{y_i}{e_i}$	Absolute percentage Error $\left|\frac{y_i}{e_i}\right|$
1	18
2	13	18	−5	−38.46	38.46
3	16	13	3	18.75	18.75
4	11	16	−5	−45.45	45.45
5	17	11	6	35.29	35.29
6	14	17	−3	−21.43	21.43
			Total	−51.30	159.38

$\begin{array}{c}MAPE=\frac{{\displaystyle {\sum }_{i=k+1}^n\left|\frac{y_i}{e_i}\right|100}}{n-k}\\ =\frac{159.38}{5}\\ =31.88\end{array}$

Thus, the mean absolute percentage error is 31.88.

To determine

Obtain the forecast for week 7.

Answer to Problem 1P

The forecast for week 7 is 14.

Explanation of Solution

The forecast for week 7 is obtained as given below:

$\begin{array}{c}{\widehat{y}}_7=y_6\\ =14\end{array}$

Thus, the forecast for week 7 is 14.