Question

 

Consider the following time series data.

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

(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

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
	- Select your answer -Model developed in part (b)Model developed in part (d)Item 22

Answer 1

b)

Value	Qtr1	Qtr2	Qtr3
5	1	0	0
2	0	1	0
1	0	0	1
3	0	0	0
8	1	0	0
4	0	1	0
4	0	0	1
6	0	0	0
10	1	0	0
8	0	1	0
6	0	0	1
8	0	0	0

Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.562657
R Square	0.3165829
Adjusted R Square	0.0603015
Standard Error	2.6614532
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	26.25	8.75	1.235294	0.358901
Residual	8	56.66667	7.083333
Total	11	82.91667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	5.6666667	1.536591	3.687818	0.006149	2.123282	9.210051
Qtr1	2	2.173067	0.920358	0.384298	-3.0111	7.011103
Qtr2	-1	2.173067	-0.46018	0.657637	-6.0111	4.011103
Qtr3	-2	2.173067	-0.92036	0.384298	-7.0111	3.011103

Estimated regression equation:

ŷ = 5.667 + (2)Qtr1 + (-1)Qtr2 + (-2)Qtr3

c)  

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

ŷ = 5.667 + (2)*1 + (-1)*0 + (-2)*0 =    7.667

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

ŷ = 5.667 + (2)*0 + (-1)*1 + (-2)*0 =    4.667

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

ŷ = 5.667 + (2)*0 + (-1)*0 + (-2)*1 =    3.667

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

ŷ = 5.667 + (2)*0 + (-1)*0 + (-2)*0 =    5.667

------------------------

d)

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

Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9906599
R Square	0.981407
Adjusted R Square	0.9707825
Standard Error	0.4692953
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	4	81.375	20.34375	92.37162	3.89E-06
Residual	7	1.541667	0.220238
Total	11	82.91667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	0.4166667	0.428406	0.972598	0.363155	-0.59635	1.429686
t	0.65625	0.04148	15.82079	9.77E-07	0.558165	0.754335
Qtr1	3.96875	0.402878	9.850991	2.36E-05	3.016094	4.921406
Qtr2	0.3125	0.392056	0.79708	0.451589	-0.61456	1.239565
Qtr3	-1.34375	0.385417	-3.48649	0.010177	-2.25512	-0.43238

Estimated regression equation:  
ŷ = 0.417 + (3.969)Qtr1 + (0.313)Qtr2 + (-1.344)Qtr3 + (0.656)t

e)

Quarter 1 forecast: x1 = 1, x2 = 0, x3 = 0, t = 13  
ŷ = 0.417 + (3.969)*1 + (0.313)*0 + (-1.344)*0 + (0.656)*13 =    12.917

Quarter 2 forecast: x1 = 0, x2 = 1, x3 = 0, t = 14  
ŷ = 0.417 + (3.969)*0 + (0.313)*1 + (-1.344)*0 + (0.656)*14 =    9.917

Quarter 3 forecast: x1 = 0, x2 = 0, x3 = 1, t = 15  
ŷ = 0.417 + (3.969)*0 + (0.313)*0 + (-1.344)*1 + (0.656)*15 =    8.917

Quarter 4 forecast: x1 = 0, x2 = 0, x3 = 0, t = 16  
ŷ = 0.417 + (3.969)*0 + (0.313)*0 + (-1.344)*0 + (0.656)*16 =    10.917

f) From excel output:

MSE for b) = 7.083

MSE for d) = 0.220