Question

Consider the following time series data.

Quarter	Year 1	Year 2	Year 3
1	4	6	7
2	0	1	4
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.
	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

Answer 1

b)

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

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.805906
R Square	0.649485
Adjusted R Square	0.518041
Standard Error	1.683251
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	42	14	4.941176	0.031488
Residual	8	22.66667	2.833333
Total	11	64.66667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	6.666667	0.971825	6.859943	0.00013	4.425633	8.9077
Qtr1	-1	1.374369	-0.72761	0.4876	-4.1693	2.1693
Qtr2	-5	1.374369	-3.63803	0.006608	-8.1693	-1.8307
Qtr3	-2	1.374369	-1.45521	0.183698	-5.1693	1.1693

Estimated regression equation:  

ŷ = 6.667 + (-1)Qtr1 + (-5)Qtr2 + (-2)Qtr3  

c)

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

ŷ = 6.667 + (-1)*1 + (-5)*0 + (-2)*0 =    5.667

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

ŷ = 6.667 + (-1)*0 + (-5)*1 + (-2)*0 =    1.667

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

ŷ = 6.667 + (-1)*0 + (-5)*0 + (-2)*1 =    4.667

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

ŷ = 6.667 + (-1)*0 + (-5)*0 + (-2)*0 =    6.667

e)

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

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.988008
R Square	0.9761598
Adjusted R Square	0.9625368
Standard Error	0.4692953
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	4	63.125	15.78125	71.65541	9.24E-06
Residual	7	1.541667	0.220238
Total	11	64.66667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3.4166667	0.428406	7.9753	9.3E-05	2.4036473	4.429686
t	0.40625	0.04148	9.79382	2.45E-05	0.3081648	0.504335
Qtr1	0.21875	0.402878	0.542968	0.604002	-0.733906	1.171406
Qtr2	-4.1875	0.392056	-10.6809	1.38E-05	-5.114565	-3.26044
Qtr3	-1.59375	0.385417	-4.13514	0.004376	-2.505116	-0.68238

Estimated regression equation:  

ŷ = 3.417 + (0.219)Qtr1 + (-4.188)Qtr2 + (-1.594)Qtr3 + (0.406)t