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

A.) 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

B.)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 (a) 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

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

D.) Justify your answer

Answer 1

Actual Demand y	t	Q1	Q2	Q3
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

a)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.562657013
R Square	0.316582915
Adjusted R Square	0.060301508
Standard Error	2.661453237
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	26.25	8.75	1.235294118	0.358900532
Residual	8	56.66666667	7.083333333
Total	11	82.91666667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	5.666666667	1.536590743	3.687817783	0.006149497	2.123282059
Q1	2	2.173067468	0.920357987	0.384298458	-3.011102568
Q2	-1	2.173067468	-0.460178993	0.657636986	-6.011102568
Q3	-2	2.173067468	-0.920357987	0.384298458	-7.011102568

y^ = 5.667 + 2Q1 - 1 Q2 - 2Q3

b)

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.990659899
R Square	0.981407035
Adjusted R Square	0.970782484
Standard Error	0.469295318
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	4	81.375	20.34375	92.37162162	3.88693E-06
Residual	7	1.541666667	0.220238
Total	11	82.91666667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	0.417	0.428406053	0.972598	0.363154591	-0.596352675
t	0.656	0.041480238	15.82079	9.77012E-07	0.558164824
Q1	3.969	0.402878254	9.850991	2.36179E-05	3.016094309
Q2	0.313	0.392055911	0.79708	0.451588999	-0.614564915
Q3	-1.344	0.385416667	-3.48649	0.010176777	-2.255115597

y^ = 0.417 + 3.969 Q1 + 0.313 Q2 -1.344 Q3 + 0.656t

c)

using 3 quarter	error^2			using 3 q amd t	error^2
7.667	7.112889			5.042	0.001764
4.667	7.112889			2.042	0.001764
3.667	7.112889			1.041	0.001681
5.667	7.112889			3.041	0.001681
7.667	0.110889			7.666	0.111556
4.667	0.444889			4.666	0.443556
3.667	0.110889			3.665	0.112225
5.667	0.110889			5.665	0.112225
7.667	5.442889			10.29	0.0841
4.667	11.10889			7.29	0.5041
3.667	5.442889			6.289	0.083521
5.667	5.442889			8.289	0.083521
MSE	4.722222			MSE	0.128475

MSE for a) part = 4.722

MSE for b) part = 0.128

D)

we choose model 2

as its MSE is lower

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