Question

Consider the following time series data.

Quarter	Year 1	Year 2	Year 3
1	2	4	5
2	4	5	8
3	1	3	4
4	7	9	10

(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.

Answer 1

a)

Type of pattern: Linear trend and a seasonal pattern.

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b)

Value	Qtr1	Qtr2	Qtr3
2	1	0	0
4	0	1	0
1	0	0	1
7	0	0	0
4	1	0	0
5	0	1	0
3	0	0	1
9	0	0	0
5	1	0	0
8	0	1	0
4	0	0	1
10	0	0	0

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.85756
R Square	0.735409
Adjusted R Square	0.636187
Standard Error	1.683251
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	63	21	7.411765	0.010705
Residual	8	22.66667	2.833333
Total	11	85.66667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	8.666667	0.971825	8.917926	1.98E-05	6.425633	10.9077
Qtr1	-5	1.374369	-3.63803	0.006608	-8.1693	-1.8307
Qtr2	-3	1.374369	-2.18282	0.060595	-6.1693	0.1693
Qtr3	-6	1.374369	-4.36564	0.002394	-9.1693	-2.8307

Estimated regression equation:

ŷ = 8.667 + (-5)Qtr1 + (-3)Qtr2 + (-6)Qtr3

c)

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

ŷ = 8.667 + (-5)*1 + (-3)*0 + (-6)*0 = 3.667

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

ŷ = 8.667 + (-5)*0 + (-3)*1 + (-6)*0 = 5.667

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

ŷ = 8.667 + (-5)*0 + (-3)*0 + (-6)*1 = 2.667

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

ŷ = 8.667 + (-5)*0 + (-3)*0 + (-6)*0 = 8.667

d)

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

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9909611
R Square	0.9820039
Adjusted R Square	0.9717204
Standard Error	0.4692953
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	4	84.125	21.03125	95.49324	3.469E-06
Residual	7	1.541667	0.220238
Total	11	85.66667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	5.4166667	0.428406	12.64377	4.47E-06	4.4036473	6.429686
t	0.40625	0.04148	9.79382	2.45E-05	0.3081648	0.504335
Qtr1	-3.78125	0.402878	-9.38559	3.24E-05	-4.733906	-2.82859
Qtr2	-2.1875	0.392056	-5.57956	0.000834	-3.114565	-1.26044
Qtr3	-5.59375	0.385417	-14.5135	1.76E-06	-6.505116	-4.68238

Estimated regression equation:

ŷ = 5.417 + (0.406)t + (-3.781)Qtr1 + (-2.188)Qtr2 + (-5.594)Qtr3

e)

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

ŷ = 5.417 + (0.406)*13 + (-3.781)*1 + (-2.188)*0 + (-5.594)*0 = 6.917

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

ŷ = 5.417 + (0.406)*14 + (-3.781)*0 + (-2.188)*1 + (-5.594)*0 = 8.917

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

ŷ = 5.417 + (0.406)*15 + (-3.781)*0 + (-2.188)*0 + (-5.594)*1 = 5.917

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

ŷ = 5.417 + (0.406)*16 + (-3.781)*0 + (-2.188)*0 + (-5.594)*0 = 11.917

f)

MSE for b) : 2.833

MSE for d) : 0.220