Question

Consider the following time series data.

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

(a)	Choose the correct time series plot.
	(i) (ii) (iii) (iv)
	- Select your answer -Plot (i)Plot (ii)Plot (iii)Plot (iv)Item 1
	What type of pattern exists in the data?
	- Select your answer -Positive trend pattern, no seasonalityHorizontal pattern, no seasonalityNegative trend pattern, no seasonalityPositive trend pattern, with seasonalityHorizontal pattern, with seasonalityItem 2
(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
	- Select your answer -Model developed in part (b)Model developed in part (d)Item 22
	Justify your answer.
	The input in the box below will not be graded, but may be reviewed and considered by your instructor.

Answer 1

Year	Quarter	Ft	Q1	Q2	Q3	t
1	1	5	1	0	0	1
1	2	1	0	1	0	2
1	3	3	0	0	1	3
1	4	7	0	0	0	4
2	1	8	1	0	0	5
2	2	3	0	1	0	6
2	3	6	0	0	1	7
2	4	10	0	0	0	8
3	1	10	1	0	0	9
3	2	7	0	1	0	10
3	3	8	0	0	1	11
3	4	12	0	0	0	12

a)

Positive trend pattern with seasonality

b)

Regression equation of Y with Q1, Q2 and Q3

Excel > Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.717137166
R Square	0.514285714
Adjusted R Square	0.332142857
Standard Error	2.661453237
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	60	20	2.823529412	0.106888246
Residual	8	56.66666667	7.083333333
Total	11	116.6666667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	9.666666667	1.536590743	6.290983277	0.000235061	6.123282059	13.21005127	6.123282059	13.21005127
Q1	-2	2.173067468	-0.920357987	0.384298458	-7.011102568	3.011102568	-7.011102568	3.011102568
Q2	-6	2.173067468	-2.76107396	0.024633879	-11.01110257	-0.988897432	-11.01110257	-0.988897432
Q3	-4	2.173067468	-1.840715973	0.102932302	-9.011102568	1.011102568	-9.011102568	1.011102568

Regression equation:

Y = 9.667-2.000*Q1-6.000*Q2-4.000*Q3

c)

Year	Quarter	Ft = 9.667-2.000*Q1-6.000*Q2-4.000*Q3	Q1	Q2	Q3
4	1	7.667	1	0	0
4	2	3.667	0	1	0
4	3	5.667	0	0	1
4	4	9.667	0	0	0

d)

Regression equation of Y with Q1, Q2, Q3 and t

Excel > Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.993370885
R Square	0.986785714
Adjusted R Square	0.979234694
Standard Error	0.469295318
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	4	115.125	28.78125	130.6824324	1.18135E-06
Residual	7	1.541666667	0.220238095
Total	11	116.6666667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	4.416666667	0.428406053	10.30953377	1.74932E-05	3.403647325	5.429686009	3.403647325	5.429686009
Q1	-0.03125	0.402878254	-0.077566857	0.940343205	-0.983905691	0.921405691	-0.983905691	0.921405691
Q2	-4.6875	0.392055911	-11.95620285	6.51625E-06	-5.614564915	-3.760435085	-5.614564915	-3.760435085
Q3	-3.34375	0.385416667	-8.675675676	5.41239E-05	-4.255115597	-2.432384403	-4.255115597	-2.432384403
t	0.65625	0.041480238	15.82078687	9.77012E-07	0.558164824	0.754335176	0.558164824	0.754335176

Regression Equation:

Y = 4.417-0.031*Q1-4.688*Q2-3.344*Q3+0.656*t

e)

Year	Quarter	Ft = 4.417-0.031*Q1-4.688*Q2-3.344*Q3+0.656*t	Q1	Q2	Q3	t
4	1	12.914	1	0	0	13
4	2	8.913	0	1	0	14
4	3	10.913	0	0	1	15
4	4	14.913	0	0	0	16

f)

Model developed in part (b) MSE = 7.083

Model developed in part (d) MSE = 0.220

Model developed in part (b) MSE > Model developed in part (d) MSE

the model developed in part (d) more effective

So, prefer model developed in part (d)