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

(a) Create the correct time series plot. Which type of pattern exists in the data?

(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 calculations.

(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 calculations.

(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 = 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)

(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.

(f) Find MSE of the model developed in part (b) and the model developed in part (d). Which one is the more effective model? If required, round your intermediate calculations and final answer to three decimal places.

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If possible, for the tasks that require to use Excel, can you please help show me how to do those steps in Excel, especially how to create the time series plot in part a?

Thank you very much.

Answer 1

a)

data

t	y
1	4
2	0
3	3
4	5
5	6
6	1
7	5
8	7
9	7
10	4
11	6
12	8

select data , go to insert -> recommended chart

in left side you will see graph of below type

there is positive trend and seasonality

b)

y	t	Q1	Q2	Q3
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

regress y on Q1,Q2 and Q3

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.805906034
R Square	0.649484536
Adjusted R Square	0.518041237
Standard Error	1.683250823
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	3	42	14	4.941176	0.031487607
Residual	8	22.66667	2.833333
Total	11	64.66667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	6.666666667	0.971825	6.859943	0.00013	4.42563347
Q1	-1	1.374369	-0.72761	0.4876	-4.169299541
Q2	-5	1.374369	-3.63803	0.006608	-8.169299541
Q3	-2	1.374369	-1.45521	0.183698	-5.169299541

y^= 6.667 - 1 Q1 -5 Q2 -2Q3

c)

quarter	forecast
1	5.667
2	1.667
3	4.667
4	6.667

d)

now regress y on t, q1,q2 and q3

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.988007993
R Square	0.976159794
Adjusted R Square	0.962536819
Standard Error	0.469295318
Observations	12
ANOVA
	df	SS	MS	F	Significance F
Regression	4	63.125	15.78125	71.65541	9.24E-06
Residual	7	1.541666667	0.220238
Total	11	64.66666667
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	3.416666667	0.428406053	7.9753	9.3E-05	2.403647
t	0.40625	0.041480238	9.79382	2.45E-05	0.308165
Q1	0.21875	0.402878254	0.542968	0.604002	-0.73391
Q2	-4.1875	0.392055911	-10.6809	1.38E-05	-5.11456
Q3	-1.59375	0.385416667	-4.13514	0.004376	-2.50512

y^= 3.417 +0.219 Q1 -4.188 Q2 -1.594 Q3 + 0.406 t

e)

t	forecast
13	8.917
14	4.917
15	7.917
16	9.917