Question

Ch.8 #5

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

1)  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

2) 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_____________

3) 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

4) Compute the quarterly forecasts for next year based on the model you developed in part (d).

Quarter 1 forecast _____________

Quarter 2 forecast_____________

Quarter 3 forecast_____________

Quarter 4 forecast_____________

5) 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

Justify your answer.

Answer 1

a.

Actual Demand y	t	Q1	Q2	Q3	quarter
4	1	1	0	0	5.6666
2	2	0	1	0	3.6666
3	3	0	0	1	4.6666
5	4	0	0	0	6.6666
6	5	1	0	0	5.6666
3	6	0	1	0	3.6666
5	7	0	0	1	4.6666
7	8	0	0	0	6.6666
7	9	1	0	0	5.6666
6	10	0	1	0	3.6666
6	11	0	0	1	4.6666
8	12	0	0	0	6.6666
	13	1	0	0	5.6666
	14	0	1	0	3.6666
	15	0	0	1	4.6666
	16	0	0	0	6.6666

after regression on Quareter only

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
Regression	3	15	5	1.764706
Residual	8	22.66666667	2.833333
Total	11	37.66666667
	Coefficients	Standard Error	t Stat	P-value
Intercept	6.666666667	0.971825316	6.859943	0.00013
Q1	-1	1.374368542	-0.72761	0.4876
Q2	-3	1.374368542	-2.18282	0.060595
Q3	-2	1.374368542	-1.45521	0.183698

Value = 6.666666667 -   Qtr1t - 3 Qtr2t -2 Qtr3t

c)

Quarter 1 forecast 5.6666
Quarter 2 forecast 3.6666
Quarter 3 forecast 4.6666
Quarter 4 forecast 6.6666

d)

when we include time also

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	41.00676	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	3.416666667	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
Q1	0.21875	0.402878	0.542968	0.604002	-0.73391	1.171406
Q2	-2.1875	0.392056	-5.57956	0.000834	-3.11456	-1.26044
Q3	-1.59375	0.385417	-4.13514	0.004376	-2.50512	-0.68238

Value =  3.416666667  + 0.21875*  Qtr1t -2.1875* Qtr2t -1.59375 Qtr3t + 0.40625* t

e)

8.91666667

6.91666667

7.91666667

9.91666667

f)

(y - y_d)	(y-y_t)
1.6666	0.041666667
1.6666	0.041666667
1.6666	0.041666667
1.6666	0.041666667
0.3334	0.333333333
0.6666	0.666666667
0.3334	0.333333333
0.3334	0.333333333
1.3334	0.291666667
2.3334	0.708333333
1.3334	0.291666667
1.3334	0.291666667
1.222233333	0.284722222

MSE

Model developed in part (b)   1.222233333

Model developed in part (d)   0.284722222

(MSE 0.284722222 < MSE 1.222233333 ) so Model in d) is more effective as MSE is less

Hope this will be helpful. Thanks and God Bless You :)