Question

The following times series shows the demand for a particular product over the past 10 months.

Month	Value
1	324
2	311
3	303
4	314
5	323
6	313
7	302
8	315
9	312
10	326

a. Use α = 0.2 to compute the exponential smoothing values for the time series. Compute MSE, MAPE and a forecast for month 11.

b. Calculate MSE and MAPE for three month moving average ?

c. Compare the three-month moving average forecast with the exponential smoothing forecast using α = 0.2. Which appears to provide the better forecast based on MSE?

Answer 1

a. Use α = 0.2 to compute the exponential smoothing values for the time series. Compute MSE, MAPE and a forecast for month 11.

For exponential smoothing we will assume the first month forecast value is same as actual.

For next forecast values we hhvae the formula

Eg: For month 4 we have F4 = A3 * 0.2 + F3*0.8

Month	Value	Forecast
1	324	324
2	311	324.000
3	303	321.400
4	314	317.720
5	323	316.976
6	313	318.181
7	302	317.145
8	315	314.116
9	312	314.293
10	326	313.834
11		316.267

MSE = mean squared error

where the error = actual - forecast

So we square the error.

MSE = ......We divide 9 because we have 9 pirs of the actual and forecast.

Month	Value	Forecast	Error	Error^2
1	324	324
2	311	324.000	-13.000	169.000
3	303	321.400	-18.400	338.560
4	314	317.720	-3.720	13.838
5	323	316.976	6.024	36.289
6	313	318.181	-5.181	26.841
7	302	317.145	-15.145	229.360
8	315	314.116	0.884	0.782
9	312	314.293	-2.293	5.256
10	326	313.834	12.166	148.010
11		316.267
Total				967.936
Mean				107.548

MAPE = mean absolute percentage error

Where percentage error = Absolute error / actual

MAPE =

Month	Value	Forecast	Error	Abs error	APE
1	324	324
2	311	324.000	-13.000	13	4.2%
3	303	321.400	-18.400	18.4	6.1%
4	314	317.720	-3.720	3.72	1.2%
5	323	316.976	6.024	6.024	1.9%
6	313	318.181	-5.181	5.181	1.7%
7	302	317.145	-15.145	15.145	5.0%
8	315	314.116	0.884	0.884	0.3%
9	312	314.293	-2.293	2.293	0.7%
10	326	313.834	12.166	12.166	3.7%
11		316.267
Total					24.7%
Mean					2.7%

b. Calculate MSE and MAPE for three month moving average ?

Moving average is where we first take the total of previous 'n' periods and then divide by 'n'.

So Eg: we have F4 = (sum (A1 + A2+ A3)/ 3

Month	Value	Moving Total	Forecast	Error	Error^2	Abs error	APE
1	324
2	311
3	303
4	314	938	312.667	1.333	1.778	1.333	0.4%
5	323	928	309.333	13.667	186.778	13.667	4.2%
6	313	940	313.333	-0.333	0.111	0.333	0.1%
7	302	950	316.667	-14.667	215.111	14.667	4.9%
8	315	938	312.667	2.333	5.444	2.333	0.7%
9	312	930	310.000	2.000	4.000	2.000	0.6%
10	326	929	309.667	16.333	266.778	16.333	5.0%
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