Question

Consider the following time series data.

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

(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 -Only randomnessRandomness & Linear trendRandomness & SeasonalityRandomness, Linear trend & 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.
	Value =  +  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.
	Quarter 1 forecast Quarter 2 forecast Quarter 3 forecast Quarter 4 forecast
(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)
	Value =  +  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.
	Quarter 1 forecast Quarter 2 forecast Quarter 3 forecast Quarter 4 forecast
(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

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

a)

b)

Excel > Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.631054743
R Square	0.398230088
Adjusted R Square	0.172566372
Standard Error	1.683250823
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	15	5	1.764705882	0.231424712
Residual	8	22.66666667	2.833333333
Total	11	37.66666667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	6.666666667	0.971825316	6.859943406	0.000129675	4.42563347	8.907699864	4.42563347	8.907699864
Q1	-1	1.374368542	-0.727606875	0.487599617	-4.169299541	2.169299541	-4.169299541	2.169299541
Q2	-3	1.374368542	-2.182820625	0.060595435	-6.169299541	0.169299541	-6.169299541	0.169299541
Q3	-2	1.374368542	-1.45521375	0.183698114	-5.169299541	1.169299541	-5.169299541	1.169299541

Ft = 6.667-1.000*Q1-3.000*Q2-2.000*Q3

c)

	Quarter	Ft = 6.67-1*Q1-3*Q2-2*Q3	Q1	Q2	Q3
Year 4	1	5.670	1	0	0
	2	3.670	0	1	0
	3	4.670	0	0	1
	4	6.670	0	0	0

d)

Excel > Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9793216
R Square	0.959070796
Adjusted R Square	0.93568268
Standard Error	0.469295318
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	4	36.125	9.03125	41.00675676	6.04337E-05
Residual	7	1.541666667	0.220238095
Total	11	37.66666667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	3.416666667	0.428406053	7.975299706	9.29664E-05	2.403647325	4.429686009	2.403647325	4.429686009
Q1	0.21875	0.402878254	0.542968	0.60400226	-0.733905691	1.171405691	-0.733905691	1.171405691
Q2	-2.1875	0.392055911	-5.57956133	0.000833612	-3.114564915	-1.260435085	-3.114564915	-1.260435085
Q3	-1.59375	0.385416667	-4.135135135	0.004375743	-2.505115597	-0.682384403	-2.505115597	-0.682384403
t	0.40625	0.041480238	9.793820446	2.45398E-05	0.308164824	0.504335176	0.308164824	0.504335176

Ft = 3.417+0.219*Q1-2.188*Q2-1.594*Q3+0.406*t

e)

	Quarter	Ft= 3.4167+0.2188*Q1-2.1875*Q2-1.5938*Q3+0.4063*t	Q1	Q2	Q3	t
Year 4	1	8.917	1	0	0	13
	2	6.917	0	1	0	14
	3	7.917	0	0	1	15
	4	9.918	0	0	0	16

f)

part b MSE = 2.833

part d MSE = 0.220

part b MSE > part d MSE, So part (d) is more effective