Question

Quarter	Year 1	Year 2	Year 3
1	5	8	10
2	1	3	7
3	3	6	8
4	7	10	12

(A) What type of pattern exists in the data?
a. Positive trend, no seasonality
b. Horizontal trend, no seasonality
c. Vertical trend, no seasonality
d. Positive tend, with seasonality
e. Horizontal trend, with seasonality
f. Vertical trend, with seasonality

(B) Use a multiple regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data. Qtr1 = 1 if Quarter 1, 0 otherwise; Qtr2 = 1 if Quarter 2, 0 otherwise; Qtr3 = 1 if Quarter 3, 0 otherwise. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300) If the constant is "1" it must be entered in the box. Do not round intermediate calculation.

ŷ =____ + ____Qtr1 + ____Qtr2 + ____Qtr3

(C)

Compute the quarterly forecasts for next year based on the model you developed in part (b)

If required, round your answers to three decimal places. Do not round intermediate calculation.

Year Quarter Ft 4 1 4 2 4 3 4 4

(D)Use a multiple regression model to develop an equation to account for trend and seasonal effects in the data. Use the dummy variables you developed in part (b) to capture seasonal effects and create a variable t such that t = 1 for Quarter 1 in Year 1, t = 2 for Quarter 2 in Year 1,… t = 12 for Quarter 4 in Year 3. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)

ŷ =____ + ____Qtr1 + ____Qtr2 + ____Qtr3+ ____t

(E) Compute the quarterly forecasts for next year based on the model you developed in part (d).
Do not round your interim computations and round your final answer to three decimal places.

Year Quarter Period Ft 4 1 13 4 2 14 4 3 15 4 4 16

(F) Is the model you developed in part (b) or the model you developed in part (d) more effective? If required, round your intermediate calculations and final answer to three decimal places.

	Model Developed in Part (b)	Model developed in part (d)
MSE

Answer 1

a) Time-series plot:

Type of pattern : d. Positive tend, with seasonality

b)

Value	Qtr1	Qtr2	Qtr3
5	1	0	0
1	0	1	0
3	0	0	1
7	0	0	0
8	1	0	0
3	0	1	0
6	0	0	1
10	0	0	0
10	1	0	0
7	0	1	0
8	0	0	1
12	0	0	0

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.717137
R Square	0.514286
Adjusted R Square	0.332143
Standard Error	2.661453
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	60	20	2.823529	0.106888
Residual	8	56.66667	7.083333
Total	11	116.6667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	9.666667	1.536591	6.290983	0.000235	6.123282	13.21005
Qtr1	-2	2.173067	-0.92036	0.384298	-7.0111	3.011103
Qtr2	-6	2.173067	-2.76107	0.024634	-11.0111	-0.9889
Qtr3	-4	2.173067	-1.84072	0.102932	-9.0111	1.011103

Estimated regression equation:

ŷ = 9.667 + (-2)Qtr1 + (-6)Qtr2 + (-4)Qtr3

c)

Quarter 1 forecast: x1 = 1, x2 = 0, x3 = 0

ŷ = 9.667 + (-2)*1 + (-6)*0 + (-4)*0 = 7.667

Quarter 2 forecast: x1 = 0, x2 = 1, x3 = 0

ŷ = 9.667 + (-2)*0 + (-6)*1 + (-4)*0 = 3.667

Quarter 3 forecast: x1 = 0, x2 = 0, x3 = 1

ŷ = 9.667 + (-2)*0 + (-6)*0 + (-4)*1 = 5.667

Quarter 4 forecast: x1 = 0, x2 = 0, x3 = 0

ŷ = 9.667 + (-2)*0 + (-6)*0 + (-4)*0 = 9.667

d)

Value	t	Qtr1	Qtr2	Qtr3
5	1	1	0	0
1	2	0	1	0
3	3	0	0	1
7	4	0	0	0
8	5	1	0	0
3	6	0	1	0
6	7	0	0	1
10	8	0	0	0
10	9	1	0	0
7	10	0	1	0
8	11	0	0	1
12	12	0	0	0

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9933709
R Square	0.9867857
Adjusted R Square	0.9792347
Standard Error	0.4692953
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	4	115.125	28.78125	130.6824	1.181E-06
Residual	7	1.541667	0.220238
Total	11	116.6667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	4.4166667	0.428406	10.30953	1.75E-05	3.4036473	5.429686
t	0.65625	0.04148	15.82079	9.77E-07	0.5581648	0.754335
Qtr1	-0.03125	0.402878	-0.07757	0.940343	-0.983906	0.921406
Qtr2	-4.6875	0.392056	-11.9562	6.52E-06	-5.614565	-3.76044
Qtr3	-3.34375	0.385417	-8.67568	5.41E-05	-4.255116	-2.43238

Estimated regression equation:

ŷ = 4.417 + (-0.031)Qtr1 + (-4.688)Qtr2 + (-3.344)Qtr3 + (0.656)t

e)

Quarter 1 forecast: x1 = 1, x2 = 0, x3 = 0, t = 13

ŷ = 4.417 + (-0.031)*1 + (-4.688)*0 + (-3.344)*0 + (0.656)*13 = 12.917

Quarter 2 forecast: x1 = 0, x2 = 1, x3 = 0, t = 14

ŷ = 4.417 + (-0.031)*0 + (-4.688)*1 + (-3.344)*0 + (0.656)*14 = 8.917

Quarter 3 forecast: x1 = 0, x2 = 0, x3 = 1, t = 15

ŷ = 4.417 + (-0.031)*0 + (-4.688)*0 + (-3.344)*1 + (0.656)*15 = 10.917

Quarter 4 forecast: x1 = 0, x2 = 0, x3 = 0, t = 16

ŷ = 4.417 + (-0.031)*0 + (-4.688)*0 + (-3.344)*0 + (0.656)*16 = 14.917

f)

MSE for b) = 7.083

MSE fro d) = 0.220