Question

(a) Explain how you determine to fit either multiplicative or additive
decomposition model to a time series data.
(b) The following table gives quarterly sales figures of a well-known brand of
designer bag in a shop in City center in the last two years.
Year 201
7

201
8

201
9

Quarte
r
1 2 3 4 1 2 3 4 1 2 3 4
Sales 10 15 25 34 18 19 33 38 14 27 29 46
You have been requested by the shop owner to forecast sales for 2020 (ignore
the lockdown). Le t us suppose you have decided to use multiple
decomposition method to carry out the forecast, and not use any statistical
software.
(i) Compute appropriate four-period moving averages for these data.
(ii) Compute centered moving averages for the data.
(iii) Calculate sn t x ir t values for the data.
(iv) Calculate estimates of the seasonal factors of the quarterly sales data.
(v) Compute the deseasonalized observations.
(vi) Assuming that a linear trend TR t = describes the deseasonalized
observations, with computed least squares point estimates of to be
18.02 and 1.09, respectively, compute forecasts for the 2020 quarters.

Answer 1

ANSWER::

				four-period	Centered
				Moving	Moving	Ratio to	Seasonal	Data
t	Year	Quarter	Data	Average	Average	CMA	Indexes	Deseasonalized
1	1	1	10				0.604	16.6
2	1	2	15				0.826	18.2
3	1	3	25	21	22.000	1.136	1.169	21.4
4	1	4	34	23	23.500	1.447	1.401	24.3
5	2	1	18	24	25.000	0.720	0.604	29.8
6	2	2	19	26	26.500	0.717	0.826	23.0
7	2	3	33	27	26.500	1.245	1.169	28.2
8	2	4	38	26	27.000	1.407	1.401	27.1
9	3	1	14	28	27.500	0.509	0.604	23.2
10	3	2	27	27	28.000	0.964	0.826	32.7
11	3	3	29	29			1.169	24.8
12	3	4	46				1.401	32.8

The graphed data is:

The forecast for 2020 are:

Period	Forecast
13	19.4371
14	27.4812
15	40.1948
16	49.6869

(OR) TRY THIS ANSWER

b)

i)

Year	Qtr	Sales	four quarter moving average
1	1	10
	2	15
	3	25	21
	4	34	23
2	1	18	24
	2	19	26
	3	33	27
	4	38	26
3	1	14	28
	2	27	27
	3	29	29
	4	46

ii)

Year	Qtr	Sales	four quarter moving average	centered moving average
1	1	10
	2	15
	3	25	21	22
	4	34	23	23.5
2	1	18	24	25
	2	19	26	26.5
	3	33	27	26.5
	4	38	26	27
3	1	14	28	27.5
	2	27	27	28
	3	29	29
	4	46

iii)

Year	Qtr	Sales	four quarter moving average	centered moving average	Seasonal irregular value
1	1	10
	2	15
	3	25	21	22	1.136
	4	34	23	23.5	1.447
2	1	18	24	25	0.720
	2	19	26	26.5	0.717
	3	33	27	26.5	1.245
	4	38	26	27	1.407
3	1	14	28	27.5	0.509
	2	27	27	28	0.964
	3	29	29
	4	46

Iv)

Quarter	Seasonal irregular values	Seasonal Index
1	0.720	0.509	0.61
2	0.717	0.964	0.84
3	1.245	1.136	1.19
4	1.407	1.447	1.43
total			4.07

v)

Year	Qtr	Sales	four quarter moving average	centered moving average	Seasonal irregular value	Seasonal Index	deseasonalized
1	1	10				0.61	16.27
	2	15				0.84	17.84
	3	25	21	22	1.136	1.19	20.99
	4	34	23	23.5	1.447	1.43	23.82
2	1	18	24	25	0.720	0.61	29.29
	2	19	26	26.5	0.717	0.84	22.60
	3	33	27	26.5	1.245	1.19	27.71
	4	38	26	27	1.407	1.43	26.63
3	1	14	28	27.5	0.509	0.61	22.78
	2	27	27	28	0.964	0.84	32.12
	3	29	29			1.19	24.35
	4	46				1.43	32.23

vi)

deseasonalized sales=   18.02   +   1.090   *t

forecast for next Quarter period
Quarter	deseasonalized trend forecast	seasonal index	Quarterly forecast
Q1=	32.190	0.61	19.782
Q2=	33.280	0.84	27.976
Q3=	34.370	1.19	40.929
Q4=	35.460	1.43	50.605

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