Question

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

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

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

b.   Develop a three-week moving average for this time series. Compute MSE and a forecast for month 11.

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

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

Answer 1

Answer:

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%
11		953	317.667
Total					680.000		16.0%
Mean					97.143		2.3%

Here for mean we divide by 7 becuase we have 7 pairs of actual and forecast value.

c)

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%

d) 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?

The error represents how far our forecast is away from the actual value. So if it is greater then the forecast is not very accurate.

Since MSE expo > moving avg

Better forecast is provided by the moving average.

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