Question

Consider the following time series data.

Quarter	Year 1	Year 2	Year 3
1	3	6	8
2	2	4	8
3	4	7	9
4	6	9	11

.

(a)  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

.

(b) 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

.

(c)  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 Which is better model developed in part (B) or (D) Justify your answer with a 2 sentence response

Answer 1

(a) y = 8.667 - 3*Qtr 1 - 4*Qtr 2 - 2*Qtr 3

(b) y = 3.417 - 1.031*Qtr 1 - 2.688*Qtr 2 - 1.344*Qtr 3 + 0.656*t

(c)

	Model developed in part (b)	Model developed in part (d)
MSE	7.083	0.220

The model developed in part (d) is better.

yt	Qtr 1	Qtr 2	Qtr 3	t
3	1	0	0	1
2	0	1	0	2
4	0	0	1	3
6	0	0	0	4
6	1	0	0	5
4	0	1	0	6
7	0	0	1	7
9	0	0	0	8
8	1	0	0	9
8	0	1	0	10
9	0	0	1	11
11	0	0	0	12
	R²	0.317
	Adjusted R²	0.060
	R	0.563
	Std. Error	2.661
	n	12
	k	3
	Dep. Var.	yt
ANOVA table
Source	SS	df	MS	F	p-value
Regression	26.2500	3	8.7500	1.24	.3589
Residual	56.6667	8	7.0833
Total	82.9167	11
Regression output				confidence interval
variables	coefficients	std. error	t (df=8)	p-value	95% lower	95% upper
Intercept	8.667
Qtr 1	-3.000	2.1731	-1.381	.2048	-8.0111	2.0111
Qtr 2	-4.000	2.1731	-1.841	.1029	-9.0111	1.0111
Qtr 3	-2.000	2.1731	-0.920	.3843	-7.0111	3.0111
	R²	0.981
	Adjusted R²	0.971
	R	0.991
	Std. Error	0.469
	n	12
	k	4
	Dep. Var.	yt
ANOVA table
Source	SS	df	MS	F	p-value
Regression	81.3750	4	20.3438	92.37	3.89E-06
Residual	1.5417	7	0.2202
Total	82.9167	11
Regression output				confidence interval
variables	coefficients	std. error	t (df=7)	p-value	95% lower	95% upper
Intercept	3.417
Qtr 1	-1.031	0.4029	-2.560	.0376	-1.9839	-0.0786
Qtr 2	-2.688	0.3921	-6.855	.0002	-3.6146	-1.7604
Qtr 3	-1.344	0.3854	-3.486	.0102	-2.2551	-0.4324
t	0.656	0.0415	15.821	9.77E-07	0.5582	0.7543