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 (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

b)

Value	Qtr1	Qtr2	Qtr3
4	1	0	0
2	0	1	0
3	0	0	1
5	0	0	0
6	1	0	0
3	0	1	0
5	0	0	1
7	0	0	0
7	1	0	0
6	0	1	0
6	0	0	1
8	0	0	0

Regression ? X Input ок | Input Y Range: $A$1:$A$13 Cancel Input X Range: $B$1:$D$13 1 Help Labels Confidence Level: Constant is Zero 95 % $G$1 Output options O Output Range: O New Worksheet Ply: O New Workbook Residuals Residuals Standardized Residuals U Residual Plots Line Fit Plots Normal Probability Normal Probability Plots

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.6310547
R Square	0.3982301
Adjusted R Square	0.1725664
Standard Error	1.6832508
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	15	5	1.764706	0.231425
Residual	8	22.66667	2.833333
Total	11	37.66667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	6.6666667	0.971825	6.859943	0.00013	4.425633	8.9077
Qtr1	-1	1.374369	-0.72761	0.4876	-4.1693	2.1693
Qtr2	-3	1.374369	-2.18282	0.060595	-6.1693	0.1693
Qtr3	-2	1.374369	-1.45521	0.183698	-5.1693	1.1693

Estimated regression equation:

ŷ = 6.667 + (-1)Qtr1 + (-3)Qtr2 + (-2)Qtr3

c)

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

ŷ = 6.667 + (-1)*1 + (-3)*0 + (-2)*0 = 5.667

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

ŷ = 6.667 + (-1)*0 + (-3)*1 + (-2)*0 = 3.667

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

ŷ = 6.667 + (-1)*0 + (-3)*0 + (-2)*1 = 4.667

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

ŷ = 6.667 + (-1)*0 + (-3)*0 + (-2)*0 = 6.667

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

d)

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

Regression ? X Input Input Y Range: OK $A$1:$A$13 Cancel Input X Range: $B$1:$E$13 Help Labels O Confidence Level: Constant is Zero 95 % Output options O Output Range: $H$10 O New Worksheet Ply: O New Workbook Residuals Residuals Standardized Residuals Residual Plots Line Fit Plots Normal Probability Normal Probability Plots

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.9793216
R Square	0.9590708
Adjusted R Square	0.9356827
Standard Error	0.4692953
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	4	36.125	9.03125	41.00676	6.04E-05
Residual	7	1.541667	0.220238
Total	11	37.66667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3.4166667	0.428406	7.9753	9.3E-05	2.403647	4.429686
t	0.40625	0.04148	9.79382	2.45E-05	0.308165	0.504335
Qtr1	0.21875	0.402878	0.542968	0.604002	-0.73391	1.171406
Qtr2	-2.1875	0.392056	-5.57956	0.000834	-3.11456	-1.26044
Qtr3	-1.59375	0.385417	-4.13514	0.004376	-2.50512	-0.68238

Estimated regression equation:

ŷ = 3.417 + (0.406)t + (0.219)Qtr1 + (-2.188)Qtr2 + (-1.594)Qtr3

e)

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

ŷ = 3.417 + (0.406)*13 + (0.219)*1 + (-2.188)*0 + (-1.594)*0 = 8.917

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

ŷ = 3.417 + (0.406)*14 + (0.219)*0 + (-2.188)*1 + (-1.594)*0 = 6.917

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

ŷ = 3.417 + (0.406)*15 + (0.219)*0 + (-2.188)*0 + (-1.594)*1 = 7.917

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

ŷ = 3.417 + (0.406)*16 + (0.219)*0 + (-2.188)*0 + (-1.594)*0 = 9.917

f)

MSE fr part b) = 2.833

MSE for part d) = 0.220