Question

This data shows the series of quarterly shipments in millions of US dollars of US hosehold appliances between 1985 and 1989.

Quarter	Shipments
Q1-1985	4009
Q1-1986	4123
Q1-1987	4493
Q1-1988	4595
Q1-1989	4245
Q2-1985	4321
Q2-1986	4522
Q2-1987	4806
Q2-1988	4799
Q2-1989	4900
Q3-1985	4224
Q3-1986	4657
Q3-1987	4551
Q3-1988	4417
Q3-1989	4585
Q4-1985	3944
Q4-1986	4030
Q4-1987	4485
Q4-1988	4258
Q4-1989	4533

(a) which of the following methods would be suitable for forecasting this series if applied to raw data?

-naive forecasts

-moving average

-simple exponential smoothing

-double exponential smoothing

-holt-winters exponential smoothing

(b) apply a moving average with window span w=4 to the data. Use all but the last year as the training period. Create a time ploy of the moving average series.

- What does the MA (4) chart reveal?

- Use the MA (4) model to forecast appliance sales in Q1-1990

-  Use the MA (4) model to forecast appliance sales in Q1-1991

- is the Q1-1990 forecast most likely to underestimate, overestimate, or accurately estimate the actual sales on Q1-1990? Explain?

- Managment Feels most comfortable with moving averages. the analyst therefore plans to use this method for forecasting future quaters. what else should be considered before using the MA(4) to forecast future quarterly shipments of household appliances.

(c) we now focus on forcasting beyond 1989. in the following continue to use all but the last year as the training period and the last four quarters as the validation period. apply holt winters exponential smoothing to the training period.

- compute MAPE values for the training and validation periods using holt winters exponential smoothing.

- Draw two time plots: one for the actual forecasted values and theo ther for the residuals. the x-acis should include the training and validation periods. comment on the model fit in the training and validation periods.

- if we optimize the smoothing constants in the holt winters method are the optimal values likely to be close to zero? why or why not?

Answer 1

answering Part 1 only as it has a lot of sub-parts as per company policies.

a)
an upward trend is observed in the data. Seasonality is also present in the data
hence I prefer Holt Winter method

b)

Quarter	t	shipment (millions $)	MA(4)	abs(error)	error = Ft- At
Q1 1985	1	4009
Q2 1985	2	4321
Q3 1985	3	4224
Q4 1985	4	3944
Q1 1986	5	4123	4124.5	1.5	1.5
Q2 1986	6	4522	4153	369	-369
Q3 1986	7	4657	4203.25	453.75	-453.75
Q4 1986	8	4030	4311.5	281.5	281.5
Q1 1987	9	4493	4333	160	-160
Q2 1987	10	4806	4425.5	380.5	-380.5
Q3 1987	11	4551	4496.5	54.5	-54.5
Q4 1987	12	4485	4470	15	-15
Q1 1988	13	4595	4583.75	11.25	-11.25
Q2 1988	14	4799	4609.25	189.75	-189.75
Q3 1988	15	4417	4607.5	190.5	190.5
Q4 1988	16	4258	4574	316	316
Q1 1989	17	4245	4517.25	272.25	272.25
Q2 1989	18	4900	4429.75	470.25	-470.25
Q3 1989	19	4585	4455	130	-130
Q4 1989	20	4533	4497	36	-36
Q1 1990	21		4565.75
Q2 1990			4565.75
Q3 1990			4565.75
Q4 1990			4565.75
Q1 1991			4565.75

the seasonal variations are smoothed by moving average (4). But I cant predict values beyond Q1 1990
i)

ii) forecast appliance sales in Q1-1990
4565.75

iii) forecast appliance sales in Q1-1990
4565.75

iv) TS
-5.802356119

mad = sum(abs(At-Ft))/n
208.234375
TS<2 implies under-estimating the forecast by using MA(4)

v) auto correlation and normality should be checked

excel sheet:

Quarter	t	shipment (millions $)	MA(4)	abs(error)	error = Ft- At
Q1 1985	1	4009
Q2 1985	2	4321
Q3 1985	3	4224
Q4 1985	4	3944
Q1 1986	5	4123	=AVERAGE(C3:C6)	=ABS(F7)	=D7-C7
Q2 1986	6	4522	=AVERAGE(C4:C7)	=ABS(F8)	=D8-C8
Q3 1986	7	4657	=AVERAGE(C5:C8)	=ABS(F9)	=D9-C9
Q4 1986	8	4030	=AVERAGE(C6:C9)	=ABS(F10)	=D10-C10
Q1 1987	9	4493	=AVERAGE(C7:C10)	=ABS(F11)	=D11-C11
Q2 1987	10	4806	=AVERAGE(C8:C11)	=ABS(F12)	=D12-C12
Q3 1987	11	4551	=AVERAGE(C9:C12)	=ABS(F13)	=D13-C13
Q4 1987	12	4485	=AVERAGE(C10:C13)	=ABS(F14)	=D14-C14
Q1 1988	13	4595	=AVERAGE(C11:C14)	=ABS(F15)	=D15-C15
Q2 1988	14	4799	=AVERAGE(C12:C15)	=ABS(F16)	=D16-C16
Q3 1988	15	4417	=AVERAGE(C13:C16)	=ABS(F17)	=D17-C17
Q4 1988	16	4258	=AVERAGE(C14:C17)	=ABS(F18)	=D18-C18
Q1 1989	17	4245	=AVERAGE(C15:C18)	=ABS(F19)	=D19-C19
Q2 1989	18	4900	=AVERAGE(C16:C19)	=ABS(F20)	=D20-C20
Q3 1989	19	4585	=AVERAGE(C17:C20)	=ABS(F21)	=D21-C21
Q4 1989	20	4533	=AVERAGE(C18:C21)	=ABS(F22)	=D22-C22
Q1 1990	21		=AVERAGE($C$19:$C$22)
Q2 1990			=AVERAGE($C$19:$C$22)
Q3 1990			=AVERAGE($C$19:$C$22)
Q4 1990			=AVERAGE($C$19:$C$22)
Q1 1991			=AVERAGE($C$19:$C$22)