Question

The following time series shows the sales of a particular product over the past 12 months.

Month	Sales
1	105
2	135
3	120
4	105
5	90
6	120
7	145
8	140
9	100
10	80
11	100
12	110

(a)

Construct a time series plot.

(b)

Use α = 0.3 to compute the exponential smoothing forecasts for the time series. (Round your answers to two decimal places.)

Month t	Time Series Value Yt	Forecast Ft
1	105
2	135
3	120
4	105
5	90
6	120
7	145
8	140
9	100
10	80
11	100
12	110

(c)

Use a smoothing constant of α = 0.5 to compute the exponential smoothing forecasts. (Round your answers to two decimal places.)

Month t	Time Series Value Yt	Forecast Ft
1	105
2	135
3	120
4	105
5	90
6	120
7	145
8	140
9	100
10	80
11	100
12	110

Does a smoothing constant of 0.3 or 0.5 appear to provide more accurate forecasts based on MSE?

A smoothing constant of 0.5 is better than a smoothing constant of 0.3 since the MSE is greater for 0.5 than for 0.3.

A smoothing constant of 0.3 is better than a smoothing constant of 0.5 since the MSE is greater for 0.3 than for 0.5.

A smoothing constant of 0.3 is better than a smoothing constant of 0.5 since the MSE is less for 0.3 than for 0.5.

A smoothing constant of 0.5 is better than a smoothing constant of 0.3 since the MSE is less for 0.5 than for 0.3.

Answer 1

solution:

a) Plot the points on the graph.taking months on X-axis and sales on Y- axis.

The time series plot would be:

  

b) Exponential Smoothing Forecast

F(t+1) =  Yt + (1-)Ft

Where F(t+1) is the forecast of timeseries for period t+1

Yt is the actual values of time series in period t

Ft is the forecast of time series for period t

   is the smoothing constant [ 0<= <=1]

The following table shows forecast calculations for calculating forecast and MSE

Let   = 0.3 and 1- = 0.7

F2 = 0.3*105 + 0.7*105 = 105     F7 = 0.3*120 + 0.7*105.8 = 110.06

F3 = 0.3*135 + 0.7*105 = 114 F8 = 0.3*145 + 0.7*110.06 = 120.54

F4 = 0.3*120 + 0.7*114 = 115.8 F9 = 0.3*140 + 0.7*120.54 = 126.38

F5 = 0.3*105 + 0.7*115.8 = 112.56 F10 = 0.3*100 + 0.7*126.38 = 118.50

F6 = 0.3*90 + 0.7*112.56 = 105.8 F11 = 0.3*80 + 0.7*118.50 = 106.95

F12 = 0.3*100 + 0.7*106.95 = 104.90

Month	Sales (A)	Forecast(N)	Forecast Error(A-N)	Absolute value of forecast Error |A-N|	Squared forecast error |A-N|^2
1	105
2	135	105	30	30	900
3	120	114	6	6	36
4	105	115.8	-10.8	10.8	116.64
5	90	112.56	-22.56	22.56	508.95
6	120	105.80	14.2	14.2	201.64
7	145	110.06	34.94	34.94	1220.80
8	140	120.54	19.46	19.46	378.69
9	100	126.38	-26.38	26.38	695.90
10	80	118.50	-38.50	38.50	1482.25
11	100	106.95	-6.95	6.95	48.30
12	110	104.90	5.1	5.1	26.01
		Totals	4.51	214.89	5,615.18

Therefore, MSE = squared forecast error/ No.of Months forecasted = 5615.18 / 11 =~ 510.47

c)

The following table shows forecast calculations for calculating forecast and MSE

Let   = 0.5 and 1- = 0.5

F2 = 0.5*105 + 0.5*105 = 105 F7 = 0.5*120 + 0.5*101.25 = 110.63

F3 = 0.5*135 + 0.5*105 = 120 F8 = 0.5*145 + 0.5*110.63 = 127.82

F4 = 0.5*120 + 0.5*120 = 120 F9 = 0.5*140 + 0.5*127.82 = 133.91

F5 = 0.5*105 + 0.5*120 = 112.5 F10 = 0.5*100 + 0.5*133.91 = 116.95

F6 = 0.5*90 + 0.5*112.5 = 101.25    F11 = 0.5*80 + 0.5*116.95 = 98.48

F12 = 0.5*100 + 0.5*98.48 = 99.24

Month	Sales (A)	Forecast(N)	Forecast Error(A-N)	Absolute value of forecast Error |A-N|	Squared forecast error |A-N|^2
1	105
2	135	105	30	30	900
3	120	120	0	0	0
4	105	120	-15	15	225
5	90	112.50	-22.50	22.50	506.25
6	120	101.25	18.75	18.75	351.56
7	145	110.63	34.37	34.37	1181.30
8	140	127.82	12.18	12.18	148.35
9	100	133.91	-33.91	33.91	1149.90
10	80	116.95	-36.95	36.95	1365.30
11	100	98.48	1.52	1.52	2.31
12	110	99.24	10.76	10.87	118.16
		Totals	-0.78	216.83	5,948.13

Therefore, MSE = squared forecast error/ No.of Months forecasted = 5948.13 / 11 =~ 540.74

c) The Exponential smoothing forecast using = .3 provides a better forecast than The Exponential smoothing forecast using = .5.Since, it has smaller MSE

Option-c is correct: A smoothing constant of 0.3 is better than a smoothing constant of 0.5 since the MSE is less for 0.3 than for 0.5.