Question

Consider the following time series data: Month 1 2 3 4 5 6 7 Value 23 13 21 13 19 21 17 (a) Choose the correct time series plot. (i) (ii) (iii) (iv)
	- Select your answer -Graph (i)Graph (ii)Graph (iii)Graph (iv)Item 1
	What type of pattern exists in the data?
	- Select your answer -Positive trend patternHorizontal patternVertical patternNegative trend patternItem 2
(b)	Develop a three-month moving average for this time series. Compute MSE and a forecast for month 8.
	If required, round your answers to two decimal places. Do not round intermediate calculation.
	MSE:
	The forecast for month 8:
(c)	Use α = 0.2 to compute the exponential smoothing values for the time series. Compute MSE and a forecast for month 8.
	If required, round your answers to two decimal places. Do not round intermediate calculation.
	MSE:
	The forecast for month 8:
(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?
	- Select your answer -3-month moving averageexponential smoothingItem 7
(e)	Use trial and error to find a value of the exponential smoothing coefficient α that results in the smallest MSE.
	If required, round your answer to two decimal places.
	α =

Answer 1

1. The graphs are not seen but here is a picture of what the graph should look like. Please match it correctly. with the ones given

Here the x-axis has the months and the y-axis has the values.

3-month moving average

Here first we find the total of previous 'n years and then it is divided by 'n'.

For e.g.: For month = 4 with n = 3

Total =( 1st + 2nd + 3rd ) / 3

Month	Value	Total	Avg (forecast)	Error (value -forecast)	Error^2
1	23
2	13
3	21
4	13	57	19	-6	36
5	19	47	15.667	3.333	11.111
6	21	53	17.667	3.333	11.111
7	17	53	17.667	-0.667	0.444
8		57	19
Total					58.667
MSE					14.667

MSE = .....here it is divide by 4 because we only have 4 pairs of actual and forecast

Exponential smoothing with smoothing constant = 0.2

Month	Value	Forecast	Error	Error ^2
1	23	23
2	13	23	-10	100
3	21	21	0	0
4	13	21	-8	64
5	19	19.4	-0.4	0.16
6	21	19.32	1.68	2.822
7	17	19.656	-2.656	7.054
8		19.1248
Total				174.037
MSE				29.006

Here MSE =   .here it is divide by 6 because we only have 4 pairs of actual and forecast

Since we haven't been given the forecast for motnh = 1, we take it same as the actual.

(a)	What type of pattern exists in the data?
	- Select your answer -Positive trend patternHorizontal patternVertical patternNegative trend pattern It is not positive or negative since the points are not sloping upward or downward.
(b)	Develop a three-month moving average for this time series. Compute MSE and a forecast for month 8.
	If required, round your answers to two decimal places. Do not round intermediate calculation.
	MSE: 14.667
	The forecast for month 8: 19  (ans explained above)
(c)	Use α = 0.2 to compute the exponential smoothing values for the time series. Compute MSE and a forecast for month 8.
	If required, round your answers to two decimal places. Do not round intermediate calculation.
	MSE: 29.006
	The forecast for month 8: 19.1248
(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?
	- Select your answer -3-month moving average exponential smoothingItem 7 This is because MSE represents error lack of accuracy. So the smaller the MSE more the accuracy.
(e)	Use trial and error to find a value of the exponential smoothing coefficient α that results in the smallest MSE.
	If required, round your answer to two decimal places.
	α = 0.3 Since if we look at less than 0.3 or greater than 0.3 we see that MSE starts increasing so the smallest is givne at 0.3.

At α = 0.1

Month	Value	Forecast	Error	Error ^2
1	23	23
2	13	23	-10	100
3	21	22	-1	1
4	13	21.9	-8.9	79.21
5	19	21.01	-2.01	4.0401
6	21	20.809	0.191	0.036
7	17	20.8281	-3.8281	14.654
8		20.44529
Total				198.941
MSE				33.157

α =0.2

MSE = 29.006

α =0.3

Month	Value	Forecast	Error	Error ^2
1	23	23
2	13	23	-10	100
3	21	20	1	1
4	13	20.3	-7.3	53.29
5	19	18.11	0.89	0.7921
6	21	18.377	2.623	6.880
7	17	19.1639	-2.1639	4.682
8		18.51473
Total				166.645
MSE				27.774

α =0.4

Month	Value	Forecast	Error	Error ^2
1	23	23
2	13	23	-10	100
3	21	19	2	4
4	13	19.8	-6.8	46.24
5	19	17.08	1.92	3.6864
6	21	17.848	3.152	9.935
7	17	19.1088	-2.1088	4.447
8		18.26528
Total				168.309
MSE				28.051